Multiple Dedekind zeta functions

Ivan Emilov Horozov

doi:10.1515/crelle-2014-0055

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Multiple Dedekind zeta functions

Ivan Emilov Horozov

Published/Copyright: July 26, 2014

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 722

Abstract

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler’s multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series [6]. We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein–Kronecker series [9]. Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1;1,1). Based on such computations, we state two conjectures about MDZV.

0 Introduction

Multiple Dedekind zeta functions generalize Dedekind zeta functions in the same way the multiple zeta functions generalize the Riemann zeta function. Let us recall known definitions of the above functions. The Riemann zeta function is defined as

ζ⁢(s)=∑n>01ns,

where n is an integer. Multiple zeta functions are defined as

ζ⁢(s1,…,sm)=∑0<n1<…<nm1n1s1⁢⋯⁢nmsm,

where n1,…,nm are integers. Special values of the Riemann zeta function ζ⁢(k) and of the multiple zeta functions ζ⁢(k1,…,km) were defined by Euler [5]. The Riemann zeta function is closely related to the ring of integers.

The Dedekind zeta function ζK⁢(s) is an analogue of the Riemann zeta function, which is closely related to the algebraic integers 𝒪K in a number field K. It is defined as

ζK⁢(s)=∑𝔞≠(0)1NK/ℚ⁢(𝔞)s,

where the sum is over all ideals 𝔞 different from the zero ideal (0) and N⁢(𝔞)=#⁢|𝒪K/𝔞| is the norm of the ideal 𝔞.

A definition of multiple Dedekind zeta functions should combine ideas from multiple zeta functions and from Dedekind zeta functions.

There is a definition of multiple Dedekind zeta functions due to Masri [15]. Let us recall his definition. Let K1,…,Km be number fields and let 𝒪Ki, for i=1,…,m, be the corresponding rings of integers. Let 𝔞i, for i=1,…,m, be ideals in 𝒪Ki, respectively. Then he defines

ζ⁢(K1,…,Kd;s1,…,sm)=∑0<N⁢(𝔞1)<…<N⁢(𝔞m)1N⁢(𝔞1)s1⁢⋯⁢N⁢(𝔞m)sm.

We propose a different definition. The advantage of our definition is that it leads to more properties: analytic, topological and algebraic-geometric. Let us give an explicit formula for a multiple Dedekind zeta function, in a case when it is easier to formulate. Let K be a number field with ring of integers 𝒪K. Let UK be the group of units in 𝒪K. Let C be a cone inside a fundamental domain of 𝒪K modulo UK. (More precisely, C has to be a positive unimodular simple cone as defined in Section 2.2. A fundamental domain for 𝒪K modulo UK can be written as a finite union of unimodular simple cones.) For such a cone C, we define a multiple Dedekind zeta function

ζK;C⁢(s1,…,s1;…;sm,…,sm)=∑α1,…,αm∈C1N⁢(α1)s1⁢N⁢(α1+α2)s2⁢⋯⁢N⁢(α1+⋯+αm)sm.

The key new ingredient in the definition of multiple Dedekind zeta functions is the definition of iterated integrals on a membrane. This is a higher-dimensional analogue of iterated path integrals. In the iterated integrals on a membrane the iteration happens in n-directions. Such iterated integrals were defined in [10] generalizing Manin’s non-commutative modular symbol [14] to higher dimensions in some cases. For Hilbert modular surfaces this is developed in [11].

Structure of the paper

In Section 1.1, we recall definitions of multiple zeta values and of polylogarithms by giving many explicit formulas. In Section 1.2, we generalize the previous formulas to multiple Dedekind zeta values over the Gaussian integers via many examples.

In Section 2.1, we give two definitions of iterated integrals on a membrane. The first definition is more intuitive. It can be used to generalize the first few formulas for MDZV over the Gaussian integers from Section 1.2. The second definition is the one needed for the definition of multiple Dedekind zeta values. It is needed in order to express special values of the multiple Eisenstein series via MDZV, when the modular parameter has a value in an imaginary quadratic field.

In Section 2.2, we use some basic algebraic number theory (see [12]), in order to construct the functions that we integrate. We use an idea of Zagier and Shintani (see [18, 16, 2]) for defining a cone. We associate a product of geometric series to every unimodular simple cone. This is the type of functions that we integrate. Lemma 2.16 shows that a fundamental domain for the non-zero integer 𝒪K-{0} modulo the units UK can be written as a finite union of unimodular simple cones.

In Section 3, we define Dedekind polylogarithms associated to a positive unimodular simple cone. Theorem 3.2 expresses Dedekind zeta values in terms of Dedekind polylogarithms. The heart of the section is Definition 3.4 of multiple Dedekind zeta values (MDZV) as an iterated integral on a membrane and Definition 3.7 of multiple Dedekind zeta functions (MDZF) in terms of an integral representation. Theorems 3.6 and 3.9 express MDZV and MDZF as an infinite sum. At the end of Section 3, we give many examples. Examples 1–2 are the simplest multiple Dedekind zeta values. Example 3 expresses partial Eisenstein–Kronecker series associated to an imaginary quadratic ring as multiple Dedekind zeta values (see for instance [9, Section 8.1]). Example 4 considers multiple Eisenstein–Kronecker series (for an alternative definition see [9, Section 8.2]). Example 5 gives the simplest multiple Dedekind zeta functions. Example 6 is a double Dedekind zeta function.

In Section 4, we prove an analytic continuation of multiple Dedekind zeta functions, which allows us to consider special values of multiple Eisenstein series, examined by Gangl, Kaneko and Zagier (see [6]), as values of multiple Dedekind zeta functions (see Examples 7–9 in Section 4.1). Examples 10 and 11 are particular cases of analytic continuation and of a multiple residue at (1;1) and at (1,1;1,1), respectively. The proof of analytic continuation is based on a generalization of Example 11 and a theorem of Gelfand–Shilov (Theorem 4.1). At the end of Section 4.4, based on Examples 10 and 11, we state two conjectures about MDZV.

1 Examples

We are going to present several examples of Riemann zeta values and multiple zeta values in order to introduce key examples of multiple Dedekind zeta value as iterated integrals. Instead of considering the ring of integers in a general number field, which we will do in the later sections, we will examine only the ring of Gaussian integers. Also, here we will ignore questions about convergence. Such questions will be addressed in Section 2.2.

1.1 Classical cases

Let us recall the m-th polylogarithm and its relation to Riemann zeta values.

If the first polylogarithm is defined as

Li1⁢(x1)=∫0x1d⁢x01-x0=∫0x1(1+x0+x02+⋯)⁢𝑑x0=x1+x122+x133+⋯

and the second polylogarithm is

Li2⁢(x2)=∫0x2Li1⁢(x1)⁢d⁢x1x1=x2+x2222+x2332+⋯

(note that ζ⁢(2)=Li2⁢(1)), then the m-th polylogarithm is defined by iteration

(1.1)Lim⁢(xm)=∫0xmLim-1⁢(xm-1)⁢d⁢xm-1xm-1.

This is a presentation of the m-th polylogarithm as an iterated integral (see for example [4, 3, 8]). By a direct computation it follows that

Lim⁢(x)=x+x22m+x33m+⋯

and the relation

ζ⁢(m)=Lim⁢(1)

is straightforward. Using equation (1.1), we can express the m-th polylogarithm as

Lim⁢(xm)=∫0<x0<x1<…<xmd⁢x01-x0∧d⁢x1x1∧⋯∧d⁢xm-1xm-1.

Let xi=e-ti. Then the m-th polylogarithm can be written in the variables t0,…,tm in the following way:

(1.2)Lim⁢(e-tm)=∫t0>t1>…>tmd⁢t0∧…∧d⁢tm-1et0-1.

This is achieved, first, by changing the variables in the differential forms

d⁢x01-x0=d⁢(-t0)et0-1 and d⁢xixi=d⁢(-ti),

and second, by reversing the bounds of integration 0<x0<…<xm versus t0>…>tm, which absorbs the sign. As an infinite sum, we have

(1.3)Lim⁢(e-t)=∑n>0e-n⁢tnm.

In Section 1.2, we present a key analogy of equations (1.2) and (1.3) leading to Dedekind polylogarithms over the Gaussian integers. Equations (1.2) and (1.3) will be generalized to Dedekind polylogarithms in Section 3.1 and to multiple Dedekind zeta values in Section 3.2.

Below we present similar formulas for multiple polylogarithms with exponential variables. We will construct their generalizations in Section 1.2.

Let us recall the definition of double logarithm

Li1,1⁢(1,x2)=∫0x2Li1⁢(x1)⁢d⁢x11-x1=∫0x2(∑n1=1∞x1n1n1)⁢(∑n2=1∞x1n2)⁢d⁢x1x1

=∑n1,n2=1∞x2n1+n2n1⁢(n1+n2).

Let xi=e-ti. Then Li1,1⁢(1,e-t2) can be written as an iterated integral in terms of the three variables t0,t1,t2 in the following way:

Li1,1⁢(1,e-t2)=∫t0>t1>t2>0d⁢t0∧d⁢t1(et0-1)⁢(et1-1).

As an infinite sum, we have

(1.4)Li1,1⁢(1,e-t)=∑n1,n2=1∞e-(n1+n2)⁢tn1⁢(n1+n2).

An example of a multiple zeta value is

ζ⁢(1,2)=∑n1,n2=1∞1n1⁢(n1+n2)2=∫01Li1,1⁢(x2)⁢d⁢x2x2.

Thus, an integral representation of ζ⁢(1,2) is

(1.5)ζ⁢(1,2)=∫t0>t1>t2>0d⁢t0(et0-1)∧d⁢t1(et1-1)∧d⁢t2.

Similarly,

(1.6)ζ⁢(2,2)=∫t0>t1>t2>t3>0d⁢t0(et0-1)∧d⁢t1∧d⁢t2(et2-1)∧d⁢t3.

1.2 Dedekind polylogarithms over the Gaussian integers

In this subsection, we are going to construct analogues of polylogarithms (and of some multiple polylogarithms), which we call Dedekind (multiple) polyologarithms over the Gaussian integers. We will denote by fm the m-th Dedekind polylogarithm, which will be an analogue the m-th polylogarithm Lim⁢(e-t) with an exponential variable. Each of the analogues will have an integral representation, resembling an iterated integral and an infinite sum representation, resembling the classical Dedekind zeta values over the Gaussian integers. We also draw diagrams that represent integrals in order to give a geometric view of the iterated integrals on membranes in dimension 2. We will give examples of multiple Dedekind zeta values (MDZV) over the Gaussian integers, using the Dedekind (multiple) polylogarithms.

We are going to generalize equations (1.3) and (1.4) for (multiple) polylogarithms to their analogue over the Gaussian integers. We will recall some properties and definitions related to Gaussian integers. For more information one may consider [12].

By Gaussian integers we mean all numbers of the form a+i⁢b, where a and b are integers and i=-1. The ring of Gaussian integers is denoted by ℤ⁢[i]. We call the set

C=ℕ⁢{1+i,1-i}={α∈ℤ⁢[i]:α=a⁢(1+i)+b⁢(1-i),a,b∈ℕ}

a cone, where ℕ denotes the positive integers. Note that 0 does not belong to the cone C, since the coefficients a and b are positive integers. We are going to use two sequences of inequalities

t1>u1>v1>w1 and t2>u2>v2>w2

when we deal with a small number of iterations. The reason for introducing them is to make the examples easier to follow. However, for generalizations to higher order of iteration we will use the following notation for the n sequences of inequalities:

t1,1>t1,2>t1,3>t1,4>⋯ and t2,1>t2,2>t2,3>t2,4>⋯ and so on,

where n is the degree of the number field. We are going to define a function f1, which will be an analogue of Li1⁢(e-t). Let

(1.7)f0⁢(C;t1,t2)=∑α∈Cexp⁡(-α⁢t1-α¯⁢t2),

f1⁢(C,u1,u2)=∫u1∞∫u2∞f0⁢(C;t1,t2)⁢𝑑t1∧d⁢t2.

We can draw the following diagram for the integral representing f1.

The diagram represents that the integrant is f0⁢(C;t1,t2)⁢d⁢t1∧d⁢t2, depending on the variables t1 and t2, subject to the restrictions +∞>t1>u1 and +∞>t2>u2.

We need the following lemma, whose proof is straightforward:

Lemma 1.1

The following hold.

We have
∫u∞e-k⁢t⁢𝑑t=e-k⁢uk.
Let N⁢(α)=α⁢α¯. Then
∫u1∞∫u2∞exp⁡(-α⁢t1-α¯⁢t2)⁢𝑑t1∧d⁢t2=exp⁡(-α⁢u1-α¯⁢u2)N⁢(α).

Using the above lemma, we obtain

f1⁢(C;u1,u2)=∑α∈Cexp⁡(-α⁢u1-α¯⁢u2)N⁢(α).

We define a Dedekind dilogarithm f2 by

(1.8)f2⁢(C;v1,v2)=∫v1∞∫v2∞f1⁢(C;u1,u2)⁢𝑑u1∧d⁢u2

=∫t1>u1>v1t2>u2>v2f0⁢(C;t1,t2)⁢𝑑t1∧d⁢t2∧d⁢u1∧d⁢u2.

We can associate a diagram to the integral representation of the Dedekind dilogarithm f2 (see equation (1.8)).

The diagram represents that the variables under the integral are t1, t2, u1, u2, subject to the conditions +∞>t1>u1>v1 and +∞>t2>u2>v2. Also, the function f0 in the diagram depends on the variables t1 and t2.

Similarly to equation (1.1), we define inductively the m-th Dedekind polylogarithm over the Gaussian integers

(1.9)fm⁢(C;u1,u2)=∫u1∞∫u2∞fm-1⁢(C;t1,t2)⁢𝑑t1∧d⁢t2.

The above integral is the key example of an iterated integral over a membrane, which is the topic of Section 2.1.

From equation (1.9), we can derive an analogue of the infinite sum representation of a polylogarithm (see equation (1.3)):

(1.10)fm⁢(C;u1,u2)=∑α∈Cexp⁡(-α⁢u1-α¯⁢u2)N⁢(α)m.

The above equation gives an infinite sum representation of the m-th Dedekind polylogarithm over the Gaussian integers.

We derive one relation between the Dedekind m-polylogarithm fm, a Dedekind zeta value over the Gaussian integers and a Riemann zeta value. For arithmetic over the Gaussian integers one can consider [12].

Lemma 1.2

For the Dedekind polylogarithm fm, associated to the above cone C, we have

fm⁢(C;0,0)=2-m⁢(ζℚ⁢(i)⁢(m)-ζ⁢(2⁢m)),

where ζQ⁢(i)⁢(m) is a Dedekind zeta value and ζ⁢(2⁢m) is a Riemann zeta value.

Proof.

We are going to prove the following equalities, which give the lemma:

fm⁢(C;0,0)=∑α∈C1N⁢(α)m=2-m⁢(∑(α)≠(0)⊂ℤ⁢[i]-∑α∈ℕ)⁢1N⁢((α))m

=2-m⁢(ζℚ⁢(i)⁢(m)-ζ⁢(2⁢m)),

The first equality follows from (1.10). The second and the third equalities relate our integral to classical zeta values. The second equality uses two facts: (1) For the Gaussian integers the norm of an element α, N⁢(α), is equal to the norm of the principal ideal generated by α, denoted by N⁢((α)), namely N⁢(α)=N⁢((α)). Recall that for the Gaussian integers the norm of an element α is N⁢(α)=α⁢α¯, and the norm of a principal ideal N⁢((α)) is equal to the number of elements in the quotient module

N⁢((α))=#⁢|ℤ⁢[i]/(α)|,

where

(α)=α⁢ℤ⁢[i]={μ∈ℤ⁢[i]:μ=α⁢β⁢ for some ⁢β∈ℤ⁢[i]}

is view as a ℤ⁢[i]-submodule of ℤ⁢[i]. (2) The set of non-zero principal ideals can be parameterized by the non-zero integers modulo the units. Since the units are ±1,±i, we have that (α)⊂ℤ⁢[i], (α)≠(0), can be parameterized by elements of the Gaussian integers with positive real part and non-negative imaginary part, which we will denote by C0. Multiplying each element of C0 by 1-i, we obtain the union of the cone C and the set {a+a⁢i:a∈ℕ}. Summing over the set C0 gives the Dedekind zeta value. Hence summing over the set (1-i)⁢C0 gives 2-m⁢ζℚ⁢(i)⁢(m). Such a sum can be separated to a sum over C, which contributes fm and a sum over the set {a+a⁢i:a∈ℕ}, which gives 2-m⁢ζ⁢(2⁢m). ∎

Now we can define an analogue of the double logarithm Li1,1⁢(1,e-t) over the Gaussian integers, using the following integral representation:

f1,1⁢(C,C;v1,v2)=∫v1∞∫v2∞f1⁢(C;u1,u2)⁢f0⁢(C;u1,u2)⁢𝑑u1∧d⁢u2,

called a Dedekind double logarithm. Such an integral will be considered as an example of an iterated integral over a membrane in Section 2.1. As an analog for equation (1.8), we can express f1,1 only in terms of f0 by

f1,1⁢(C,C;v1,v2)=∫t1>u1>v1t2>u2>v2(f0⁢(C;t1,t2)⁢d⁢t1∧d⁢t2)∧(f0⁢(C;u1,u2)⁢d⁢u1∧d⁢u2).

It allows us to associate a diagram to the Dedekind double logarithm f1,1.

The variables t1, t2, u1, u2 in the diagram are variables in the integrant. They are subject to the conditions t1>u1>v1 and t2>u2>v2. Also, the lower left function f0 in the diagram depends on the variables t1 and t2 and the upper right function f0 depends on u1 and u2.

The similarity between f1,1⁢(C,C;v1,v2) and Li1,1⁢(1,e-t) can be noticed by the infinite sum representation in the following:

Lemma 1.3

We have

f1,1⁢(C,C;v1,v2)=∑α,β∈Cexp⁡(-(α+β)⁢v1-(α¯+β¯)⁢v2)N⁢(α)⁢N⁢(α+β).

Proof.

We have

f1,1⁢(C,C;v1,v2)=∫v1∞∫v2∞f1⁢(C;u1,u2)⁢f0⁢(C;u1,u2)⁢𝑑u1∧d⁢u2

=∫v1∞∫v2∞∑α∈Cexp⁡(-α⁢u1-α¯⁢u2)N⁢(α)⁢∑β∈Cexp⁡(-β⁢u1-β¯⁢u2)⁢d⁢u1∧d⁢u2

=∫v1∞∫v2∞∑α,β∈Cexp⁡(-(α+β)⁢u1-(α¯+β¯)⁢u2)N⁢(α)⁢d⁢u1∧d⁢u2

=∑α,β∈Cexp⁡(-(α+β)⁢v1-(α¯+β¯)⁢v2)N⁢(α)⁢N⁢(α+β).∎

Similarly to the Dedekind double logarithm f1,1, we define a multiple Dedekind polylogarithm

f1,2⁢(C,C;w1,w2)=∫w1∞∫w2∞f1,1⁢(C,C;v1,v2)⁢𝑑v1∧d⁢v2.

We can associate the following diagram to the multiple Dedekind polylogarithm f1,2.

The diagram represents the following: The variables of the integrant are t1, t2, u1, u2, v1, v2. The variables are subject to the conditions t1>u1>v1>w1 and t2>u2>v2>w2. The lower left function f0 depends on the variables t1 and t2. The middle function f0 depends on u1 and u2. Thus, the diagram represents the following integral:

(1.11)f1,2⁢(C,C;w1,w2)=∫Dw1,w2(f0⁢(C;t1,t2)⁢d⁢t1∧d⁢t2)

∧(f0⁢(C;u1,u2)⁢d⁢u1∧d⁢u2)∧(d⁢v1∧d⁢v2),

where the domain of integration is

Dw1,w2={(t1,t2,u1,u2,v1,v2)∈ℝ6:t1>u1>v1>w1⁢ and ⁢t2>u2>v2>w2}.

A direct computation leads to

f1,2⁢(C,C;w1,w2)=∑α,β∈Cexp⁡(-(α+β)⁢w1-(α¯+β¯)⁢w2)N⁢(α)⁢N⁢(α+β)2.

We define a multiple Dedekind zeta value as

(1.12)ζℚ⁢(i);C,C⁢(1,2;1,2)=f1,2⁢(C,C;0,0)=∑α,β∈C1N⁢(α)⁢N⁢(α+β)2.

For a comparison of formula (1.12) to the general Definition 3.4, one can use Remark 2.4.

Now let us give a relation between multiple Dedekind zeta values and iterated integrals. We shall use the following pair of inequalities in the following sections:

t1,1>t1,2>t1,3 and t2,1>t2,2>t2,3

instead of

t1>u1>v1>w1 and t2>u2>v2>w2,

since using such notation it is easier to write higher order iterated integrals. In this notation, from equation (1.9), we obtain

(1.13)f2⁢(C;t1,3,t2,3)=∫t1,1>t1,2>t1,3t2,1>t2,2>t2,3(f0⁢(C;t1,1,t2,1)⁢d⁢t1,1∧d⁢t2,1)

∧(d⁢t1,2∧d⁢t2,2).

and

(1.14)f1,1⁢(C,C;t1,3,t2,3)=∫t1,1>t1,2>t1,3t2,1>t2,2>t2,3(f0⁢(C;t1,1,t2,1)⁢d⁢t1,1∧d⁢t2,1)

∧(f0⁢(C;t1,2,t2,2)⁢d⁢t1,2∧d⁢t2,2).

In the next section, we generalize the (iterated) integrals appearing in equations (1.9), (1.11), (1.13), and (1.14), called iterated integrals on membranes (see Definition 2.1).

The next two examples are needed in order to relate multiple Dedekind zeta values to values of Eisenstein series and values of multiple Eisenstein series (see [6], see also Examples 7–9 at the end of Section 3). The integrals below will present another type of iterated integrals on membranes (see Definition 2.3) leading to a multiple Dedekind zeta values.

We define the following iterated integral to be a multiple Dedekind zeta value:

(1.15)ζℚ⁢(i);C⁢(3;2)=∫t1>u1>v1>0t2>u2>0f0⁢(C;t1,t2)⁢𝑑t1∧d⁢u1∧d⁢v1∧d⁢t2∧d⁢u2.

The reason for such a definition is its infinite sum representation

(1.16)ζℚ⁢(i);C⁢(3;2)=∑α∈C1α3⁢α¯2,

which can be achieved essentially in the same way as for the other multiple Dedekind zeta values. We can associate the following diagram to the integral representation of ζℚ⁢(i);C⁢(3;2) in equation (1.15).

The arrows in the diagram signify the direction of decrease of the variables in differential 1-forms. It is important to consider the variables only in the horizontal direction and then only in vertical direction. In horizontal direction, we have f0⁢(C;t1,t2)⁢d⁢t1 followed by d⁢u1 and d⁢v1. Integration with respect to the variables t1,u1,v1 leads to 1/α3 in the summation of equation (1.16). In vertical direction, we have a double iteration. First we have f0⁢(C;t1,t2)⁢d⁢t2 followed by u2. That leads to 1/α¯2 in the summation in equation (1.16).

Diagrams associated to the integral representation of ζℚ⁢(i);C⁢(3;2) (equations (1.15)) are not unique. Alternatively, we could have used one of the following diagrams.

Consider the following diagram, associated to a more complicated MDZV, for the purpose of establishing notation.

The diagram encodes that +∞>t1>u1>v1>w1>0 and +∞>t2>u2>v2>0. Consider the horizontal direction of the diagram and equation (1.6). We have f0⁢d⁢t1, followed by d⁢u1, f0⁢d⁢v1 and d⁢w1. That gives an analogue of ζ⁢(a,b)=ζ⁢(2,2) in horizontal direction for (a,b)=(2,2). Consider the vertical direction of the diagram and equation (1.5). We have f0⁢d⁢t2, followed by f0⁢d⁢u2 and d⁢v2. That gives an analogue of ζ⁢(c,d)=ζ⁢(1,2) in vertical direction, for (c,d)=(1,2). We write

ζℚ⁢(i);C,C⁢(a,b;c,d)=ζℚ⁢(i);C,C⁢(2,2;1,2)

for the multiple Dedekind zeta function associated to the above diagram. We leave the proof of the following statement to the reader:

ζℚ⁢(i);C,C⁢(2,2;1,2)=∑α,β∈C1α2⁢α¯1⁢(α+β)2⁢(α¯+β¯)2.

In Section 3, we use iterated integrals on membranes to define multiple Dedekind zeta values associated to any number field.

2 Arithmetic and geometric tools

2.1 Iterated integrals on a membrane

Let D be a domain defined in terms of the real variables ti,j for i=1,…,n and j=1,…,m, by

D={(t1,1,…,tn,m)∈ℝn⁢m:ti,1>ti,2>…>ti,m>0⁢ for ⁢i=1,…,n}.

For each j=1,…,m, let ωj be a differential n-form on ℂn. Let

g:(0,+∞)n→ℂn

be a smooth map, whose pull-back sends the coordinate-wise foliation on ℂn to a coordinate-wise foliation on (0,+∞)n. We will call such a map a membrane. One should think of the n-forms g*⁢ωj as an analogue of f0⁢(C;t1,t2)⁢d⁢t1∧d⁢t2 from equation (1.7).

Definition 2.1

Let g be a membrane. An iterated integral on g, in terms of n-forms ωj, for j=1,…,m, is defined as

(2.1)∫gω1⁢…⁢ωm=∫D⋀j=1mg*⁢ωj⁢(t1,j,…,tn,j).

Definition 2.2

A shuffle between two ordered sets

S1={1,…,p}

and

S2={p+1,…,p+q}

is a permutation τ of the union S1∪S2 such that

for a,b∈S1, we have τ⁢(a)<τ⁢(b) if a<b,
for a,b∈S2, we have τ⁢(a)<τ⁢(b) if a<b.

We shall denote the set of shuffles between two ordered sets of orders p and q, respectively, by Sh⁢(p,q).

The definition of an iterated integral on a membrane is associated with the following objects:

g:(0,+∞)n→ℂn, a membrane (that is a smooth map, whose pull-back sends the coordinate-wise foliation on ℂn to a coordinate-wise foliation on (0,+∞)n),
ω1,…,ωm differential n-forms on ℂn,
mi copies of differential 1-forms d⁢zi on ℂn, for i=1,…,n,
a shuffle τi∈Sh⁢(m,mi) for each i=1,…,n,
τ=(τ1,…,τn), the set of n shuffles τ1,…,τn.

Definition 2.3

Given the above data, we define an iterated integral on a membrane g, involving n-forms and 1-forms, as

(2.2)∫(g,τ)ω1⁢…⁢ωm⁢(d⁢z1)m1⁢…⁢(d⁢zn)mn

=∫D(∏j=1mϕj⁢(g⁢(t1,τ1⁢(j),…,tn,τn⁢(j))))⁢⋀i=1n⋀j=1m+mid⁢ti,j,

where ϕj is a function defined by ωj=ϕj⁢d⁢z1∧…∧d⁢zn, also ti,j=g*⁢zi,j. Here, the variables ti,j belong to the domain

D={(t1,1,…,tn,m+mn)∈∏i=1nℝm+mi:ti,1>ti,2>…>ti,m+mi>0}.

Remark 2.4

Comparing the Definitions 2.1 and 2.3, one can notice that there is no sign occurring. The reason for that is the following:

In Definition 2.1 we use a domain D, whose coordinates are ordered by
t1,1,…,tn,1,t1,2,…,tn,2,…,t1,m,…,tn,m.
It is the same as the order of the differential 1-forms under the integral in equation (2.1).
In Definition 2.3 we use a domain D, whose coordinates are ordered by
t1,1,…,t1,m+m1,t2,1,…,t2,m+m2,…,tn,1,…,tn,m+mn.
It is the same as the order of the differential 1-forms under the integral in equation (2.2).

Thus, if m1=⋯=mn=0, both definitions lead to the same value, since the permutation of the differential forms coincides with the permutation of the coordinates of the domain of integration. Thus, the change of orientation of the domain of integration coincides with the sign of permutation acting on the differential forms.

Theorem 2.5

Theorem 2.5 (Homotopy invariance)

The iterated integrals on membranes from Definition 2.3 are homotopy invariant when the homotopy preserves the boundary of the membrane.

Proof.

Let g be a homotopy between the two membranes g0 and g1. Let

Ω=(∏j=1mϕj⁢(z1,τ1⁢(j),…,zn,τn⁢(j)))⁢⋀i=1n⋀j=1m+mid⁢zi,j.

Note that Ω is a closed form, since ωi is a form of top dimension and since d⁢zi,j is closed. By the Stokes Theorem, we have

0=∫s=0s=1∫Dg*⁢𝑑Ω

(2.3)=∫(g1,τ)Ω-∫(g0,τ)Ω

(2.4)±∫s=0s=1∑i=1n∑j=1m+mi∫D|(zi,j=zi,j+1)g*⁢Ω

(2.5)±∫s=0s=1∑i=1n∫D|(zi,m+mi=0)g*⁢Ω.

We want to show that the difference in the terms in (2.3) is zero. It is enough to show that each of the terms (2.4) and (2.5) are zero. If zi,j=zi,j+1, then the wedge of the corresponding differential forms will vanish. Thus the terms in (2.4) are zero. If zi,m+mi=0, then d⁢ti,m+mi=0, defined via the pull-back g*. Then the terms (2.5) are equal to zero. ∎

2.2 Cones and geometric series

Let n=[K:ℚ] be the degree of the number field K over ℚ. Let 𝒪K be the ring of integers in K, and let Uk be the group of units in K. We are going to use an idea of Shintani [2] by examining Dedekind zeta functions in terms of a cone inside the ring of integers.

We define the notion cone for any number ring. The meaning of cones is roughly the following: summation over the elements of finitely many cones would give multiple Dedekind zeta values or multiple Dedekind zeta functions.

Definition 2.6

We define a coneC to be

C=ℕ⁢{e1,…,ek}={α∈𝒪K:α=a1⁢e1+⋯⁢ak⁢ek⁢ for ⁢ei∈𝒪K⁢ and ⁢ai∈ℕ}

with generators e1,…,ek.

For the next definition, we are going to use that a number field K can be viewed as an n-dimensional vector space over the rational numbers ℚ.

Definition 2.7

An unimodular cone is a cone with generators e1,…,ek such that e1,…,ek as elements of K are linearly independent over ℚ, when we view the field K as a vector space over ℚ.

Note that if C is an unimodular cone, then 0∉C, since e1,…,ek are linearly independent over ℚ and the coefficients a1,…,an are positive integers.

Definition 2.8

We call C an unimodular simple cone if for any embedding σi of K into the complex numbers, σi:K→ℂ, and a suitable branch of the functions arg⁡(z), we have that the closure of the set arg⁡(σ⁢(C)) is an interval [θ0,θ1] such that its lengths is less than π, namely, θ1-θ0∈[0,π).

In particular, the cone

C={α∈ℤ⁢[i]:α=a⁢(1+i)+b⁢(1-i),a,b∈ℕ},

considered in Section 1.2, is an unimodular simple cone, since arg⁡(σ1⁢(α))∈(-π/4,π/4) and arg⁡(σ2⁢(α))∈(-π/4,π/4), for each α∈C. The maps σ1 and σ2 are complex conjugates of each other.

Definition 2.9

For an unimodular simple cone C with generators e1,…,ek, we define a dual cone of C to be

C*={(z1,…,zn)∈ℂn:Re⁡(zi⁢σi⁢(ej))>0⁢ for ⁢i=1,…,n⁢ and ⁢j=1,…,k}.

Clearly, if C is an unimodular simple cone, then the dual cone C* is a non-empty set. One can prove that by considering each coordinate of C*, separately.

Definition 2.10

For an unimodular simple cone C, we define a function

(2.6)f0⁢(C;z1,…,zn)=∑α∈Cexp⁡(-∑i=1nσi⁢(α)⁢zi),

where σ1,…,σn are all embeddings of the number field K into the complex numbers ℂ and the domain of the function f0 is the dual cone C*.

Lemma 2.11

The function f0 is uniformly convergent for (z1,…,zn) in any compact subset B of the dual cone C* of an unimodular simple cone C.

Proof.

From Definition 2.9, we have Re⁡(σi⁢(ej)⁢zi)>0. Let

(2.7)yj=∏i=1nexp⁡(-σi⁢(ej)⁢zi).

Then |yj|<1 on the domain B. Moreover, |yi| achieves a maximum on the compact subset B. Let |yj|≤cj<1 on the compact set B for some constant cj. Then the rate of convergence of the geometric sequence in yj is uniformly bounded by cj on the compact set B. Therefore, we have a uniform convergence of the geometric series in yj. The function f0⁢(C;z1,…,zn) is a product of k geometric series in the variables y1,…,yk each of which is uniformly bounded in absolute value by the constants c1,…,ck on the domain B, respectively. Then, we obtain that

(2.8)f0⁢(C;z1,…,zn)=∏j=1kyj1-yj.∎

Corollary 2.12

The function f0⁢(C;z1,…,zn) has analytic continuation to all values of z1,…,zn, except at

∑i=1nσi⁢(ej)⁢zi∈2⁢π⁢i⁢ℤ,

for j=1,…,k.

Proof.

Using the product formula (2.8) in terms of geometric series in yj, we see that the right hand side of (2.8) makes sense for all yj≠1. This gives analytic continuation from the domain C* to the domain consisting of points (y1,…,yn) with yi≠1. ∎

Definition 2.13

We call C a positive unimodular simple cone if C is an unimodular simple cone and the product of the positive real coordinates is in C*, namely (ℝ>0)n⊂C* as subsets of ℂn.

Lemma 2.14

If C is an unimodular simple cone, then for some α∈OK we have that

α⁢C={α⁢β:β∈C}

is a positive unimodular simple cone.

Proof.

In order to find such an elements α, we need to recall properties of real or complex embeddings of a number field K.

The degree of a number field n=[K:ℚ] is the dimension of K as vector space over ℚ. Then there are exactly n distinct embeddings K→ℂ. Let the first r1 embeddings, σ1,…,σr1, be the ones whose image is inside the real numbers. They are called real embeddings. Let the next r2 embeddings be complex embeddings, which are not pair-wise complex conjugates of each other. Let us denote them by σr1+1,…,σr1+r2. Let the last r2 embeddings be the complex conjugates of previously counted complex embeddings, namely,

σr1+r2+i⁢(β)=σr1+i⁢(β)¯,

for i=1,…,r2. We also have that n=r1+2⁢r2.

Let Vℝ be an n-dimensional real vector subspace of ℂn defined in the following way:

Vℝ={(z1,…,zn)∈ℂn:(z1,…,zr1)∈ℝr1,(zr1+1,…,zr1+r2)∈ℂr2 and

(zr1+r2+1,…,zr1+2⁢r2)=(z¯r1+1,…,z¯r1+r2)}.

Now, we proceed with the proof of the lemma in six steps.

Step 1: K is dense in Vℝ.
Step 2: Vℝ∩C* in non-empty.
Step 3: Vℝ∩C* is an open subset of Vℝ.
Step 4: K∩C* is non-empty.
Step 5: 𝒪K∩C* is non-empty.
Step 6: α⁢C is a positive unimodular simple cone for any α∈𝒪K∩C*.

(Step 1) Recall the product of the n embeddings of K to the space Vℝ,

∏i=1nσi:K→Vℝ,

mapping β∈K to (σ1⁢(β),…,σn⁢(β))∈Vℝ has a dense image.

(Step 2) Indeed, let zi be the i-th coordinate of C*. The first r1 coordinates z1,…,zr1 of C* can be real numbers (positive or negative), since σi⁢(K)⊂ℝ for i=1,…,r1. Thus, the first ri coordinates can be both in Vℝ and in C*. For the coordinates zr1+1,…,zr1+r2 of C* there are no restrictions when we intersect C* with Vℝ. For the last r2 coordinates of C* we must have that zr1+r2+i=z¯r1+i in order for the coordinates to be in the intersection C*∩Vℝ. Since σr1+r2+i⁢(β)=σr1+i⁢(β)¯, we have the conditions on the (r1+i)-coordinate and on the (r1+r2+i)-coordinate of a point in C* to be in Vℝ are

Re⁡(zr1+i⁢σr1+i⁢(β))>0,

for β∈C and zr1+r2+i=z¯r1+i. The last condition implies that

Re⁡(zr1+r2+i⁢σr1+r2⁢i⁢(β))=Re⁡(zr1+i⁢σr1+i⁢(β)¯)>0.

Thus, such a point (z1,…,zn) is in C*∩Vℝ.

(Step 3) It is true, since C* is an open subsets of ℂn

(Step 4) Since K is dense in Vℝ (Step 1) and Vℝ∩C* is open in Vℝ (Steps 2 and 3), we have that K∩C* is non-empty.

(Step 5) If α∈K∩C*, then for some positive integer L, we have that L⁢α∈𝒪K, and also L⁢α∈C*, since C* is invariant under rescaling by a positive (real) number L.

(Step 6) Let (t1,…,tn)∈ℝ>0n and let β∈C. Put zi=ti⁢σi⁢(α). Then (z1,…,zn)∈C* and

Re⁡(ti⁢σi⁢(α⁢β))=Re⁡(ti⁢σi⁢(α)⁢σi⁢(β))=Re⁡(zi⁢σi⁢(β))>0.∎

2.3 Cones and ideals

In this subsection, we are going to examine union of cones that give a fundamental domain of the ring of integers 𝒪K modulo the group of units UK. We also examine a fundamental domain of an ideal 𝔞 modulo the group of units UK.

Definition 2.15

We define M as a fundamental domain of 𝒪K-{0}⁢⁢mod⁢⁢Uk. For an ideal 𝔞, let

M⁢(𝔞)=M∩𝔞.

Lemma 2.16

For any ideal a the set M⁢(a) can be written as a finite disjoint union of unimodular simple cones.

Proof.

It is a simple observation that M⁢(𝔞) can be written as a finite union of unimodular cones. We have to show that we can subdivide each of the unimodular cones into finite union of unimodular simple cones.

Let σ1,…,σr1 be the real embeddings of the number field K and let σr1+1,…,σr1+r2 be the non-conjugate complex embeddings of K. We define

T={-1,1}r1×(S1)r2.

Let C be an unimodular cone. We define a map J by

J:C→T,

α↦(σ1(α))|σ1(α))|,…,σr1+r2(α))|σr1+r2(α))|).

Denote by C¯ the closure of the image of J in T. Then one can cut the cone C into finitely many cones Ci such that for Ci and any embedding σ of K into ℂ, we have that arg⁡(σ⁢(C))∈[θ0,θ1], for θ1-θ0∈[0,π). Then Ci is an unimodular simple cone. Thus, the cones Ci are finitely many unimodular simple cones, whose (disjoint) union gives the set M⁢(𝔞). ∎

For an element α in a ring of integers 𝒪K, denote by (α) the principal ideal generated by the element α. Then NK/ℚ⁢((α)) denoted the norm of the principal ideal generated by α. We have that NK/ℚ⁢((α)) is a positive integer equal to the number of elements in the quotient 𝒪K/(α). Also NK/ℚ⁢(α) is the norm of the algebraic number α. This is equal to the product of all of its Galois conjugates, which is an integer, possibly a negative integer. We always have that NK/ℚ⁢((α))=|NK/ℚ⁢(α)|.

However, for elements of an unimodular simple cone, we can say more.

Lemma 2.17

Let C be an unimodular simple cone. Then for every α∈C, we have

NK/ℚ⁢((α))=ϵ⁢(C)⁢NK/ℚ⁢(α),

where ϵ⁢(C)=±1 depends only on the cone C, not on α.

Proof.

Note that on the left we have a norm of an ideal and on the right we have a norm of a number. Since C is a simple cone, we have that for all real embeddings σ:K→ℝ, the signs of σ⁢(α) and σ⁢(β) are the same for all α and β in C. Let ϵσ be the sign of σ⁢(α) for each real embedding σ. Then the product over all real embeddings of ϵσ is equal to ϵ⁢(C). ∎

3 Multiple Dedekind zeta functions

3.1 Dedekind polylogarithms

Let us recall the Dedekind zeta values

ζK⁢(m)=∑𝔞≠(0)1NK/ℚ⁢(𝔞)m,

where 𝔞 is an ideal in 𝒪K.

We are going to express the summation over elements, which belong to a finite union of positive unimodular simple cones. We will define a Dedekind polylogarithm associates to a positive unimodular simple cone. The key result in this subsection will be that a Dedekind zeta value can be expressed as a ℚ-linear combination of values of the Dedekind polylogarithms.

We also define a partial Dedekind zeta function by summing over ideals in a given ideal class [𝔞]:

ζK,[𝔞]⁢(m)=∑𝔟∈[𝔞]NK/ℚ⁢(𝔟)-m.

Let us consider a partial Dedekind zeta functions ζK,[𝔞]-1⁢(m), corresponding to an ideal class [𝔞]-1, where 𝔞 is an integral ideal. For every integral ideal 𝔟 in the class [𝔞]-1, we have that

𝔞⁢𝔟=(α),

where α∈𝔞. Then

NK/ℚ⁢(𝔟)=NK/ℚ⁢(𝔞)-1⁢NK/ℚ⁢((α)).

Let

M⁢(𝔞)=⋃i=1n⁢(𝔞)Ci⁢(𝔞),

where n⁢(𝔞) is a positive integer and Ci⁢(𝔞) are unimodular simple cones. Let αi be an element of the intersection of 𝒪K with the dual cone Ci⁢(𝔞)*. Then αi⁢Ci⁢(𝔞) is a positive unimodular simple cone (see Lemma 2.14).

Then,

(3.1)ζK,[𝔞]-1⁢(m)=∑𝔟∈[𝔞]-1NK/ℚ⁢(𝔟)-m

=NK/ℚ⁢(𝔞)m⁢∑i=1n⁢(𝔞)ϵ⁢(Ci⁢(𝔞))m⁢N⁢(αi)m⁢∑α∈αi⁢Ci⁢(𝔞)NK/ℚ⁢(α)-m,

where ϵ⁢(Ci⁢(𝔞))=±1, depending on the cone, NK/ℚ⁢(𝔞) is a norm of the ideal 𝔞 and N⁢(αi) is the norm of the algebraic integer αi.

We are going to give an example of higher-dimensional iteration in order to illustrate the usefulness of this procedure. For a positive unimodular simple cone C, we define

fm⁢(C;u1,…,un)=∫u1∞…⁢∫un∞fm-1⁢(C;t1,…,tn)⁢𝑑t1∧…∧d⁢tn,

where ti∈(ui,+∞). This is an iteration, giving the simplest type of iterated integrals on a membrane. We start the induction on m from m=0. Recall that f0 was introduced in Definition 2.10.

Note that a norm of an algebraic number α can be expresses as a product of its embeddings in the complex numbers σ1⁢(α),…,σn⁢(α),

NK/ℚ⁢(α)=σ1⁢(α)⁢⋯⁢σn⁢(α).

Integrating term by term, we can express fm as an infinite sum

fm⁢(C;t1,…,tn)=∑α∈Cexp⁡(-∑i=1nσi⁢(α)⁢ti)NK/ℚ⁢(α)m.

Note that a cone C is a linear combination of its generators so that the coefficients of the generators are positive integers. In particular, 0 is not an element of an unimodular simple cone C, since then the generators are linearly independent over ℚ. Thus, there is no division by 0.

Definition 3.1

We define an m-th Dedekind polylogarithm, associated to a number field K and a positive unimodular simple cone C, to be

LimK⁢(C;X1,…,Xn)=fm⁢(C;-log⁡(X1),…,-log⁡(Xn)).

Theorem 3.2

Dedekind zeta value at s=m>1 can be written as a finite Q-linear combination of Dedekind polylogarithms evaluated at (X1,…,Xn)=(1,…,1).

Proof.

If 𝔞1,…,𝔞h are integral ideals in 𝒪K, representing all the ideal classes, then using equation (3.1), we obtain

ζK⁢(m)=∑j=1hNK/ℚ⁢(𝔞j)m⁢∑i=1n⁢(𝔞)jϵ⁢(Ci⁢(𝔞j))m⁢N⁢(αi,j)m⁢fm⁢(αi,j⁢Ci⁢(𝔞j);0,…,0),

where Ci⁢(𝔞j) are unimodular simple cones such that

⋃i=1n⁢(𝔞)jCi⁢(𝔞j)=M⁢(𝔞𝔧)

and ϵ⁢(Ci⁢(𝔞j))=±1, depending on the cone (see Definition 2.15 and Lemma 2.16 above). Let αi,j∈Ci⁢(𝔞j)*∩𝒪K be an algebraic integer in the dual cone of Ci⁢(𝔞j). Then by Lemma 2.14 we have that αi,j⁢Ci⁢(𝔞j) is a positive unimodular simple cone. The iterated integrals are hidden in the functions fm. Consider Definition 2.1 with differential forms

ω1=f0⁢(αi,j⁢Ci⁢(𝔞j);z1,…,zn)⁢d⁢z1∧…∧d⁢zn,ω2=ω3=…=ωm=d⁢z1∧…∧d⁢zn,

and let g be the inclusion of (0,∞)n in ℂn. Then the corresponding iterated integral on a membrane gives fm⁢(αi,j⁢Ci⁢(𝔞j);t1,…,tn). ∎

3.2 Multiple Dedekind zeta values

We recall an integral representation of a multiple zeta value

ζ⁢(k1,k2,…,km)=∑0<n1<⋯<nm1n1k1⁢⋯⁢nmkm

has the following integral representation (see for example [8]):

(3.2)ζ⁢(k1,k2,…,km)=∫0<x1<⋯<xk1+⋯+kmd⁢x11-x1∧(d⁢x2x2∧⋯∧d⁢xk1xk1)

∧d⁢xk1+11-xk1+1∧(d⁢xk1+2xk1+2∧⋯∧d⁢xk1+k2xk1+k2)

∧⋯∧d⁢xk1+⋯+km-1+11-xk1+⋯+km-1+1

∧(d⁢xk1+⋯+km-1+2xk1+⋯+km-1+2∧⋯∧d⁢xk1+⋯+kmxk1+⋯+km).

Note that there are essentially two types of differential 1-forms under the integral: d⁢x/(1-x) and d⁢x/x. If we set xi=e-ti, then we obtain the following formula, needed for the generalization to multiple Dedekind zeta values:

(3.3)ζ⁢(k1,k2,…,km)=∫t1>⋯>tk1+⋯+km>0d⁢t1et1-1∧(d⁢t2∧⋯∧d⁢tk1)

∧d⁢tk1+1etk1+1-1∧(d⁢tk1+2∧⋯∧d⁢tk1+k2)

∧⋯∧d⁢tk1+⋯+km-1+1exk1+⋯+km-1+1-1

∧(d⁢tk1+⋯+km-1+2∧⋯∧d⁢tk1+⋯+km).

Note that in equation (3.3), we have used m copies of d⁢t/(et-1). To find their order, first we shuffle a set S with m elements (corresponding to m copies of d⁢t/(et-1)) with another set S1 consisting of m1=-m+k1+⋯+km elements (corresponding to m1 copies of dt). We choose a shuffle τ1∈Sh⁢(m,m1) such that τ1⁢(1)=1. The reason is that the first differential form in the iterated integral in equation (3.2) has to be d⁢x/(1-x), which is needed for convergence. The corresponding 1-form in equation (3.3) is d⁢t/(et-1). The relation between the shuffle τ1 and the set of integers k1,…,km is the following:

1=τ1⁢(1),

k1+1=τ1⁢(2),

k1+k2+1=τ1⁢(3),

⋮

k1+⋯+km-1+1=τ1⁢(m),

k1+⋯+km-1+km=number of differential 1-forms.

The integers 1,k1+1,k1+k2+1,…,k1+⋯+km-1+1 are the values of the index i, where the analogue of the form d⁢ti/(eti-1) appears under the integral, not the form d⁢ti (see equation (3.3))

In order to define multiple Dedekind zeta values, we will use n shuffles of pairs of ordered sets, where n=[K:ℚ] is the degree on the number field.

Let m1,…,mn be positive integers. (The positive integer mi will denote the number of times the differential form d⁢zi occurs.) We define the following ordered sets:

S={1,2,…,m},Si={m+1,…,m+mi}.

Definition 3.3

Denote by Sh1⁢(p,q) the subset of all shuffles τ∈Sh⁢(p,q) of the two sets {1,…,p} and {p+1,…,p+q} such that τ⁢(1)=1

For the definition of multiple Dedekind zeta values at the positive integers, we use Definition 2.3, where we take the n-forms to be

ωj=f0⁢(Cj,z1,…,zn)⁢d⁢z1∧…∧d⁢zn,

for j=1,…,m, where C1,…,Cm are positive unimodular simple cones, and the 1-forms to be d⁢zi on ℂn occurring mi times, for i=1,…,n.

Definition 3.4

Definition 3.4 (Multiple Dedekind zeta values)

For i=1,…,n, let τi∈Sh1⁢(m,mi). We define the integers ki,j and mi in terms of the shuffle τi via the following relations:

(3.4)1=τi⁢(1),

(3.5)ki,1+1=τi⁢(2),

(3.6)ki,1+ki,2+1=τi⁢(3),

⁢⋮

(3.7)ki,1+⋯+ki,m-1+1=τi⁢(m),

(3.8)ki,1+⋯+ki,m-1+ki,m=m+mi.

We define multiple Dedekind zeta values at the positive integers by

ζK;C1,…,Cm⁢(k1,1,…,k1,m;…;kn,1,…,kn,m)=∫(g,τ)ω1⁢…⁢ωm⁢(d⁢z1)m1⁢…⁢(d⁢zn)mn.

Remark 3.5

Remark 3.5 (On convergence of MDZV)

Assume for now that each of the cones C1,…,Cm has the number 1∈𝒪K as a generator. Then for every α∈Cj and for all i and j, we have that |σi⁢(α)|>|σi⁢(1)|=1, since the cones are positive. Consider the integral representing an MDZV

ζK;C1,…,Cm⁢(k1,1,…,k1,m;…;kn,1,…,kn,m).

For the domain of iteration, let the lower bounds for each chain of inequalities be uj instead of 0; that is

ti,1>ti,2>…>ti,m+mj>ui.

That integral would give a Dedekind multiple polylogarithm

f=fk1,1,…,k1,m;…;kn,1,…,kn,m⁢(C1,…,Cm;u1,…,un).

Note that for ki,j≥2, we have that the Dedekind multiple polylogarithm f is absolutely convergent for fixed positive values of u1,…,un. It is straightforward to show that if we let u1,…,un vary, then the Dedekind multiple polylogarithm is uniformly convergent. Then we can interchange the order of integration and summation. Moreover, the limit when ui→0 is the MDZV we started with.

We return to the assumption on the cones. We can embed a positive unimodular simple cone into a finite disjoint union of positive unimodular simple cones having 1 as a generator for which we have uniform convergence for the corresponding Dedekind multiple polylogarithms. From that we conclude that for any positive unimodular simple cones C1,…,Cm we have uniform convergence for the corresponding Dedekind multiple polylogarithm with limit the MDZV.

Theorem 3.6

For the general form of a multiple Dedekind zeta value, we need

a number field K,
positive unimodular simple cones C1,…,Cm in 𝒪K,
elements αj∈Cj for j=1,…,m,
complex embeddings of the elements αi,j=σi⁢(αj).

Then a multiple Dedekind zeta value has the following representation as an infinite sum:

(3.9)ζK;C1,…,Cm⁢(k1,1,…,k1,m;…;kn,1,…,kn,m)

=∑α1∈C1⋯⁢∑αm∈Cm∏i=1n∏j=1m(αi,1+⋯+αi,j)-ki,j.

Proof.

There are n different embedding σ1,…,σn of K into ℂ. Given ki,1,…,ki,m we find mi using equation (3.8). Then we find τi by the values at 1,2,…,m obtained from equations (3.5)–(3.7).

Now we use Definition 3.4 of a multiple Dedekind zeta value in terms of iterated integrals on a membrane from Definition 2.3. We are going to follow closely equation (3.3). The variable t1,1 enters as a variable of the function f0⁢(C1;…), and the variables t1,2,…,t1,k1,1 appear as differential 1-forms d⁢t1,2,…,d⁢t1,k1,1, since τ1⁢(2)=k1,1+1 (see equation (3.5)). Recall that σ1:K→ℂ is an embedding of K into the complex numbers and α1,j=σ1⁢(αj). Thus integrating with respect to t1,1,…,t1,k1,1 gives us a denominator α1,1k1,1, associated to each α1∈C1. Then the variable t1,k1,1+1 enters as a variable in the function f0⁢(C2;…), again since τ1⁢(2)=k1,1+1, and the variables t1,k1,1+2,…,t1,k1,1+k1,2 appear as differential 1-forms d⁢t1,k1,1+2,…,d⁢t1,k1,1+k1,2, since τ1⁢(3)=k1,1+k1,2+1 (see equation (3.6)). Thus integrating with respect to t1,1,…,t1,k1,1 gives us a denominator

α1,1k1,1⁢(α1,1+α1,2)k1,2,

associated to each α1∈C1 and each α2∈C2. Continuing this process to the variable t1,m+m1, we obtain a denominator

∏j=1m(α1,1+⋯+α1,j)k1,j,

associated to each m-tuple (α1,…,αm), where αj∈Cj. There are n different embeddings σ1,…,σn of K into ℂ, where n=[K:ℚ] is the degree of the number field. So far we have considered the contribution of the first embedding. The contribution of the first and the second embedding is obtained in essentially the same way as for the first embedding. It gives a denominator

∏j=1m(α1,1+⋯+α1,j)k1,j⁢(α2,1+⋯+α2,j)k2,j,

associated to each m-tuple (α1,…,αm), where αj∈Cj. Similarly, after integrating with respect to all the variables ti,j, we obtain a denominator

∏i=1n∏j=1m(αi,1+⋯+αi,j)ki,j,

associated to each m-tuple (α1,…,αm), where αj∈Cj. Then the numerators are all equal to 1, since the lower bound for the variables under the exponents in f0⁢(Cj;…) is 0. This finishes the computation of the integral of one summand. After summing over the points on the cones, we obtain the explicit formula of the theorem. ∎

The following examples of MDZV give analogues of Dedekind zeta function and of (multiple) Eisenstein–Kronecker series.

Example 1

Let C be a positive unimodular simple cone in the ring of integers 𝒪K of a number filed K. If m=1 and if all values ki,1 are equal to k, then

ζK;C⁢(k;…;k)=∑α∈C1NK/ℚ⁢(α)k.

Note that the number 0 does not belong to any unimodular simple cone C.

Example 2

Let m=2, and let

k=k1,1=⋯=kn,1

and

l=k1,2=⋯=kn,2

be positive integers greater than 1. Finally, let C1 and C2 be positive unimodular simple cones in the ring of integers 𝒪K of a number field K. Then the corresponding multiple Dedekind zeta value can also be written as a sum:

(3.10)ζK;C1,C2⁢(k,l;k,l;…;k,l)=∑α∈C1,β∈C21NK/ℚ⁢(α)k⁢NK/ℚ⁢(α+β)l.

Example 3

Let K be an imaginary quadratic field. Let C be a positive unimodular simple cone in 𝒪K. We can represent the cone C as an ℕ-module: C=ℕ⁢{μ,ν}, μ,ν∈𝒪K. Put z=μ/ν. Consider then

ζK;C⁢(k;k)=∑α∈C1N⁢(α)k=|ν|-2⁢k⁢∑a,b∈ℕ1|a⁢z+b|2⁢k,

where the last sum is a portion of the k-th Eisenstein–Kronecker series. Such a series could be found in [17]

Ek⁢(z)=∑a,b∈ℤ(a,b)≠(0,0)1|a⁢z+b|2⁢k.

Example 4

With the notation of Example 3, we obtain an analogue of values of multiple Eisenstein–Kronecker series

ζK;C,C⁢(k,l;k,l)=|ν|-k-l⁢∑a,b,c,d∈ℕ1|a⁢z+b|2⁢k⁢|(a+c)⁢z+(b+d)|2⁢l.

An alternative generalization was considered in [9, Section 8.2].

3.3 Multiple Dedekind zeta functions

We will try to give some intuition behind the integral representation of the multiple zeta functions (see [8]). After that we will generalize the construction to define the number field analogues – multiple Dedekind zeta functions. In order to do that, we give two examples – one for ζ⁢(3) and another for ζ⁢(1,3).

We have

ζ⁢(3)=∫0<x1<x2<x3<1d⁢x11-x1∧d⁢x2x2∧d⁢x3t3

=∫t1>t2>t3>0d⁢t1∧d⁢t2∧d⁢t3et1-1

=∫0∞t12⁢d⁢t1Γ⁢(3)⁢(et1-1).

The first equality is due to Kontsevich, and the second uses the change of variables xi=e-ti. Both representations were examined in more details in Section 1. The last equality uses the equation

(3.11)∫b>t1>t2>…>tn>a𝑑t1∧d⁢t2∧…∧d⁢tn=(b-a)nΓ⁢(n+1),

whose proof we leave for the reader.

Similarly,

ζ⁢(1,3)=∫0<x1<x2<x3<x4<1d⁢x11-x1∧d⁢x21-x2∧d⁢x3x3∧d⁢x4x4

=∫t1>t2>t3>t4>0d⁢t1∧d⁢t2∧d⁢t3∧d⁢t4(et1-1)⁢(et2-1)

=∫t1>t2>0d⁢t1Γ⁢(1)⁢(et1-1)∧t23-1⁢d⁢t2Γ⁢(3)⁢(et2-1)

=∫(0,∞)2u11-1⁢u23-1⁢d⁢u1∧d⁢u2Γ⁢(1)⁢Γ⁢(3)⁢(eu1+u2-1)⁢(eu2-1).

The first two equalities are of the same type as in the previous example. For the third equality we use equation (3.11). For the last equality we use the change of variable

t2=u2,

t1=u1+u2,

where u1>0 and u2>0. Following [8], we can interpolate the multiple zeta values by

ζ⁢(s1,…,sd)=1Γ⁢(s1)⁢⋯⁢Γ⁢(sd)⁢∫(0,∞)du1s1-1⁢⋯⁢udsd-1⁢d⁢u1∧…∧d⁢ud(eu1+…+ud-1)⁢(eu2+…+ud-1)⁢⋯⁢(eud-1).

If we denote

f0⁢(ℕ;t)=∑a∈ℕe-a⁢t,

then

f0⁢(ℕ,t)=1et-1

and

ζ⁢(s1,…,sd)=1Γ⁢(s1)⁢⋯⁢Γ⁢(sd)⁢∫(0,∞)d⋀j=1df0⁢(ℕ;ui+…+ud)⁢ujsj-1⁢d⁢uj.

Let n=[K:ℚ] be the degree of the number field. We recall Definition 2.10 of f0,

f0⁢(C;t1,t2,…,tn)=∑α∈Ce-∑i=1nσi⁢(α)⁢ti,

where σi:K→ℂ run through all embeddings of the field K into the complex numbers. Let C1 and C2 be two unimodular simple cones. We want to raise an algebraic integer α1 to a complex power s1 as a portion of the multiple Dedekind zeta function. We define

α1s1=es1⁢log⁡(α1)

for one element α∈C and α1=σ1⁢(α). Choose a branch of the logarithmic function by making a cut of the complex plane at the negative real numbers. Since C is a positive unimodular simple cone, we have that ℝ>0n⊂C*, the function σi composed with log is well defined on a positive unimodular simple cone C.

Then, we define a double Dedekind zeta function as

(3.12)ζK;C1,C2⁢(s1,1,…,sn,1;s1,2,…,sn,2)

=1Γ⁢(s1,1)⁢⋯⁢Γ⁢(sn,2)⁢∫(0,+∞)2⁢nf0⁢(C1;(u1,1+u1,2),…,(un,1+un,2))

×f0⁢(C2;u1,2,…,un,2)⁢⋀i=1nui,1si,1-1⁢d⁢ui,1∧⋀i=1nui,2si,2-1⁢d⁢ui,2.

This definition combines both double zeta function and multiple Dedekind zeta values with double iteration. More generally, we can interpolate all multiple Dedekind zeta values into multiple Dedekind zeta functions so that multiple zeta functions are particular cases. Again, in the above formula ui,jx is defined by

ui,jx=ex⁢log⁡(ui,j),

along the branch of logarithm described above.

Definition 3.7

Definition 3.7 (Multiple Dedekind zeta functions)

Let n=[K;ℚ] be the degree of the number field. Let C1,…,Cm be m positive unimodular simple cones in 𝒪K. We define multiple Dedekind zeta functions by the integral

ζK;C1,…,Cm⁢(s1,1,…,sn,1;…;s1,m,…,sn,m)

=∏(i,j)=(1,1)(n,m)1Γ⁢(si,j)⁢∫(0,+∞)m⁢n⋀j=1mf0⁢(Cj;(u1,j+…+u1,m),…,(un,j+…+un,m))

∧⋀i=1nui,jsi,j-1⁢d⁢ui,j

when Re⁡(si,j)>1.

Remark 3.8

One can deduce convergence of MDZF for Re⁡(si,j)>1 following the same steps as for the MDZV given in Remark 3.5.

Theorem 3.9

Theorem 3.9 (Infinite sum representation)

For the general form of a multiple Dedekind zeta function, we need

a number field K,
positive unimodular simple cones C1,…,Cm in 𝒪K,
elements αj∈Cj for j=1,…,m,
complex embeddings of the elements αi,j=σi⁢(αj).

Then, a multiple Dedekind zeta function has the following infinite sum representation:

ζK;C1,…,Cm⁢(s1,1,…,sn,1;…;s1,m,…,sn,m)

=∑α1∈C1⋯⁢∑αm∈Cm∏i=1n∏j=1m(αi,1+…+αi,j)-si,j

when Re⁡(si,j)>1.

Proof.

We have

ζK;C1,…,Cm⁢(s1,1,…,sn,1;…;s1,m,…,sn,m)

=∏(i,j)=(1,1)(n,m)1Γ⁢(si,j)⁢∫(0,+∞)m⁢n⋀j=1mf0⁢(Cj;(u1,j+…+u1,m),…,(un,j+…+un,m))

∧⋀i=1nui,jsi,j-1⁢d⁢ui,j

=∏(i,j)=(1,1)(n,m)1Γ⁢(si,j)⁢∑α1∈C1⋯⁢∑αm∈Cm∫(0,+∞)m⁢n⋀j=1m⋀i=1ne-(αi,1+…+αi,j)⁢ui,j⁢ui,jsi,j-1⁢d⁢ui,j

=∑α1∈C1⋯⁢∑αm∈Cm∏i=1n∏j=1m(αi,1+…+αi,j)-si,j,

proving the theorem. ∎

The following examples give a bridge between Dedekind zeta function and values of Eisenstein series (Example 5), and between multiple Dedekind zeta function and values of multiple Eisenstein series. More about values of multiple Eisenstein series will appear in Examples7–9.

Example 5

Let K be any number field, let m=1 and let C be a positive unimodular simple cone in 𝒪K. Then

(3.13)ζK;C⁢(s1,1,…,sn,1)=∑α∈C1∏i=1nαisi,1,

where αi=σi⁢(α) is the i-th embedding in the complex numbers. In particular, if all variables si,1, for i=1,…,n, have the same value s, then

(3.14)ζK;C⁢(s,…,s)=∑α∈C1NK/ℚ⁢(α)s.

Example 6

Now, let m=2. Then we have a double iteration. Let K be any number field. Let C1 and C2 be two positive unimodular simple cones. Then

(3.15)ζK;C1,C2⁢(s1,1,…,sn,1;s1,2,…,sn,2)=∑α∈C1,β∈C21∏i=1nαisi,1⁢(αi+βi)si,2.

In particular, if

sj=s1,j=…=sn,j

for j=1,2, then

(3.16)ζK;C1,C2⁢(s1,…,s1;s2,…,s2)=∑α∈C1,β∈C21NK/ℚ⁢(α)s1⁢NK/ℚ⁢(α+β)s2.

4 Analytic properties and special values

4.1 Applications to multiple Eisenstein series

Assuming the analytic continuation (Theorem 4.2), we can consider values of the multiple Dedekind zeta functions when one or more of the arguments are zero, which allows us to express special values of multiple Eisenstein series (see [6]) as multiple Dedekind zeta values. This is presented in the following three examples.

Example 7

Let K be an imaginary quadratic field. Let C be a positive unimodular simple cone in 𝒪K. We can represent the cone C as an ℕ-module: C=ℕ⁢{μ,ν}, μ,ν∈𝒪K. Put z=μ/ν. Consider ζK;C⁢(k1,1,k2,1) at k1,1=k and k2,1=0. Then

ζK;C⁢(k,0)=∑α∈C1α1k=ν-k⁢∑a,b∈ℕ1(a⁢z+b)k,

where the last sum is a portion of the k-th Eisenstein series, and

Ek⁢(τ)=∑a,b∈ℤ(a,b)≠(0,0)1(a⁢z+b)k.

is an analogue of Eisenstein series.

Example 8

Let K be an imaginary quadratic field. Let C be a positive unimodular simple cone in 𝒪K. We can represent C as

C=ℕ⁢{μ,ν}={α∈𝒪K:α=a⁢μ+b⁢ν,a,b∈ℕ}.

Put z=μ/ν. Then, we obtain a value of multiple Eisenstein series

ζK;C,C⁢(k,0;l,0)=ν-k-l⁢∑a,b,c,d∈ℕ1(a⁢z+b)k⁢((a+c)⁢z+(b+d))l.

Example 9

Similarly, one can define analogue of values of the above Eisenstein series over real quadratic field K, by setting

Ek⁢(z)=νk⁢ζK;C⁢(k,0)=∑α∈C1α1k,

where C=ℕ⁢{μ,ν} is a positive unimodular simple cone in a real quadratic ring of integers 𝒪K.

4.2 Examples of analytic continuation and multiple residues

The following examples of analytic continuations are based on a theorem of Gelfand–Shilov. The constructions in Example 11 is central for this subsection. Using Example 11, we change the variables in such a way that we can apply Gelfand–Shilov’s theorem (Theorem 4.1) that gives analytic continuation of MDZF. Moreover, in Example 11 we compute a multiple residue at (1,1;1,1). In Section 4.4, we generalize this method to other multiple residues and we state two conjectures – one about the values of the multiple residues, again based on Examples 10 and 11, and other about more general MDZV.

Let us recall the theorem of Gelfand–Shilov.

Theorem 4.1

Theorem 4.1 ([7])

Let ϕ⁢(x) be a test function on R which decreases rapidly (exponentially) when x→∞ and let

x+={xif x>0,0if x≤0.

Then the value of the distribution x+s-1⁢d⁢xΓ⁢(s) on the test function ϕ, namely,

∫ℝϕ⁢(x)⁢x+s-1⁢d⁢xΓ⁢(s),

is an analytic function in the variable s.

In Examples 10 and 11, we express a multiple Dedekind zeta function (MDZF) as a test function times a distribution when si,j>1 up to Γ-factors. Then Theorem 4.1 tells us that we have an analytic continuation of the MDZF to all complex values of si,j after multiplying by a suitable Γ-factors. Using this method, we compute the multiple residue at (1,…,1).

Example 10

Let K be a quadratic field and let C=ℕ⁢{α,β} be a positive unimodular simple cone. Let α1, α2 and β1, β2 be the images under the two embeddings into ℂ of α and β, respectively. We will compute the residue of

ζK;C⁢(s1,s2)=∑μ∈C1μ1s1⁢μ2s2

at the hyperplane s1+s2=2 and evaluated at s1=s2=1.

We have

ζK;C⁢(s1,s2)=1Γ⁢(s1)⁢Γ⁢(s2)⁢∫0∞∫0∞t1s1-1⁢t2s2-1⁢d⁢t1∧d⁢t2(eα1⁢t1+α2⁢t2-1)⁢(eβ1⁢t1+β2⁢t2-1).

Set t1=x1⁢(1-x2) and t2=x1⁢x2. Then Γ⁢(s1+s2-2)-1⁢ζK;C⁢(s1,s2) is the value of the distribution

D⁢x=x1+s1+s2-3⁢x2+s2-1⁢(1-x2)+s1-1⁢d⁢x1⁢d⁢x2Γ⁢(s1+s2-2)⁢Γ⁢(s1)⁢Γ⁢(s2)

at the test function

ϕ=x12(ex1⁢(α1+(α2-α1)⁢x2)-1)⁢(ex1⁢(β1+(β2-β1)⁢x2)-1).

Using Theorem 4.1 we obtain that Γ⁢(s1+s2-2)-1⁢ζK;C⁢(s1,s2) is an analytic function. The residue of ζK;C⁢(s1,s2) at s1+s2=2 is

∫011(α1+(α2-α1)⁢x2)⁢(β1+(β2-β1)⁢x2)⋅x2+s2-1⁢(1-x2)+s1-1⁢d⁢x2Γ⁢(s1)⁢Γ⁢(s2).

Evaluating at s2=1, the integral becomes

∫01d⁢x2(α1+(α2-α1)⁢x2)⁢(β1+(β2-β1)⁢x2).

Thus, the residue of ζK;C⁢(s1,s2) at s1+s2=2 evaluated at (s1,s2)=(1,1) is given by the above integral. After evaluating it, we obtain

(Ress1+s2=2⁡ζK;C⁢(s1,s2))|(s1,s2)=(1,1)=log⁡(α2α1)-log⁡(β2β1)|α1β1α2β2|.

In particular, if β=1, we obtain

(4.1)(Ress1+s2=2⁡ζK;C⁢(s1,s2))|(s1,s2)=(1,1)=log⁡(α2)-log⁡(α1)α2-α1.

Note that if K is a real quadratic field and α is a generator of the group of units, then

|log(α2)-log(α1)|=2|log(α1)|

is two times the regulator of the number field K and α2-α1 is an integer multiple of the discriminant of K. For a definition of a discriminant and a regulator of a number field, one may consult [12]. Equation (4.1) is true for any quadratic field, not only for real quadratic fields.

The following example gives key constructions needed for the proof of the analytic continuation of MDZF (Theorem 4.2). It is also a case study of Conjecture 4.4 about the multiple residue of a multiple Dedekind zeta function at (1,…,1).

Example 11

Let K be a quadratic extension of ℚ. Let C1=ℕ⁢{1,α}, C2=ℕ⁢{1,γ} be two positive unimodular simple cones. Let

ζK;C1,C2⁢(s1,s2;s1′,s2′)=∑μ∈C1,ν∈C21μ1s1⁢μ2s2⁢(μ1+ν1)s1′⁢(μ2+ν2)s2′.

An integral representation can be written as

(4.2)ζK;C1,C2⁢(s1,s2;s1′,s2′)⁢Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)

=∫t1>t1′>0t2>t2′>0(t1-t1′)s1-1⁢(t2-t2′)2s2-1⁢(t1′)s1′-1⁢(t2′)s2′-1⁢d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′(et1+t2-1)⁢(eα1⁢t1+α2⁢t2-1)⁢(et1′+t2′-1)⁢(eγ1⁢t1′+γ2⁢t2′-1).

We compute the residue of a double Dedekind zeta function by taking the multiple residues of six functions ζ(a),…,ζ(f) and considering their sum.

We shall write the last differential form in equation (4.2) as Dt. Then, we have

(4.3)ζK;C1,C2⁢(s1,s2;s1′,s2′)=1Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)⁢∫t1>t1′>0t2>t2′>0D⁢t.

We are going to express the above integral as a sum of six integrals, which are enumerated by all possible shuffles of t1>t1′>0 and t2>t2′>0. In other words, each of the six new integrals will be associated to each linear order among the variables t1,t2,t1′,t2′ that respect the above two inequalities among them. Thus, all possible cases are:

t1>t1′>t2>t2′>0,
t1>t2>t1′>t2′>0,
t1>t2>t2′>t1′>0,
t2>t1>t1′>t2′>0,
t2>t1>t2′>t1′>0,
t2>t2′>t1>t1′>0.

Then the domain {t1>t1′>0,t2>t2′>0} can be represented as a disjoint union of the domains of integration, given in parts (a)–(f). Then

ζK;C1,C2⁢(s1,s2;s1′,s2′)=1Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)⁢∫t1>t1′>0t2>t2′>0D⁢t

=1Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)(∫t1>t1′>t2>t2′>0Dt+∫t1>t2>t1′>t2′>0Dt

+∫t1>t2>t2′>t1′>0D⁢t+∫t2>t1>t1′>t2′>0D⁢t

+∫t2>t1>t2′>t1′>0Dt+∫t2>t2′>t1>t1′>0Dt)

=ζK;C1,C2(a)⁢(s1,s2;s1′,s2′)+ζK;C1,C2(b)⁢(s1,s2;s1′,s2′)

+ζK;C1,C2(c)⁢(s1,s2;s1′,s2′)+ζK;C1,C2(d)⁢(s1,s2;s1′,s2′)

+ζK;C1,C2(e)⁢(s1,s2;s1′,s2′)+ζK;C1,C2(f)⁢(s1,s2;s1′,s2′).

We define ζ(a),…,ζ(f) to be the above six integrals, corresponding to the domains of integration given by (a)–(f). The reason for defining them is to take multiple residues of the multiple Dedekind zeta function. It is easier to work with the functions ζ(a),…,ζ(f) for the purpose of proving analytic continuation and taking residues.

Thus, we compute the residue of a double Dedekind zeta function by taking the multiple residues of six functions ζ(a),…,ζ(f) and considering their sum.

Consider the domain of integration

t1>t1′>t2>t2′>0.

We will compute the residues of

ζK;C1,C2(a)⁢(s1,s2;s1′,s2′)⁢Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)

=∫t1>t1′>t2>t2′>0(t1-t1′)s1-1⁢(t2-t2′)2s2-1⁢(t1′)s1′-1⁢(t2′)s2′-1⁢d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′(et1+t2-1)⁢(eα1⁢t1+α2⁢t2-1)⁢(et1′+t2′-1)⁢(eγ1⁢t1′+γ2⁢t2′-1).

Let the successive differences be

u1=t1-t1′,u2=t1′-t2,

u3=t2-t2′,u4=t2′.

Their admissive values are in the interval (0,+∞). Let us make the substitution

u1=x1⁢(1-x2),u2=x1⁢x2⁢(1-x3),

u3=x1⁢x2⁢x3⁢(1-x4),u4=x1⁢x2⁢x3⁢x4.

We are going to express the above integral in terms of the variables x1, x2, x3 and x4. We have

α1⁢t1+α2⁢t2=α1⁢(u1+u2+u3+u4)+α2⁢(u3+u4)=x1⁢(α1+α2⁢x2⁢x3),

t1+t2=x1⁢(1+x1⁢x3),

γ1⁢t1′+γ2⁢t2′=γ1⁢(u2+u3+u4)+γ2⁢u4=x1⁢x2⁢(1+x3⁢x4),

t1′+t2′=x1⁢x2⁢(1+x3⁢x4),

t1-t1′=u1=x1⁢(1-x2),

t2-t2′=u3=x1⁢x2⁢x3⁢(1-x4),

t1′=u2+u3+u4=x1⁢x2,

t2′=u4=x1⁢x2⁢x3⁢x4.

For the change of variables in the differential forms, we have

d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′=d⁢u1∧d⁢u2∧d⁢u3∧d⁢u4

and

d⁢u1u1∧d⁢u2u2∧d⁢u3u3∧d⁢u4u4=d⁢x1∧d⁢x2∧d⁢x3∧d⁢x4x1⁢x2⁢x3⁢x4⁢(1-x2)⁢(1-x3)⁢(1-x4).

Then

(4.4)ζK;C1,C2(a)⁢(s1,s2;s1′,s2′)⁢Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)

=∫t1>t1′>t2>t2′>01(et1+t2-1)⁢(eα1⁢t1+α2⁢t2-1)⁢(et1′+t2′-1)⁢(eγ1⁢t1′+γ2⁢t2′-1)

×(t1-t1′)s1-1⁢(t2-t2′)2s2-1⁢(t1′)s1′-1⁢(t2′)s2′-1⁢d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′

=∫(u1,…,u4)∈(0,∞)4[(eu1+u2+2⁢(u3+u4)-1)-1

×(eα1⁢(u1+u2+u3+u4)+α2⁢(u3+u4)-1)-1

×(eu2+u3+2⁢u4-1)-1⁢(eγ1⁢(u2+u3+u4)+γ2⁢u4-1)-1

×u1s1-1u3s2-1(u2+u3+u4)s1′-1u4s2′-1du1∧du2∧du3∧du4]

=∫(u1,…,u4)∈(0,∞)4[(eu1+u2+2⁢(u3+u4)-1)-1

×(eα1⁢(u1+u2+u3+u4)+α2⁢(u3+u4)-1)-1

×(eu2+u3+2⁢u4-1)-1⁢(eγ1⁢(u2+u3+u4)+γ2⁢u4-1)-1

×u1s1u3s2(u2+u3+u4)s1′-1u4s2′u2d⁢u1u1∧d⁢u2u2∧d⁢u3u3∧d⁢u4u4]

=∫0∞∫(0,1)3[(ex1⁢(1+x2⁢x3)-1)-1(ex1⁢(α1+α2⁢x2⁢x3)-1)-1

×(ex1⁢x2⁢(1+x3⁢x4)-1)-1⁢(ex1⁢x2⁢(γ1+γ2⁢x3⁢x4)-1)-1

×[x1⁢(1-x2)]s1⁢[x1⁢x2⁢x3⁢(1-x4)]s2⁢[x1⁢x2]s1′-1

×[x1x2x3x4]s2′[x1x2(1-x3)]Ω],

where

Ω=d⁢u1u1∧d⁢u2u2∧d⁢u3u3∧d⁢u4u4=d⁢x1∧d⁢x2∧d⁢x3∧d⁢x4x1⁢x2⁢x3⁢x4⁢(1-x2)⁢(1-x3)⁢(1-x4).

We can express the last integral as a distribution evaluated at a test function. Put

ϕ⁢(x1,x2,x3,x4)

=x14⁢x22(ex1⁢(1+x2⁢x3)-1)⁢(ex1⁢(α1+α2⁢x2⁢x3)-1)⁢(ex1⁢x2⁢(1+x3⁢x4)-1)⁢(ex1⁢x2⁢(γ1+γ2⁢x3⁢x4)-1)

to be a test function. Let Dx be a distribution defined by

(4.5)D⁢x=x1+s1+s2+s1′+s2′-5Γ⁢(s1+s2+s1′+s2′-4)⁢x2+s2+s1′+s2′-3Γ⁢(s2+s1′+s2′-2)

×x3+s2+s2′-1Γ⁢(s2+s2′)⁢x4+s2′-1Γ⁢(s2′)⁢(1-x2)+s1-1⁢(1-x4)+s2-1.

Then we have

ζK;C1,C2(a)⁢(s1,s2;s1′,s2′)=Γ⁢(s1+s2+s1′+s2′-4)⁢Γ⁢(s2+s1′+s2′-2)⁢Γ⁢(s2+s2′)Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢∫ϕ⁢D⁢x.

Using Theorem 4.1 we prove analytic continuation of ζK;C1,C2(a)⁢(s1,s2;s1′,s2′) everywhere except at the poles of

Γ⁢(s1+s2+s1′+s2′-4)⁢Γ⁢(s2+s1′+s2′-2)⁢Γ⁢(s2+s2′).

We will take the residue at

s1+s2+s1′+s2′=4

and then evaluate at (s1,s2;s1′,s2′)=(1,1;1,1). Note that

ϕ⁢(0,x2,x3,x4)=1(1+x2⁢x3)⁢(α1+α2⁢x2⁢x3)⁢(1+x3⁢x4)⁢(γ1+γ2⁢x3⁢x4).

Then we can compute

(Ress1+s2+s1′+s2′=4⁡ζK;C1,C2(a)⁢(s1,s2;s1′,s2′))|(1,1;1,1)

=∫(0,1)3x3⁢d⁢x2∧d⁢x3∧d⁢x4(1+x2⁢x3)⁢(α1+α2⁢x2⁢x3)⁢(1+x3⁢x4)⁢(γ1+γ2⁢x3⁢x4).

Note that we cannot take a double residue at the point (1,1;1,1). Also, the value that we obtain is a period.

Consider the domain in integration

t1>t2>t1′>t2′>0.

We will compute the residues of

ζK;C1,C2(b)⁢(s1,s2;s1′,s2′)⁢Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)

=∫t1>t2>t1′>t2′>0(t1-t1′)s1-1⁢(t2-t2′)s2-1⁢(t1′)s1′-1⁢(t2′)s2′-1⁢d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′(et1+t2-1)⁢(eα1⁢t1+α2⁢t2-1)⁢(et1′+t2′-1)⁢(eγ1⁢t1′+γ2⁢t2′-1).

Let the successive differences be

u1=t1-t2,u2=t2-t1′,

u3=t1′-t2′,u4=t2′.

Their admissive values are (0,+∞). Let

u1=x1⁢(1-x2),u2=x1⁢x2⁢(1-x3),

u3=x1⁢x2⁢x3⁢(1-x4),u4=x1⁢x2⁢x3⁢x4.

We will express the above integral in terms of x1,…,x4. We have

α1⁢t1+α2⁢t2=α1⁢(u1+u2+u3+u4)+α2⁢(u2+u3+u4)=x1⁢(α1+α2⁢x2),

t1+t2=x1⁢(1+x2),

γ1⁢t1′+γ2⁢t2′=γ1⁢(u3+u4)+γ2⁢u4=x1⁢x2⁢x3⁢(γ1+γ2⁢x4),

t1′+t2′=x1⁢x2⁢x3⁢(1+x4),

t1-t1′=u1+u2=x1⁢(1-x2⁢x3),

t2-t2′=u2+u3=x1⁢x2⁢(1-x3⁢x4),

t1′=u3+u4=x1⁢x2⁢u3,

t2′=u4=x1⁢x2⁢x3⁢x4.

For the differential forms, we have

d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′=d⁢u1∧d⁢u2∧d⁢u3∧d⁢u4,

d⁢u1u1∧d⁢u2u2∧d⁢u3u3∧d⁢u4u4=d⁢x1∧d⁢x2∧d⁢x3∧d⁢x4x1⁢x2⁢x3⁢x4⁢(1-x2)⁢(1-x3)⁢(1-x4).

Then

ζK;C1,C2(b)⁢(s1,s2;s1′,s2′)⁢Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢Γ⁢(s2′)

=∫t1>t2>t1′>t2′>0(t1-t1′)s1-1⁢(t2-t2′)s2-1⁢(t1′)s1′-1⁢(t2′)s2′-1⁢d⁢t1∧d⁢t2∧d⁢t1′∧d⁢t2′(et1+t2-1)⁢(eα1⁢t1+α2⁢t2-1)⁢(et1′+t2′-1)⁢(eγ1⁢t1′+γ2⁢t2′-1)

=∫(u1,…,u4)∈(0,∞)4[(eu1+2⁢(u2+u3+u4)-1)-1(eα1⁢(u1+u2+u3+u4)+α2⁢(u2+u3+u4)-1)-1

×(eu3+2⁢u4-1)-1⁢(eγ1⁢(u3+u4)+γ2⁢u4-1)-1⁢(u1+u2)s1-1⁢(u2+u3)s2-1

×(u3+u4)s1′-1u4s2′-1du1∧du2∧du3∧du4]

=∫(u1,…,u4)∈(0,∞)4[(eu1+2⁢(u2+u3+u4)-1)-1(eα1⁢(u1+u2+u3+u4)+α2⁢(u2+u3+u4)-1)-1

×(eu3+2⁢u4-1)-1⁢(eγ1⁢(u3+u4)+γ2⁢u4-1)-1⁢(u1+u2)s1-1⁢(u2+u3)s2-1

×(u3+u4)s1′-1u4s2′u1u2u3d⁢u1∧d⁢u2∧d⁢u3∧d⁢u4u1⁢u2⁢u3⁢u4]

=∫0∞∫(0,1)3[(ex1⁢(1+x2⁢x3)-1)-1(ex1⁢(α1+α2⁢x2⁢x3)-1)-1(ex1⁢x2⁢(1+x3⁢x4)-1)-1

×(ex1⁢x2⁢(γ1+γ2⁢x3⁢x4)-1)-1⁢[x1⁢(1-x2⁢x3)]s1-1⁢[x1⁢x2⁢(1-x3⁢x4)]s2-1

×[x1x2x3]s1′-1[x1x2x3x4]s2′XΩ],

where

X=x13⁢x22⁢x3⁢(1-x2)⁢(1-x3)⁢(1-x4)

and

Ω=d⁢u1∧d⁢u2∧d⁢u3∧d⁢u4u1⁢u2⁢u3⁢u4=d⁢x1∧d⁢x2∧d⁢x3∧d⁢x4x1⁢x2⁢x3⁢x4⁢(1-x2)⁢(1-x3)⁢(1-x4).

Now we can express the last integral as a distribution evaluated at a test function. Put

ϕ⁢(x1,x2,x3,x4)

=x14⁢x22⁢x32(ex1⁢(1+x2)-1)⁢(ex1⁢(α1+α2⁢x2)-1)⁢(ex1⁢x2⁢x3⁢(1+x4)-1)⁢(ex1⁢x2⁢x3⁢(γ1+γ2⁢x4)-1)

to be a test function. Let Dx be a distribution defined by

(4.6)D⁢x=x1+s1+s2+s1′+s2′-5Γ⁢(s1+s2+s1′+s2′-4)⁢x2+s2+s1′+s2′-3Γ⁢(s2+s1′+s2′-2)

×x3+s2+s2′-3Γ⁢(s2+s2′-2)⁢x4+s2′-1Γ⁢(s2′)⁢(1-x2⁢x3)+s1-1⁢(1-x3⁢x4)+s2-1.

Then we have

ζK;C1,C2(b)⁢(s1,s2;s1′,s2′)=Γ⁢(s1+s2+s1′+s2′-4)⁢Γ⁢(s2+s1′+s2′-2)⁢Γ⁢(s2+s2′-2)Γ⁢(s1)⁢Γ⁢(s2)⁢Γ⁢(s1′)⁢∫ϕ⁢D⁢x.

Using Theorem 4.1 we prove analytic continuation of ζK;C1,C2(b)⁢(s1,s2;s1′,s2′) everywhere except at the poles of

Γ⁢(s1+s2+s1′+s2′-4)⁢Γ⁢(s2+s1′+s2′-2)⁢Γ⁢(s2+s2′-2).

We will take the residues at s1+s2+s1′+s2′=4 and at s1′+s2′=2. Then, we will evaluate at (s1,s2;s1′,s2′)=(1,1;1,1). Note that

ϕ⁢(0,x2,0,x4)=1(1+x2)⁢(α1+α2⁢x2)⁢(1+x4)⁢(γ1+γ2⁢x4).

Then we can compute

(Ress1′+s2′=2⁡Ress1+s2+s1′+s2′=4⁡ζK;C1,C2(b)⁢(s1,s2;s1′,s2′))|(1,1;1,1)

=∫(0,1)2d⁢x2∧d⁢x4(1+x2)⁢(α1+α2⁢x2)⁢(1+x4)⁢(γ1+γ2⁢x4)=log⁡(2⁢α1α1+α2)α1-α2⋅log⁡(2⁢γ1γ1+γ2)γ1-γ2.

For the cases (c), (d) and (e), we obtain

(Ress1′+s2′=2⁡Ress1+s2+s1′+s2′=4⁡ζK;C1,C2(c)⁢(s1,s2;s1′,s2′))|(1,1;1,1)

=∫(0,1)2d⁢x2∧d⁢x4(1+x2)⁢(α1+α2⁢x2)⁢(x4+1)⁢(γ1⁢x4+γ2)=-log⁡(2⁢α1α1+α2)α1-α2⋅log⁡(2⁢γ2γ1+γ2)γ1-γ2,

(Ress1′+s2′=2⁡Ress1+s2+s1′+s2′=4⁡ζK;C1,C2(d)⁢(s1,s2;s1′,s2′))|(1,1;1,1)

=∫(0,1)2d⁢x2∧d⁢x4(x2+1)⁢(α1⁢x2+α2)⁢(1+x4)⁢(γ1+γ2⁢x4)=-log⁡(2⁢α2α1+α2)α1-α2⋅log⁡(2⁢γ1γ1+γ2)γ1-γ2

and

(Ress1′+s2′=2⁡Ress1+s2+s1′+s2′=4⁡ζK;C1,C2(e)⁢(s1,s2;s1′,s2′))|(1,1;1,1)

=log⁡(2⁢α2α1+α2)α1-α2⋅log⁡(2⁢γ2γ1+γ2)γ1-γ2.

Case (f) is similar to case (a), namely, there is no double residue at the point (1,1;1,1). Thus, we obtain

Ress1′+s2′=2Ress1+s2+s1′+s2′=4ζK;C1,C2(s1,s2;s1′,s2′))|(1,1;1,1)

=Ress1′+s2′=2Ress1+s2+s1′+s2′=4(ζK;C1,C2(b)(s1,s2;s1′,s2′)+ζK;C1,C2(c)(s1,s2;s1′,s2′)

+ζK;C1,C2(d)(s1,s2;s1′,s2′)+ζK;C1,C2(e)(s1,s2;s1′,s2′))|(1,1;1,1)

=log⁡(α2)-log⁡(α1)α2-α1⋅log⁡(γ2)-log⁡(γ1)γ2-γ1.

Note that if K is a real quadratic field and α is a generator of the group of units, then

|log(α2)-log(α1)|=2log|α1|

is two times the regulator of the number field K and α2-α1 is an integer multiple of the discriminant of the field K. The above formula is true for any quadratic field, not necessarily for a real quadratic field.

4.3 Analytic continuation of multiple Dedekind zeta functions

Theorem 4.2

Multiple Dedekind zeta functions

ζK;C1,…,Cm⁢(s1,1,…,sn,1;…;s1,m,…,sn,m)

have an analytic continuation from the region Re⁡(si,j)>1 for all i and j to si,j∈C with exception of hyperplanes. The hyperplanes are defined by a sum of several of the variables si,j without repetitions being set equal to an integer.

Proof.

Recall that f0⁢(Cj;t1,j,…,tn,j) is used to define the multiple Dedekind zeta functions, where the domain of integration is

D={(ti,j)∈ℝm⁢n:ti,1>ti,2>…>ti,m>0}.

For each i=1,…,n, let Ji={ki,1,…,ki,m} be a finite set of integers. Note that there are n sets J1,…,Jn, and each of them has m elements, |Ji|=m. Let τ run through all the shuffles of the ordered sets J1,…,Jn,

τ∈Sh⁢(J1,…,Jn).

Let t1,t2,…,tm⁢n be the variables t1,1,…,tn,m, written in decreasing order. There are finitely many ways of arranging the variables in decreasing order. More precisely, the number of such arrangements is equal to the number of shuffles in Sh⁢(J1,…,Jn). We need to consider all such shuffles in order to express the multiple Dedekind zeta function as a sum of partial multiple Dedekind zeta functions, corresponding to each shuffle τ (see Example 11). Let uk=tk-tk+1 for k=1,…,m⁢n-1 and um⁢n=tm⁢n. Let

u1=x1⁢(1-x2),

u2=x1⁢x2⁢(1-x3),

⋮

um⁢n-1=x1⁢⋯⁢xm⁢n-1⁢(1-xm⁢n),

um⁢n=x1⁢⋯⁢xm⁢n.

Then we examine the integrant that gives an MDZF in terms of the variables x1,…,xm⁢n. It consists of a product of functions f0⁢(Cj;…) times xi raised to a power and times (1-xi) raised to a power. We can write f0⁢(Cj;…) in a compact form. Its denominator is of the type (eh1-1)⁢⋯⁢(ehn-1), where h1,…,hn are polynomials in x1,…,xm⁢n. For example, in equation (4.4) for ζ(a) consider the denominator of f0⁢(C1;…). It is of the above type, where the polynomials h1 and h2 are

h1=x1⁢(1+x2⁢x3) and h2=x1⁢(α1+α2⁢x2⁢x3).

Note that in general hj is a factor of (ehj-1). We might be able to factor hj further. We find that hj=x1⁢⋯⁢xk⁢gj for some k and for some polynomial gk non-vanishing at the origin. For example in equation (4.4), we have g1=1+x2⁢x3 and g2=α1+α2⁢x2⁢x3. Similarly to MDZF, we have that a partial MDZF is an analytic function for Re⁡(si,j)>1. Using Theorem 4.1, we find that the partial MDZF together with the Γ-factors is an analytic function for all values of si,j. Similarly to the computations for ζ(a),…,ζ(f) in Example 11, we find that the Γ-factors are of the following type: Γ⁢(ψ⁢(s1,…,sn)), where ψ⁢(s1,…,sn) is a degree 1 polynomial with integer coefficients. Geometrically this means that the poles of a partial MDZF lie on hyperplanes, whose equations have integer coefficients. Expressing an MDZF as a finite sum of partial MDZFs we obtain the analytic continuation from the domain Re⁡(si,j)>1 to si,j∈ℂ with poles along hyperplanes coming from Γ-factors. ∎

4.4 Final remarks

In this final subsection, we proof that certain multiple residue of a multiple Dedekind zeta function is a period in the sense of algebraic geometry. Based on Theorem 4.3, we state two conjectures. One of the conjectures is about the exact values of the multiple residue and the other conjecture is about values of the multiple Dedekind zeta functions at other integers.

Theorem 4.3

The multiple residue of a multiple Dedekind zeta function at the point

(s1,1,…,sn,1;…;s1,m,…,sn,m)=(1,…,1)

is a period over Q in the sense of Kontsevich–Zagier [13].

Proof.

We use the notation introduced in the proof of Theorem 4.2. After we take the multiple residues of a partial MDZF at (1,…,1) along hyperplanes, we obtain an integral of a rational function, whose denominator is a product of polynomials gj, defined in the proof of Theorem 4.2. The boundaries of the integral (after taking the multiple residues) form a unit cube. Therefore, the value of the multiple residue at (1,…,1) of a partial MDZF is a period. Since an MDZF is a finite sum of partial MDZFs, we obtain that the multiple residue of an MDZF at (1,…,1) is also a period. ∎

For a more precise interpretation see Conjecture 4.4 and Examples 10 and 11.

From Examples 10 and 11, we know that a multiple residue of a multiple Dedekind zeta function is a product of residues of partial Dedekind zeta functions, for quadratic fields and double iteration. For unimodular simple cones C1,…,Cm, we consider a multiple Dedekind zeta function

ζK;C1,…,Cm⁢(s1,…,sd)=∑α1∈C1,…,αd∈Cm1N⁢(α1)s1⁢N⁢(α1+α2)s2⁢⋯⁢N⁢(α1+…+αm)sm.

We expect that

Conjecture 4.4

The multiple residue of ζK;C1,…,Cm⁢(s1,…,sm) at the point (1,…,1) is equal to

Ressm=1⁡…⁢Ress1+…+sm=m⁡ζK;C1,…,Cm⁢(s1,…,sm)=∏j=1mRess=1⁡ζK;Cj⁢(s).

The conjecture is proven for a quadratic fields K and double iteration in Examples 10 and 11.

We do expect that multiple Dedekind zeta values should be periods over ℚ.

Conjecture 4.5

Let K be a number field. For any choice of unimodular simple cones C1,…,Cm, in the ring of integers of a number field K, we have that the multiple Dedekind zeta values (see Definition 3.4)

ζK;C1,…,Cm⁢(k1,1,…,kn,1;…;k1,m,…,kn,m)

are periods over ℚ when the k1,1,…,km,n are natural numbers greater than 1.

The reasons for this conjecture are the following:

We have that multiple zeta values are periods.
Dedekind zeta values are periods.
From Theorem 4.3, we have that the multiple residue of a multiple Dedekind zeta function at (1,…,1) is a period.
The main reason is the representation of multiple Dedekind zeta values as iterated integrals on membranes. We will give a semi-algebraic relations among the variables in such integrals.

We use equations (2.7) and (2.8). Recall that

yj=∏i=1nexp⁡(-σi⁢(ej)⁢zi).

If we set

xi=exp⁡(-zi),

then a multiple Dedekind zeta value is an iterated integral on a membrane of the n-form ∏jd⁢yj1-yj and the 1-forms d⁢xixi, which mostly resembles polylogarithms. However, the relations between the variables xi and yj are (semi-)algebraic, namely,

d⁢yjyj=∑i=1nσi⁢(ej)⁢d⁢xixi,

which are not algebraic. Explicitly, they are given by

log⁡(yj)=∑i=1nσi⁢(ej)⁢log⁡(xi),

which involves the logarithmic function. Note that the logarithmic function is a homotopy invariant function on a path space. In this setting, the above logarithmic functions can be considered as functions on the path space of an affine n-space without some divisors. One may take a simplicial scheme as a model of the path space so that it restricts well onto the loop space of a scheme as a simplicial scheme. Hopefully, that would interpret the above (semi-)algebraic relations in terms of logarithms in an algebraic context.

Acknowledgements

This paper owes a lot to many people. Ronald Brown gave inspiring talk on higher cubical categories. A few days after that talk, I had a good definition of iterated integrals on a membrane. With Alexander Goncharov I had a lot of fruitful discussions. His great interest in my different approaches to multiple Dedekind zeta functions encouraged me to continue my search of the right ones. From Mladen Dimitrov I learned about the cones constructed by Shintani. Anton Deitmar asked many questions on the subject that helped me to clear up (at least to myself) the structure of the paper. Matthew Kerr clarified many questions I had about Hodge theory. Dev Sinha explained simplicial de Rham structures on mapping spaces of a manifold, in particular, a path space or a loop space of a variety. Finally, I would like to thank the referee whose effort greatly improved the presentation. I acknowledge the great hospitality of Durham University and the generous financial support from the European Network Marie Curie that allowed me to invite Ronald Brown for several days. I would like to thank Tubingen University and in particular, Anton Deitmar and his grant “Higher modular forms and higher invariants” for the financial support and great working conditions.

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Received: 2012-1-3

Revised: 2014-4-14

Published Online: 2014-7-26

Published in Print: 2017-1-1

Articles in the same Issue

https://doi.org/10.1515/crelle-2014-0055