Volume 24, Issue 1

Abstract. This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies “zeta integrals” associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a,c)$(a,c)$ with 0<a,c<1$0&lt; a, c&lt; 1$. It extends these functions to real a$a$ and c$c$, and studies limiting cases of these functions where at least one of a$a$ and c$c$ take the values 0 or 1. A main feature is that as a function of three variables (s,a,c)$(s, a, c)$, in which a$a$ and c$c$ are real, the Lerch zeta function has discontinuities at integer values of a$a$ and c$c$. For fixed s$s$, the function (s,a,c)$\zeta (s,a,c)$ is discontinuous on part of the boundary of the closed unit square in the (a,c)$(a,c)$-variables, and the location and nature of these discontinuities depend on the real part (s)$\Re (s)$ of s$s$. Analysis of this behavior is used to determine membership of these functions in Lp([0,1]2,dadc)$L^p([0,1]^2, da\,dc)$ for 1p<$1 \le p &lt; \infty $, as a function of (s)$\Re (s)$. The paper also defines generalized Lerch zeta functions associated to the oscillator representation, and gives analogous four-term functional equations for them.

Abstract. This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. The Lerch zeta function (s,a,c):=n=0e2ina(n+c)s$ \zeta (s,a,c) := \sum _{n=0}^\infty \frac{e^{2 \pi in a}}{ (n+c)^{s}} $ was introduced by Lipschitz in 1857, and is named after Lerch, who showed in 1887 that it satisfied a functional equation. Here we analytically continue (s,a,c)$\zeta (s, a, c)$ as a function of three complex variables. We show that it is well-defined as a multivalued function on the manifold :=(s,a,c)()(),${\mathcal {M}}:= \lbrace (s, a, c) \in {\mathbb {C}}\times ( {\mathbb {C}}\setminus {\mathbb {Z}}) \times ( {\mathbb {C}}\setminus {\mathbb {Z}}) \rbrace ,$ and that this analytic continuation becomes single-valued on the maximal abelian cover of ${\mathcal {M}}$. We compute the monodromy functions describing the multivalued nature of this function on ${\mathcal {M}}$, and determine various of its properties.

Abstract. The new bound toward the generalized Ramanujan conjecture for GLn(𝐀𝐐)$\textup {GL}_n({\bf A}_{\bf Q})$ is given by Kim–Sarnak [4]. We generalize it to that for GLn(𝐀F)$\textup {GL}_n({\bf A}_{F})$ with F$F$ being an imaginary quadratic field. Applying the obtained bound to the automorphic representation of GL5$\textup {GL}_5$ gives the following estimate for the first positive eigenvalue of the Laplacian on L2(3)$L^2(\Gamma \backslash {\mathbb {H}}^3)$, where $\Gamma $ is any congruence subgroups of SL2(F)$\textup {SL}_2({\mathcal {O}}_F)$ with F${\mathcal {O}}_F$ the ring of integers of F$F$: 1()0.952$\lambda _1(\Gamma ) \ge 0.952\cdots $.

Abstract. Let E$E$ be an elliptic curve over ${\mathbb {Q}}$ without complex multiplication. For each prime p$p$ of good reduction, let |E(p)|$|E({\mathbb {F}}_p)|$ be the order of the group of points of the reduced curve over p${\mathbb {F}}_p$. According to a conjecture of Koblitz, there should be infinitely many such primes p$p$ such that |E(p)|$|E({\mathbb {F}}_p)|$ is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E$E$ is a curve without local obstructions (which is the case for most elliptic curves over ${\mathbb {Q}}$). We prove in this paper that, under the GRH, there are at least 2.778CE twin x/(logx)2$2.778 C_E^{\rm twin} x / (\log x)^2$ primes p$p$ such that |E(p)|$|E({\mathbb {F}}_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant CE twin $C_E^{\rm twin}$ appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes p$p$ such that |E(p)|$|E({\mathbb {F}}_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.

Abstract. In this article we describe a canonical way to expand a certain kind of (2)n+1$(\mathbb {Z}_2)^{n+1}$-colored regular graphs into closed n$n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial n$n$-manifold can be obtained in this way. When n3$n\le 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a (2)n+1$(\mathbb {Z}_2)^{n+1}$-colored regular graph admits an n$n$-skeletal expansion, then it is realizable as the induced graph of an (n+1)$(n+1)$-dimensional closed (2)n+1$(\mathbb {Z}_2)^{n+1}$-manifold.

Abstract. Let X$X$ be a complex space and k$k$ be a positive integer such the cohomology groups H1(X,),...,Hk(X,)$H^1(X,{\cal O}), \ldots , H^{k}(X,{\cal O})$ are of finite dimension. If there is a holomorphic map f:Xk$f:X \longrightarrow ^k $ with fibres Stein, then X$X$ is Stein.

Abstract. We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called `crowns'. Moreover, there is no homomorphism from any of our digraphs onto Z$Z$. The methods are adapted from [6] and [7].

Abstract. Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein–Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein–Weiss lemma are used to provide short proofs of the Frank–Seiringer “Hardy inequalities” where fractional smoothness is measured by a Besov norm.

Abstract. We consider actions of the complex affine group Aff()$\operatorname{Aff}({\mathbb {C}})$. For the case the action exhibits a regular dicritic singularity psing()$p\in \operatorname{sing}({\mathcal {F}_{\varphi }})$ and the ambient manifold is Stein we prove a linearization theorem. In dimension three we prove a (Reeb) stability theorem for a holomorphic foliations given by an affine group action on Stein 3-manifolds.