Uniqueness theorems for L-functions in the extended Selberg class

Wen-Jie Hao; Jun-Fan Chen

doi:10.1515/math-2018-0107

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Uniqueness theorems for L-functions in the extended Selberg class

Wen-Jie Hao and Jun-Fan Chen

Published/Copyright: November 10, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we obtain uniqueness theorems of L-functions from the extended Selberg class, which generalize and complement some recent results due to Li, Wu-Hu, and Yuan-Li-Yi.

Keywords: Meromorphic function; L-function; Selberg class; Value distribution

MSC 2010: 11M36; 30D35; 30D30

1 Introduction

The Riemann hypothesis as one of the millennium problems has been given a lot of attention by many scholars for a long time. Selberg guessed that the Riemann hypothesis also holds for the L-function in the Selberg class. Such an L-function based on the Riemann zeta function as a prototype is defined to be a Dirichlet series

L(s)=∑n=1∞a(n)ns(1)

of a complex variable s = σ + it satisfying the following axioms [1]:

Ramanujan hypothesis: a(n) ≪ n^ϵ for every ϵ > 0.
Analytic continuation: There exists a nonnegative integer m such that (s − 1)^mL(s) is an entire function of finite order.
Functional equation: L satisfies a functional equation of type

ΛL(s)=ωΛL(1−s¯)¯,

where

ΛL(s)=L(s)Qs∏j=1KΓ(λjs+νj)

with positive real numbers Q, λ_j, and complex numbers ν_j, ω with Reν_j ≥ 0 and |ω| = 1.

Euler product: log⁡L(s)=∑n=1∞b(n)nswhere b(n) = 0 unless n is a positive power of a prime and b(n) ≪ n^Ø for some θ<12

It is mentioned that there are many Dirichlet series but only those satisfying the axioms (i)-(iii) are regarded as the extended Selberg class [1, 2]. All the L-functions which are studied in this article are from the extended

Selberg class. Therefore, the conclusions proved in this article are also true for L-functions in the Selberg class. Theorems in this paper will be proved by means of Nevanlinna’s Value distribution theory. Suppose that F and G are two nonconstant meromorphic functions in the complex plane C, c denotes a value in the extended complex plane ℂ∪ {∞}. If F − c and G − c have the same zeros counting multiplicities, we say that F and G share c CM. If F − c and G − c have the same zeros ignoring multiplicities, then we say that F and G share c IM. It is well known that two nonconstant meromorphic functions in ℂ are identically equal when they share five distinct values IM [3, 4 1].

Theorem 1.1 (see [1]). If two L-functions with a(1) = 1 share a complex value c ≠ ∞ CM, then they are identically equal.

Remark 1.2. In [5], the authors gave an example thatL1=1+24sandL2=1+39swhich showed that Theorem 1.1 is actually false when c = 1.

In 2011, Li [6] considered values which are shared IM and got

Theorem 1.3 (see [6]). Let L₁and L₂be two L-functions satisfying the same functional equation with a(1) = 1 and let a₁, a₂ ϵ ℂ be two distinct values. IfL1−1(aj)=L2−1(aj),j=1,2then L₁ ≡ L₂.

In 2001, Lahiri [7] put forward the concept of weighted sharing as follows.

Let k be a nonnegative integer or∞, c ϵ ℂ∪{∞}. We denote by E_k(c, f) the set of all zeros of f − c, where a zero of multiplicity m is counted m times if m ≤ k and k +1 times if m > k. If E_k(c, f) = E_k(c, g), we say that f and g share the value c with weight k (see [7]).

In 2015, Wu and Hu [8] removed the assumption that both L-functions satisfy the same functional equation in Theorem 1.3. By including weights, they had shown the following result.

Theorem 1.4 (see [8]). Let L₁and L₂be two L-functions, and let a₁, a₂ ϵ ℂ be two distinct values. Take two positive integers k₁, k₂with k₁k₂ > 1. If E_kj (a_j, L₁) = E_kj (a_j, L₂), j = 1, 2, then L₁ ≡ L₂.

In 2003, the following question was posed by C.C. Yang [9].

Question 1.5 (see [9]). Let f be a meromorphic function in the complex plane and a, b, c are three distinct values, where c ≠ 0,∞. If f and the Riemann zeta function ζ share a, b CM and c IM, will then f ≡ ζ?

The L-function is based on the Riemann zeta function as the model. It is then valuable that we study the relationship between an L-function and an arbitrary meromorphic function [10, 11, 12, 13 – 14]. This paper concerns the problem of how meromorphic functions and L-functions are uniquely determined by their c-values. Firstly, we introduced the following theorem.

Theorem 1.6 (see [10]). Let a and b be two distinct finite values and f be a meromorphic function in the complex plane with finitely many poles. If f and a nonconstant L-function L share a CM and b IM, then L ≡ f.

Then, using the idea of weighted sharing, we will prove the following theorem.

Theorem 1.7. Let f be a meromorphic function in the complex plane with finitely many poles, let L be a nonconstant L-function, and let a₁, a₂ ϵ ℂ be two distinct values. Take two positive integers k₁, k₂with k₁k₂ > 1. If E_kj (a_j, f) = E_kj (a_j, L), j = 1, 2, then L ≡ f.

Remark 1.8. Note that an L-function itself can be analytically continued as a meromorphic function in the complex plane. Therefore, an L-function will be taken as a special meromorphic function. We can also see that Theorem 1.4 is included in Theorem 1.7.

In 1976, the following question was mentioned by Gross in [15].

Question 1.9 (see [15]). Must two nonconstant entire functions f₁and f₂be identically equal if f₁and f₂share a finite set S?

Recently, Yuan, Li and Yi [16] considered this question leading to the theorem below.

Theorem 1.10 (see [16]). Let S = {ω₁, ω₂,⋯, ω_l}, where ω₁, ω₂,⋯, ω_l are all distinct roots of the algebraic equation ωⁿ + aω^m + b = 0. Here l is a positive integer satisfying 1 ≤ l ≤ n, n and m are relatively prime positive integers with n ≥ 5 and n > m, and a, b, c are nonzero finite constants, where c ≠ ω_j for 1 ≤ j ≤ l. Let f be a nonconstant meromorphic function such that f has finitely many poles in ℂ, and let L be a nonconstant L-function. If f and L share S CM and c IM, then f ≡ L.

Concerning shared set, we prove the following theorem.

Theorem 1.11. Let f be an entire function with lim_R(s)→+∞f (s) = k (k ≠ ∞) and let R(a) = 0 be a algebraic equation with n ≥ 2 distinct roots, and R(k), R(b), R(1) ≠ 0. Suppose that f (s₀) = L(s₀) = b for some s₀ ϵ ℂ. If f and a nonconstant L-function L share S CM, where S = {a : R(a) = 0}, then R(L) ≡ R(f).

Furthermore, we obtain a result which is similar to Theorem 1.10 by different means.

Theorem 1.12. Let f be an entire function with lim_R(s)→+∞f (s) = k (k ≠ ∞). Let S = {ω₁, ω₂,⋯, ω_i} ⊂ ℂ\{1, k, b}, where ω₁, ω₂,⋯, ω_i are all distinct roots of the algebraic equation ωⁿ^+m + αωⁿ + β = 0, 1 ≤ i ≤ n+m, n, m are two positive integers with n > m +2, α, β are finite nonzero constants. If f and a nonconstant L-function L share S CM and f (s₀) = L(s₀) = b for some s₀ ϵ ℂ, then f ≡ tL, where t is a constant such that t^d = 1, d = GCD(n, m).

2 Some lemmas

In this section, we present some important lemmas which will be needed in the sequel. Firstly, let f be a meromorphic function in C. The order ρ(f) is defined as follows:

ρf=lim supr→∞log⁡Tr,flog⁡r.

Lemma 2.1 (see [4], Lemma 1.22). Let f be a nonconstant meromorphic function and let k ≥ 1 be an integer. Then mr,f(k)f=S(r,f)Further if ρ(f) < +∞, then

mr,f(k)f=O(log⁡r).

Lemma 2.2 (see [4], Corollary of Theorem 1.5). Let f be a nonconstant meromorphic function. Then f is a rational function if and only iflim infr→∞T(r,f)log⁡r<∞.

Lemma 2.3 (see [4], Theorem 1.19). Let T₁(r) and T₂(r) be two nonnegative, nondecreasing real functions defined in r > r₀ > 0. If T₁(r) = O (T₂(r)) (r → ∞, r ∉ E), where E is a set with finite linear measure, then

lim supr→∞log+⁡T1(r)log⁡r≤lim supr→∞log+⁡T2(r)log⁡r

and

lim infr→∞log+⁡T1(r)log⁡r≤lim infr→∞log+⁡T2(r)log⁡r,

which imply that the order and the lower order of T₁(r) are not greater than the order and the lower order of T₂(r) respectively.

Lemma 2.4 (see [4], Theorem 1.14). Let f and g be two nonconstant meromorphic functions. If the order of f and g is ρ (f) and ρ (g) respectively, then

ρf⋅g≤maxρf,ρg,

ρf+g≤maxρf,ρg.

Lemma 2.5 (see [17], Lemma 2.7). Let R(ω) = ωⁿ + aω^m + b, where n, m are positive integers satisfying n > m, a, b are finite nonzero complex numbers. Then the algebraic equation R(ω) = 0 has at least n −1 distinct roots.

Lemma 2.6 (see [18], Lemma 8). Let s > 0 and t be relatively prime integers, and let c be a finite complex number such that c^s = 1. Then there exists one and only one common zero of ω^s − 1 and ω^t − c.

3 Proofs of the theorems

3.1 Proof of Theorem 1.7

First of all, we denote by d the degree of L. Then d=2∑j=1kλj>0, where k and λ_j are respectively the positive integer and the positive real number in the functional equation of the axiom (iii) of the definition of L-functions. According to a result due to Steuding [1], p.150, we have

T(r,L)=dπrlog⁡r+O(r).(2)

Therefore (L) = λ and S(r, L) = O(log r).

Noting that f has finitely many poles and L at most has one pole at s = 1 in the complex plane, it follows that

N(r,f)=O(log⁡r),N(r,L)=O(log⁡r).(3)

Because f and L share a₁, a₂ weighted k₁, k₂ respectively, by (3), from the first and second fundamental theorems we have

T(r,f)≤N¯r,1f−a1+N¯r,1f−a2+N¯r,f+S(r,f)=N¯r,1L−a1+N¯r,1L−a2+O(log⁡r)+S(r,f)≤Tr,1L−a1+Tr,1L−a2+O(log⁡r)+S(r,f)=2Tr,L+O(log⁡r)+S(r,f).(4)

Then from (4) and Lemma 2.3 we obtain

ρ(f)≤ρ(L).(5)

Similarly,

ρ(L)≤ρ(f).(6)

Combining (6) yields

ρ(f)=ρ(L).(7)

Thus

S(r,f)=O(log⁡r).(8)

We introduce two auxiliary functions below.

F1=L′L−a1−f′f−a1,(9)

F2=L′L−a2−f′f−a2.(10)

Next, we assume that F₁ ≠ 0 and F₂ ≠ 0. By (8) and Lemma 2.1 we get

m(r,F1)=O(log⁡r).(11)

By the assumption L and f share (a₁, k₁), (a₂, k₂), from (3), (11) we have

k2N¯(k2+1r,1L−a2≤Nr,1F1≤T(r,F1)+O(1)≤N(r,F1)+m(r,F1)+O(1)≤N¯(k1+1r,1L−a1+N¯(r,L)+N¯(r,f)+O(log⁡r)≤N¯(k1+1r,1L−a1+O(log⁡r).(12)

Similarly, from (10) and (11) we have

k1N¯(k1+1r,1L−a1≤Nr,1F2≤T(r,F2)+O(1)≤N(r,F2)+m(r,F2)+O(1)≤N¯(k2+1r,1L−a2+N¯(r,L)+N¯(r,f)+O(log⁡r)≤N¯(k2+1r,1L−a2+O(log⁡r).(13)

Combining (12) with (13) yields

N¯(k1+1r,1L−a1≤1k1N¯(k2+1r,1L−a2+O(log⁡r)≤1k1k2N¯(k1+1r,1L−a1+O(log⁡r).(14)

Since k₁k₂ > 1, from (14) we obtain

N¯(k1+1r,1L−a1=O(log⁡r).(15)

Substituting (15) into (12) implies

N¯(k2+1r,1L−a2=O(log⁡r).(16)

Set

G=L−a1f−a1.

Noting L and f share (a₁, k₁), (a₂, k₂), combining (15) with (16) yields

N¯(k1+1r,1L−a1=N¯(k1+1r,1f−a1=O(log⁡r),

N¯(k2+1r,1L−a2=N¯(k2+1r,1f−a2=O(log⁡r).

Clearly,

N¯(r,G)≤N(r,L)+N¯(k1+1r,1f−a1=O(log⁡r),(17)

N¯r,1G≤N(r,f)+N¯(k1+1r,1L−a1=O(log⁡r).(18)

Set

G1=Q(L−a1)f−a1,(19)

where Q is a rational function satisfying that G₁ is a zero-free entire function. From (17) and (10), it is easy to see that such a Q does exist. By Lemma 2.2 and Lemma 2.4 we get

ρ(G1)≤max{ρ(Q),ρ(L),ρ(f)}=1.

By the Hadamard factorization theorem [19], p.384, we know

G1=Q(L−a1)f−a1=eφ,(20)

where φ is a polynomial of degree at most deg(’) ≤ 1. We may write φ = a₀s + b₀ for some complex numbers a₀, b₀. In view of (20) and Hayman [3], p.7, we have

T(r,G1)=T(r,ea0s+b0)=O(r).(21)

By (19), the assumption that L and f share a₂, we get that every a₂-point of L has to be 1-point of G1Q−1Now (20), (21) and the first fundamental theorem yield

N¯r,1L−a2≤Nr,1G1Q−1≤Tr,1G1Q−1=Tr,G1Q−1+O(1)≤T(r,G1)+T(r,Q)+O(1)=O(r).(22)

Similarly, set

G2=L−a2f−a2.

We also get

N¯r,1L−a1=O(r).(23)

By (22), (23) and the second fundamental theorem it follows that

T(r,L)≤N¯r,1L−a1+N¯r,1L−a2+N¯(r,L)+O(log⁡r)=O(r).(24)

This contradicts (2). Thus, F₁ ≡ 0 or F₂ ≡ 0. By integration, we have from (9) that

L−a1≡A(f−a1),

where A(≠ 0) is a constant. This implies that L and f share a₁ CM. Hence by Theorem 1.6 we deduce Theorem 1.7 holds. If F₂ ≡ 0, using the same manner, we also have the conclusion.

This completes the proof of Theorem 1.7.

3.2 Proof of Theorem 1.11

First we consider the following function

G=QR(L)R(f),(25)

where

Q(s)=A(s−1)nm(26)

is a rational function satisfying that G has no zeros and no poles in ℂ; A is a nonzero finite value; m is the nonnegative integer in the axiom (ii) of the definition of L-functions.

We claim that such a Q does exist. By the condition that f and L share S CM, set

F=R(L)R(f).(27)

We can see that there can be only a pole of f or L such that F = 0 or F = ∞. Since f has no pole and L has only one possible pole at s = 1, it follows that F has no zero and only one possible pole at s = 1. Hence such a Q does exist.

Next, assume that a₁, a₂,∞, a_n are all distinct roots of R(a). Using the first fundamental theorem we get

Tr,L−ai=T(r,L)+O(1),i=1,2,⋯,n.

Noting n ≥ 2, by the second fundamental theorem we have

(n−1)T(r,f)≤∑i=1nN¯r,1f−ai+N¯(r,f)+S(r,f)=∑i=1nN¯r,1L−ai+N¯(r,f)+S(r,f)≤∑i=1nTr,1L−ai+S(r,f)=nTr,L+S(r,f),(28)

which gives

T(r,f)≤nn−1Tr,L+S(r,f).

This together with Lemma 2.3 yields

ρ(f)≤ρ(L).(29)

Similarly,

ρ(L)≤ρ(f).(30)

By (29), (30) and (2) we obtain

ρ(f)=ρ(L)=1.(31)

Also, from the first fundamental theorem we get

ρ1f−ai=ρ(f)=1,

and then by Lemma 2.2 and Lemma 2.4 we deduce

ρ(G)≤max{ρ(Q),ρ(L),ρ(f)}=1.

From the Hadamard factorization theorem [19], p.384 we see

G=eh(s),(32)

where h(s) is a polynomial of degree deg(h(s)) ≤ 1. One can write

ℜh(σ+it)=α(t)σ+β(t),(33)

a polynomial in σ with α(t), β(t) being polynomials in t. Now the claim is α(t) ≡ 0. From (25), (27) and (32) we get

F=R(L)R(f)=eh(s)Q−1.(34)

Since lim_σ_→+∞L(s) = 1, lim_σ_→+∞f (s) = k(k ≠ ∞), R(k) ≠ 0 and R(1) ≠ 0, it follows that

limσ→+∞R(L)R(f)=C,(35)

where ≥ ≠ 0 is a finite value. If α(t) ≠ 0, we obtain α(t₀) ≠ 0 for some value t₀. If α(t₀) > 0, from (34) we know that

R(L)R(f)=Q−1eℜh(σ+it).(36)

Thus from (26), (35) and (36) we can deduce that, |C| = ∞ when σ → +∞ with t = t₀, which is a contradiction. Similarly, if α(t₀) < 0, we have that, |C| = 0 when σ → +∞ with t = t₀, which is also a contradiction. Therefore α(t) ≡ 0. Now by (33) and (36) we get

R(L)R(f)=Q−1eβ(t).(37)

Combining (37) yields

limσ→+∞|Q|=eβ(t)|C|(38)

for a fixed t. Considering that the limit of |Q| as σ → +∞ is a nonzero finite constant for some value t and n ≥ 2, in view of (26) we see that m = 0, and then Q(s) ≡ A. From (38) we have e^β^(t) = |A||C|. Thus it follows by (37) that

R(L)R(f)=|C|.(39)

Since c ≠ 0 is a finite complex number, from (35) we know that

R(L)R(f)≡C.(40)

From the assumption in the theorem we have f (s₀) = L(s₀) = b for some s₀ ϵ ℂ. It now follows from (40) that ≥ = 1. Thus

R(L)R(f)≡1.(41)

That is R(L) ≡ R(f).

This completes the proof of Theorem 1.11.

3.3 Proof of Theorem 1.12

First, we have that the algebraic equation ωⁿ^+m + αωⁿ + β = 0 has at least n +m −1 > 3m+1 ≥ 4 distinct roots in view of Lemma 2.5. By Theorem 1.11, we get

Ln+m+αLn≡fn+m+αfn.(42)

Set H=fLThen by (42) we deduce

1αLm=Hn−1Hn+m−1.(43)

We discuss two cases:

Case 1. H is a constant. If H ⁿ^+m ≠ 1, by (43), we get that L is a constant, which contradicts the assumption that L is a nonconstant L-function. Therefore, H ⁿ^+m = 1, and so it follows by (43) that H ^m = H ⁿ = 1, that is fⁿ = Lⁿ and f^m = L^m. We get f^d = L^d.

Case 2. H is a nonconstant meromorphic function. Note that L has at most one pole. Now we discuss the following two subcases again.

Subcase 2.1. L has no poles. Then, from (43) we get that every 1-point of H ⁿ^+m has to be 1-point of Hⁿ. Since H ⁿ^+m = H ⁿH^m, we have any 1-point of H ⁿ^+m to be a 1-point of H^m. Because n > m + 2, it follows that H is a constant, contradicting the assumption.

Subcase 2.2. L has one and only one pole. Then by (43) we know every zero of H ⁿ^+m − 1 has to be zero of Hⁿ − 1 with one exception. Put

Hn−1=(H−1)(H−ζ1)⋯(H−ζn−1),

Hn+m−1=(H−1)(H−τ1)⋯(H−τn+m−1),

where ζ₁, ζ₂,⋯, ζ_n₋₁ are n − 1 distinct finite complex numbers satisfying ζin=1,ζi≠1,1≤i≤n−1;τ1,τ2,⋯,τn+m−1 are n + m − 1 distinct finite complex numbers satisfying τjn+m=1,τj≠1,1≤j≤n+m−1

Let m = 1. By Lemma 2.6 we see H ⁿ − 1 and H ⁿ⁺₁ − 1 have only one common zero, so H cannot be equal to any n +m −2 values of {τ₁, τ₂,⋯, τ_n_+m−1}. From n > m +2 it follows that H is a constant, contradicting the assumption.

Let m ≥ 2. If any 1-point of H ⁿ is a 1-point of Hⁿ^+m, then any 1-point of Hⁿ is a 1-point of H^m. Note that n > m + 2. This contradicts the assumption that H is nonconstant. If there is at least one ζ_i ≠ τ_j, 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n + m − 1, then H cannot be equal to any m + 1 values of {τ₁, τ₂,⋯, τ_n_+m−1}. From m ≥ 2, we know H is a constant, contradicting the assumption.

This completes the proof of Theorem 1.12.

Acknowledgement

The authors would like to thank the referees for their thorough comments and helpful suggestions.

Project supported by the National Natural Science Foundation of China (Grant No. 11301076), the Natural Science Foundation of Fujian Province, China (Grant No. 2018J01658) and Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (Grant No. SX201801).

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Received: 2018-07-06

Accepted: 2018-10-01

Published Online: 2018-11-10

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https://doi.org/10.1515/math-2018-0107

Keywords for this article

Meromorphic function; L-function; Selberg class; Value distribution

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