## 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.

Show Summary Details# Uniqueness theorems for L-functions in the extended Selberg class

#### Open Access

## Abstract

## 1 Introduction

## 2 Some lemmas

## 3 Proofs of the theorems

## 3.1 Proof of Theorem 1.7

## 3.2 Proof of Theorem 1.11

## 3.3 Proof of Theorem 1.12

## Acknowledgement

## References

## About the article

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Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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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

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

$$L(s)=\sum _{n=1}^{\mathrm{\infty}}\frac{a(n)}{{n}^{s}}$$(1)of a complex variable *s* = *σ* + *it* satisfying the following axioms [1]:

- (i)
Ramanujan hypothesis:

*a*(*n*) ≪*n*for every^{ϵ}*ϵ*> 0. - (ii)
Analytic continuation: There exists a nonnegative integer

*m*such that (*s*− 1)(^{m}L*s*) is an entire function of finite order. - (iii)
Functional equation:

*L*satisfies a functional equation of type

where

$${\mathrm{\Lambda}}_{L}(s)=L(s){Q}^{s}\prod _{j=1}^{K}\mathrm{\Gamma}({\lambda}_{j}s+{\nu}_{j})$$

with positive real numbers *Q*, *λ _{j}*, and complex numbers

- (iv)
Euler product: $\mathrm{log}L(s)=\sum _{n=1}^{\mathrm{\infty}}\frac{b(n)}{{n}^{s}}$where

*b*(*n*) = 0 unless*n*is a positive power of a prime and*b*(*n*) ≪*n*for some $\theta <\frac{1}{2}$^{Ø}

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, 41].

**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 that* ${L}_{1}=1+\frac{2}{{4}^{s}}and{L}_{2}=1+\frac{3}{{9}^{s}}$ *which 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 L*_{1} *and L*_{2} *be two L-functions satisfying the same functional equation with a*(1) = 1 *and let a*_{1}, *a*_{2} ϵ ℂ *be two distinct values. If* ${L}_{1}^{-1}({a}_{j})={L}_{2}^{-1}({a}_{j}),j=1,2$*then L*_{1} ≡ *L*_{2}.

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}*(

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 L*_{1} *and L*_{2} *be two L-functions, and let a*_{1}, *a*_{2} ϵ ℂ *be two distinct values. Take two positive integers k*_{1}*, k*_{2} *with k*_{1}*k*_{2} > 1*. If E _{kj}* (

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 a*_{1}, *a*_{2} ϵ ℂ *be two distinct values. Take two positive integers k*_{1}*, k*_{2} *with k*_{1}*k*_{2} > 1*. If E _{kj}* (

**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 f*_{1} *and f*_{2} *be identically equal if f*_{1} *and f*_{2} *share 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* = {*ω*_{1}, *ω*_{2},⋯, *ω _{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* (*s*_{0}) = *L*(*s*_{0}) = *b for some s*_{0} 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* = {*ω*_{1}, *ω*_{2},⋯, *ω _{i}*} ⊂ ℂ\{1,

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:

**Lemma 2.1** (see [4], Lemma 1.22). *Let f be a nonconstant meromorphic function and let k* ≥ 1 *be an integer. Then m* $\left(r,\frac{{f}^{(k)}}{f}\right)=S(r,f)$*Further if ρ*(*f*) < +∞*, then*

**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 if* $\underset{r\to \mathrm{\infty}}{lim\u2006inf}\frac{T(r,f)}{\mathrm{log}r}<\mathrm{\infty}.$

**Lemma 2.3** (see [4], Theorem 1.19). *Let T*_{1}(*r*) *and T*_{2}(*r*) *be two nonnegative, nondecreasing real functions defined in r* > *r*_{0} > 0*. If T*_{1}(*r*) = *O* (*T*_{2}(*r*)) (*r* → ∞, *r* ∉ *E*)*, where E is a set with finite linear measure, then*

*and*

*which imply that the order and the lower order of T*_{1}(*r*) *are not greater than the order and the lower order of T*_{2}(*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*

**Lemma 2.5** (see [17], Lemma 2.7). *Let R*(*ω*) = *ω ^{n}* +

**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

First of all, we denote by *d* the degree of *L*. Then $d=2\sum _{j=1}^{k}{\lambda}_{j}>0$, 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

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

Because *f* and *L* share *a*_{1}, *a*_{2} weighted *k*_{1}, *k*_{2} respectively, by (3), from the first and second fundamental theorems we have

Then from (4) and Lemma 2.3 we obtain

$$\rho (f)\le \rho (L).$$(5)Similarly,

$$\rho (L)\le \rho (f).$$(6)Combining (6) yields

$$\rho (f)=\rho (L).$$(7)Thus

$$S(r,f)=O(\mathrm{log}r).$$(8)We introduce two auxiliary functions below.

$${F}_{1}=\frac{{L}^{\prime}}{L-{a}_{1}}-\frac{{f}^{\prime}}{f-{a}_{1}},$$(9)$${F}_{2}=\frac{{L}^{\prime}}{L-{a}_{2}}-\frac{{f}^{\prime}}{f-{a}_{2}}.$$(10)Next, we assume that *F*_{1} ≠ 0 and *F* _{2} ≠ 0. By (8) and Lemma 2.1 we get

By the assumption *L* and *f* share (*a*_{1}, *k*_{1}), (*a*_{2}, *k*_{2}), from (3), (11) we have

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

$$\begin{array}{rl}{k}_{1}{\overline{N}}_{({k}_{1}+1}\left(r,\frac{1}{L-{a}_{1}}\right)& \le N\left(r,\frac{1}{{F}_{2}}\right)\le T(r,{F}_{2})+O(1)\le N(r,{F}_{2})+m(r,{F}_{2})+O(1)\\ & \le {\overline{N}}_{({k}_{2}+1}\left(r,\frac{1}{L-{a}_{2}}\right)+\overline{N}(r,L)+\overline{N}(r,f)+O(\mathrm{log}r)\\ & \le {\overline{N}}_{({k}_{2}+1}\left(r,\frac{1}{L-{a}_{2}}\right)+O(\mathrm{log}r).\end{array}$$(13)Combining (12) with (13) yields

$$\begin{array}{rl}{\overline{N}}_{({k}_{1}+1}\left(r,\frac{1}{L-{a}_{1}}\right)& \le \frac{1}{{k}_{1}}{\overline{N}}_{({k}_{2}+1}\left(r,\frac{1}{L-{a}_{2}}\right)+O(\mathrm{log}r)\\ & \le \frac{1}{{k}_{1}{k}_{2}}{\overline{N}}_{({k}_{1}+1}\left(r,\frac{1}{L-{a}_{1}}\right)+O(\mathrm{log}r).\end{array}$$(14)Since *k*_{1}*k*_{2} > 1, from (14) we obtain

Substituting (15) into (12) implies

$${\overline{N}}_{({k}_{2}+1}\left(r,\frac{1}{L-{a}_{2}}\right)=O(\mathrm{log}r).$$(16)Set

$$G=\frac{L-{a}_{1}}{f-{a}_{1}}.$$Noting *L* and *f* share (*a*_{1}, *k*_{1}), (*a*_{2}, *k*_{2}), combining (15) with (16) yields

Clearly,

$$\overline{N}(r,G)\le N(r,L)+{\overline{N}}_{({k}_{1}+1}\left(r,\frac{1}{f-{a}_{1}}\right)=O(\mathrm{log}r),$$(17)$$\overline{N}\left(r,\frac{1}{G}\right)\le N(r,f)+{\overline{N}}_{({k}_{1}+1}\left(r,\frac{1}{L-{a}_{1}}\right)=O(\mathrm{log}r).$$(18)Set

$${G}_{1}=\frac{Q(L-{a}_{1})}{f-{a}_{1}},$$(19)where *Q* is a rational function satisfying that *G*_{1} 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

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

$${G}_{1}=\frac{Q(L-{a}_{1})}{f-{a}_{1}}={e}^{\phi},$$(20)where *φ* is a polynomial of degree at most *deg*(*’*) ≤ 1. We may write *φ* = *a*_{0}*s* + *b*_{0} for some complex numbers *a*_{0}, *b*_{0}. In view of (20) and Hayman [3], p.7, we have

By (19), the assumption that *L* and *f* share *a*_{2}, we get that every *a*_{2}-point of *L* has to be 1-point of $\frac{{G}_{1}}{Q}-1$Now (20), (21) and the first fundamental theorem yield

Similarly, set

$${G}_{2}=\frac{L-{a}_{2}}{f-{a}_{2}}.$$We also get

$$\overline{N}\left(r,\frac{1}{L-{a}_{1}}\right)=O(r).$$(23)By (22), (23) and the second fundamental theorem it follows that

$$T(r,L)\le \overline{N}\left(r,\frac{1}{L-{a}_{1}}\right)+\overline{N}\left(r,\frac{1}{L-{a}_{2}}\right)+\overline{N}(r,L)+O(\mathrm{log}r)=O(r).$$(24)This contradicts (2). Thus, *F*_{1} ≡ 0 or *F* _{2} ≡ 0. By integration, we have from (9) that

where *A*(≠ 0) is a constant. This implies that *L* and *f* share *a*_{1} CM. Hence by Theorem 1.6 we deduce Theorem 1.7 holds. If *F* _{2} ≡ 0, using the same manner, we also have the conclusion.

This completes the proof of Theorem 1.7.

First we consider the following function

$$G=\frac{QR(L)}{R(f)},$$(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

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 *a*_{1}, *a*_{2},∞, *a _{n}* are all distinct roots of

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

which gives

$$T(r,f)\le \frac{n}{n-1}T\left(r,L\right)+S(r,f).$$This together with Lemma 2.3 yields

$$\rho (f)\le \rho (L).$$(29)Similarly,

$$\rho (L)\le \rho (f).$$(30)By (29), (30) and (2) we obtain

$$\rho (f)=\rho (L)=1.$$(31)Also, from the first fundamental theorem we get

$$\rho \left(\frac{1}{f-{a}_{i}}\right)=\rho (f)=1,$$and then by Lemma 2.2 and Lemma 2.4 we deduce

$$\rho (G)\le max\{\rho (Q),\rho (L),\rho (f)\}=1.$$From the Hadamard factorization theorem [19], p.384 we see

$$G={e}^{h(s)},$$(32)where *h*(*s*) is a polynomial of degree deg(*h*(*s*)) ≤ 1. One can write

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

Since lim_{σ}_{→+∞} *L*(*s*) = 1, lim_{σ}_{→+∞} *f* (*s*) = *k*(*k* ≠ ∞), *R*(*k*) ≠ 0 and *R*(1) ≠ 0, it follows that

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

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

Combining (37) yields

$$\underset{\sigma \to +\mathrm{\infty}}{lim}|Q|=\frac{{e}^{\beta (t)}}{|C|}$$(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}*

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

From the assumption in the theorem we have *f* (*s*_{0}) = *L*(*s*_{0}) = *b* for some *s*_{0} ϵ ℂ. It now follows from (40) that *≥* = 1. Thus

That is *R*(*L*) ≡ *R*(*f*).

This completes the proof of Theorem 1.11.

First, we have that the algebraic equation *ω ^{n}*

Set $H=\frac{f}{L}$Then by (42) we deduce

$$\frac{1}{\alpha}{L}^{m}=\frac{{H}^{n}-1}{{H}^{n+m}-1}.$$(43)We discuss two cases:

Case 1. *H* is a constant. If *H ^{n}*

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}*

Subcase 2.2. *L* has one and only one pole. Then by (43) we know every zero of *H ^{n}*

where *ζ*_{1}, *ζ*_{2},⋯, *ζ _{n}*

Let *m* = 1. By Lemma 2.6 we see *H ^{n}* − 1 and

Let *m* ≥ 2. If any 1-point of *H ^{n}* is a 1-point of

This completes the proof of Theorem 1.12.

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

**Citation Information: **Open Mathematics, Volume 16, Issue 1, Pages 1291–1299, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0107.

© 2018 Hao and Chen, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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