scholarly journals Two Linnik-type problems for automorphic L-functions

2011 ◽  
Vol 151 (2) ◽  
pp. 219-227 ◽  
Author(s):  
JIANYA LIU ◽  
YAN QU ◽  
JIE WU
Keyword(s):  
Dirichlet Series ◽  
Half Plane ◽  
Implied Constant ◽  
Cuspidal Representation ◽  
Sign Change ◽  
Series Expression

AbstractLet m ≥ 2 be an integer, and π an irreducible unitary cuspidal representation for GLm(), whose attached automorphic L-function is denoted by L(s, π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s, π) in the half-plane ℜs > 1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then the first sign change in the sequence {λπ(n)}n=1∞ occurs at some n ≪ Qπ1 + ϵ, where Qπ is the conductor of π, and the implied constant depends only on m and ϵ. This improves the previous bound with the above exponent 1 + ϵ replaced by m/2 + ϵ. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −(L′/L)(s, π) in the half-plane ℜs > 1.

Fourth power moment of coefficients of automorphic L-functions for GL(m)

2017 ◽  
Vol 29 (5) ◽  
pp. 1199-1212
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Lower Bound ◽  
Dirichlet Series ◽  
Half Plane ◽  
Upper Bound ◽  
Automorphic Representation ◽  
Fourth Power ◽  
Power Moment ◽  
Cuspidal Automorphic Representation ◽  
Series Expression

AbstractLet π be a unitary cuspidal automorphic representation for {\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})}, and let {L(s,\pi)} be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane {\Re s>1}, i.e.L(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}.In this paper we are interested in the upper bound of the fourth power moment of {\lambda_{\pi}(n)}, i.e. {\sum_{n\leq x}\lambda_{\pi}(n)^{4}}. If {m=2}, we are able to consider the sixteenth power moment of {\lambda_{\pi}(n)}. As an application, we consider the lower bound of {\sum_{n\leq x}\lvert\lambda_{\pi}(n)\rvert}, which improves previous results.

Cancellation in algebraic twisted sums on GL_ m

2021 ◽  
Vol 33 (4) ◽  
pp. 1061-1082
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Dirichlet Series ◽  
Central Character ◽  
Multiplicative Functions ◽  
Cuspidal Representation ◽  
Series Coefficient

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

scholarly journals Growth estimates for a Dirichlet series and its derivative

2020 ◽  
Vol 53 (1) ◽  
pp. 3-12
Author(s):  
S.I. Fedynyak ◽  
P.V. Filevych
Keyword(s):  
Continuous Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Conjugate Function ◽  
Nonnegative Sequence

Let $A\in(-\infty,+\infty]$, $\Phi$ be a continuous function on $[a,A)$ such that for every $x\in\mathbb{R}$ we have$x\sigma-\Phi(\sigma)\to-\infty$ as $\sigma\uparrow A$, $\widetilde{\Phi}(x)=\max\{x\sigma -\Phi(\sigma)\colon \sigma\in [a,A)\}$ be the Young-conjugate function of $\Phi$, $\overline{\Phi}(x)=\widetilde{\Phi}(x)/x$ for all sufficiently large $x$, $(\lambda_n)$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum a_ne^{s\lambda_n}$ be a Dirichlet series absolutely convergent in the half-plane $\operatorname{Re}s<A$, $M(\sigma,F)=\sup\{|F(s)|\colon \operatorname{Re}s=\sigma\}$ and $G(\sigma,F)=\sum |a_n|e^{\sigma\lambda_n}$ for each $\sigma<A$. It is proved that if $\ln G(\sigma,F)\le(1+o(1))\Phi(\sigma)$, $\sigma\uparrow A$, then the inequality$$\varlimsup_{\sigma\uparrow A}\frac{M(\sigma,F')}{M(\sigma,F)\overline{\Phi}\,^{-1}(\sigma)}\le1$$holds, and this inequality is sharp. % Abstract (in English)

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

scholarly journals The multiple Dirichlet product and the multiple Dirichlet series

2017 ◽  
Vol 13 (08) ◽  
pp. 2181-2193 ◽  
Author(s):  
Tomokazu Onozuka
Keyword(s):  
Dirichlet Series ◽  
Free Region ◽  
Series Expression ◽  
Multiple Dirichlet Series ◽  
Dirichlet Product

First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.

scholarly journals Generalized types of the growth of Dirichlet series

2015 ◽  
Vol 7 (2) ◽  
pp. 172-187 ◽  
Author(s):  
T.Ya. Hlova ◽  
P.V. Filevych
Keyword(s):  
Dirichlet Series ◽  
Half Plane ◽  
Maximum Modulus ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximal Term ◽  
Necessary And Sufficient ◽  
Nonnegative Sequence

Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Euler Products ◽  
Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

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