scholarly journals Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2036
Author(s):  
Rui Zhang ◽  
Xue Han ◽  
Deyu Zhang
Keyword(s):  
Error Term ◽  
Asymptotic Formulas ◽  
Short Intervals ◽  
Mean Error ◽  
Riesz Mean ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Power Moments

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2,Z) and let L(s,sym2f)=∑n=1∞cnn−s,ℜs>1 denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form Dρ(x;sym2f)=L(0,sym2f)Γ(ρ+1)xρ+Δρ(x;sym2f) and derive the asymptotic formulas for ∫T−HT+HΔρk(x;sym2f)dx, when k≥3.

scholarly journals On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

2008 ◽  
Vol 51 (1) ◽  
pp. 148-160 ◽  
Author(s):  
Yoshio Tanigawa ◽  
WenGuang Zhai ◽  
DeYu Zhang
Keyword(s):  
Error Term ◽  
Mean Error ◽  
Riesz Mean ◽  
Higher Power ◽  
Power Moments

scholarly journals The third-power moment of the Riesz mean error term of symmetric square $ L $-function

2021 ◽  
Vol 6 (9) ◽  
pp. 9436-9445
Author(s):  
Rui Zhang ◽  
◽  
Xiaofei Yan
Keyword(s):  
Error Term ◽  
Power Moment ◽  
The Third ◽  
Mean Error ◽  
Riesz Mean ◽  
Symmetric Square

EULER–HADAMARD PRODUCTS AND POWER MOMENTS OF SYMMETRIC SQUARE L-FUNCTIONS

2013 ◽  
Vol 09 (03) ◽  
pp. 621-639 ◽  
Author(s):  
GORAN DJANKOVIĆ
Keyword(s):  
Modular Forms ◽  
Matrix Theory ◽  
Asymptotic Formulas ◽  
Hadamard Products ◽  
Central Value ◽  
General Power ◽  
Symmetric Square ◽  
Power Moments ◽  
Holomorphic Modular Forms ◽  
Work Done

In this paper we prove asymptotic formulas for general moments of partial Euler products and the first and the second moments of partial Hadamard products related to central values of the family of L-functions associated to the symmetric square lifts of holomorphic modular forms for SL2(ℤ). Then using a hybrid Euler–Hadamard product formula for the central value, we relate these results with conjectures for general power moments of L-functions in this family and with Random Matrix Theory interpretations. This continues the work done previously by Gonek–Hughes–Keating and Bui–Keating for other families of L-functions.

On the mean square of an arithmetical error term of the Selberg class in short intervals

2016 ◽  
Vol 12 (06) ◽  
pp. 1675-1701
Author(s):  
Xiaodong Cao ◽  
Yoshio Tanigawa ◽  
Wenguang Zhai
Keyword(s):  
Dirichlet Series ◽  
Error Term ◽  
Selberg Class ◽  
Mean Square ◽  
Short Intervals ◽  
Type Formula ◽  
The Mean

Let [Formula: see text] be a Dirichlet series in the Selberg class of degree [Formula: see text] and let [Formula: see text] be the arithmetical error term of [Formula: see text]. We derive two kinds of the mean square estimates of [Formula: see text] in short intervals of Jutila’s type. Our method is based on the truncated Tong-type formula of [Formula: see text]. We also give several applications of these estimates in the arithmetical problems.

Power moments of automorphic L-function attached to Maass forms

2016 ◽  
Vol 12 (02) ◽  
pp. 427-443
Author(s):  
Huafeng Liu ◽  
Shuai Li ◽  
Deyu Zhang
Keyword(s):  
Lower Bounds ◽  
Cusp Form ◽  
Asymptotic Formulas ◽  
Maass Forms ◽  
Maass Cusp Form ◽  
Power Moments

Let [Formula: see text] be a normalized Maass cusp form for [Formula: see text]. For [Formula: see text], we define [Formula: see text] [Formula: see text] as the supremum of all numbers [Formula: see text] such that [Formula: see text] where [Formula: see text] is the automorphic [Formula: see text]-function attached to [Formula: see text]. In this paper, we shall establish the lower bounds of [Formula: see text] for [Formula: see text] and obtain asymptotic formulas for the second, fourth and sixth powers of [Formula: see text].

The Bombieri–Vinogradov Theorem on Higher Rank Groups and its Applications

2019 ◽  
Vol 72 (4) ◽  
pp. 928-966
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Maass Form ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
Higher Rank ◽  
Image Position ◽  
Holomorphic Hecke Eigenforms

AbstractWe study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$ where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$ where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

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