Sums of Fourier coefficients of a Maass form for SL3(ℤ) twisted by exponential functions

2014 ◽  
Vol 26 (1) ◽  
Author(s):  
Xiumin Ren ◽  
Yangbo Ye
Keyword(s):  
Fourier Coefficients ◽  
Exponential Functions ◽  
Maass Form

On exponential sums for coefficients of general L-functions

2021 ◽  
pp. 1-22
Author(s):  
Fei Hou
Keyword(s):  
Functional Equation ◽  
General Formula ◽  
Fourier Coefficients ◽  
Exponential Sums ◽  
Exponential Functions ◽  
Maass Forms ◽  
Maass Form ◽  
Rapid Decay

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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 $\unicode[STIX]{x1D70C}(n)d(n-a)$.

A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

2019 ◽  
Vol 101 (3) ◽  
pp. 401-414
Author(s):  
HENGCAI TANG
Keyword(s):  
Trace Formula ◽  
Fourier Coefficients ◽  
Summation Formula ◽  
Divisor Function ◽  
Maass Forms ◽  
Maass Form ◽  
Kuznetsov Trace Formula ◽  
Image Position ◽  
Voronoi Summation ◽  
Shifted Convolution Sum

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)

2016 ◽  
Vol 161 (2) ◽  
pp. 339-356 ◽  
Author(s):  
YUJIAO JIANG ◽  
GUANGSHI LÜ ◽  
XIAOFEI YAN
Keyword(s):  
Fourier Coefficients ◽  
Mean Value ◽  
Implied Constant ◽  
Symmetric Power ◽  
Mean Value Theorem ◽  
Maass Forms ◽  
Maass Form ◽  
Formula Group ◽  
Image Position

AbstractLet F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF(n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish $$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$ where Q = xϑ−ϵ with some ϑ > 0, the implied constant depends on F, A, ϵ.

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