Integral power sums of Fourier coefficients of symmetric square $L$-functions

2019 ◽  
Vol 147 (7) ◽  
pp. 2847-2856
Author(s):  
Xiaoguang He
Keyword(s):  
Fourier Coefficients ◽  
Power Sums ◽  
Integral Power ◽  
Symmetric Square

scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

scholarly journals Additive twists of Fourier coefficients of symmetric-square lifts

2012 ◽  
Vol 132 (7) ◽  
pp. 1626-1640 ◽  
Author(s):  
Xiaoqing Li ◽  
Matthew P. Young
Keyword(s):  
Fourier Coefficients ◽  
Symmetric Square

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Guangshi Lü
Keyword(s):  
Zeta Function ◽  
Mean Values ◽  
Power Sums ◽  
Dedekind Zeta Function ◽  
Integral Power ◽  
Cubic Field ◽  
Cubic Extension

AbstractAfter Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$.

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

scholarly journals Integral power sums of Hecke eigenvalues

2011 ◽  
Vol 150 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Y.-K. Lau ◽  
G.-S. Lü ◽  
J. Wu
Keyword(s):  
Power Sums ◽  
Hecke Eigenvalues ◽  
Integral Power

Image Reconstruction with the Fourier Coefficients for Magnetic Induction Tomography

2020 ◽  
Vol 16 (2) ◽  
pp. 156-163
Author(s):  
Jingwen Wang ◽  
Xu Wang ◽  
Dan Yang ◽  
Kaiyang Wang
Keyword(s):  
Inverse Problem ◽  
Image Reconstruction ◽  
Magnetic Induction ◽  
Fourier Coefficients ◽  
Signal Reconstruction ◽  
Reconstruction Algorithm ◽  
Reconstruction Method ◽  
Measurement Equation ◽  
Image Reconstruction Method ◽  
Magnetic Induction Tomography

Background: Image reconstruction of magnetic induction tomography (MIT) is a typical ill-posed inverse problem, which means that the measurements are always far from enough. Thus, MIT image reconstruction results using conventional algorithms such as linear back projection and Landweber often suffer from limitations such as low resolution and blurred edges. Methods: In this paper, based on the recent finite rate of innovation (FRI) framework, a novel image reconstruction method with MIT system is presented. Results: This is achieved through modeling and sampling the MIT signals in FRI framework, resulting in a few new measurements, namely, fourier coefficients. Because each new measurement contains all the pixel position and conductivity information of the dense phase medium, the illposed inverse problem can be improved, by rebuilding the MIT measurement equation with the measurement voltage and the new measurements. Finally, a sparsity-based signal reconstruction algorithm is presented to reconstruct the original MIT image signal, by solving this new measurement equation. Conclusion: Experiments show that the proposed method has better indicators such as image error and correlation coefficient. Therefore, it is a kind of MIT image reconstruction method with high accuracy.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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