• Corpus ID: 119653545

# The twisted second moment of modular half integral weight $L$--functions.

@article{Dunn2019TheTS,
title={The twisted second moment of modular half integral weight \$L\$--functions.},
author={Alexander Dunn and Alexandru Zaharescu},
journal={arXiv: Number Theory},
year={2019}
}
• Published 8 March 2019
• Mathematics
• arXiv: Number Theory
Given a half-integral weight holomorphic Kohnen newform $f$ on $\Gamma_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}} \big | L(1/2,f,\chi) \big |^2. \end{equation*} Our result is unconditional, it does not rely on the Ramanujan-Petersson conjecture for the form $f$. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution…
On the distribution of modular square roots of primes
• Mathematics
• 2020
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be
The distribution of spacings between the fractional parts of $\boldsymbol{n^d\alpha}$
• Mathematics
• 2020
We study the distribution of spacings between the fractional parts of $n^d\alpha$. For $\alpha$ of high enough Diophantine type we prove a necessary and sufficient condition for $n^d\alpha\mod 1, Bilinear Forms With Modular Square Roots and Twisted Second Moments of Half Integral Weight Dirichlet Series • Mathematics International Mathematics Research Notices • 2021 We establish new results on equations and bilinear forms with modular square roots. The main motivation and application of these results is our new bound on the fourth moment of the error term in Bounds on bilinear forms with Kloosterman sums • Mathematics • 2022 . We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by ´E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, ´E. Fouvry, E. Kowalski, The Distribution of Spacings Between the Fractional Parts of ndα • Mathematics • 2021 We study the distribution of spacings between the fractional parts of$n^d\alpha $. For$\alpha $of high enough Diophantine type we prove a necessary and sufficient condition for$n^d\alpha \mod
Energy bounds for modular roots and their applications
• Mathematics
• 2021
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory

## References

SHOWING 1-10 OF 57 REFERENCES
The Second Moment of Twisted Modular L-Functions
• Mathematics
• 2014
AbstractWe prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment $$\underset{\chi \bmod {q}}{\left. \sum \right.^{\ast}} L(1/2, f_1\otimes \chi) Modular Forms of Half Integral Weight The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty
Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms
• Mathematics
• 2012
Introduction Preliminaries Bi-$K_\infty$-invariant functions on $\operatorname{GL}_2(\mathbf{R})$ Maass cusp forms Eisenstein series The kernel of $R(f)$ A Fourier trace formula for
The second moment theory of families of L-functions
• Mathematics
• 2018
For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the
Second Moments and simultaneous non-vanishing of GL(2) automorphic L-series
• Mathematics
• 2013
We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q,
The second moment of Dirichlet twists of Hecke L-functions
• Mathematics
• 2008
Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average
The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
• Mathematics
• 2000
We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying
A Burgess-like subconvex bound for twisted L-functions
• Mathematics
• 2007
Abstract Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = ½. It is proved that ,
On the Fourth Power Moment of the Riemann Zeta-Function
• Mathematics
• 1995
Abstract The function E 2 ( R ) is used to denote the error term in the asymptotic formula for the fourth power moment of the Riemann zeta-function on the half-line. In this paper we prove several
Quantum unique ergodicity for half-integral weight automorphic forms
• Mathematics
Duke Mathematical Journal
• 2020
We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both