No Arabic abstract
Let $f(n)$ be a multiplicative function with $|f(n)|leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, chi$ be a non-principal Dirichlet character modulo $q$. Let $varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{frac12+varepsilon}ll Nll q^A$. In this paper, we shall prove that $$ sum_{nleq N}f(n)chi(n+a)ll Nfrac{loglog q}{log q} $$ and that $$ sum_{nleq N}f(n)chi(n+a_1)cdotschi(n+a_t)ll Nfrac{loglog q}{log q}, $$ where $tgeq 2, a_1, ldots, a_t$ are distinct integers modulo $q$.
In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincare series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.
In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q rightarrow infty$ using Steinhaus random multiplicative functions, stating properties of this random process. This is motivated by Kowalski and Sawins work on Kloosterman paths.
Shifted convolution sums play a prominent role in analytic number theory. We investigate pointwise bounds, mean-square bounds, and average bounds for shifted convolution sums for Hecke eigenforms.
We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL(2) holomorphic cusp forms, GL(2) Maass forms, and GL(3) Maass forms. Building on previous works by the authors, we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL(2) holomorphic cusp forms and Maass forms.
We produce nontrivial asymptotic estimates for shifted sums of the form $sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions $n-h, n, n+h$ such that $a(n-h)a(n)a(n+h) eq 0$.