Related papers: On a rigidity property for quadratic Gauss sums

Let $\rho$ f,$\lambda$ be the residual Galois representation attached to a newform f and a prime ideal $\lambda$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the…

Let $F$ be a number field, let $N\geq 3$ be an integer, and let $k$ be a finite field of characteristic $\ell$. We show that if $\rb:G_F\longrightarrow GL_N(k)$ is a continuous representation with image of $\rb$ containing $SL_N(k)$ then,…

Let $\chi$ be a real non-principal character modulo a prime $q$ and $L(s,\chi)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,\chi )}{L(s,\chi)}\leq c \log q,$$ where $c$ can be…

Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially big integer set $S$ and, in particular, size of any arithmetic progression in $S$ having the beginning at zero, is at…

We investigate fractional sums of arithmetic functions over products of two or three integers, with emphasis on fixed greatest common divisors and multiplicative weights. Let $f$ be an arithmetic function satisfying $f(n) \ll n^\alpha$ for…

We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known…

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove…

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet…

Let $\mathbf{P} \subset [H_0,H]$ be a set of primes, where $\log H_0 \geq (\log H)^{2/3 + \epsilon}$. Let $\mathscr{L} = \sum_{p \in \mathbf{P}} 1/p$. Let $N$ be such that $\log H \leq (\log N)^{1/2-\epsilon}$. We show there exists a subset…

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both…

Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all…

A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq…

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the…

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an…

It is proved that \[ \sum_{\chi \bmod q}N(\sigma , T, \chi) \lesssim_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma, T, \chi)$ denote the number of zeros $\rho = \beta + it$ of $L(s, \chi)$ in the rectangle $\sigma \leq \beta…

Let $F$ be a locally compact non-Archimedean field, and $\bf G$ a connected quasi-split reductive group over $F$. We are interested in complex irreducible smooth generic representations $\pi$ of ${\bf G}(F)$. When $F$ has positive…

Let $p$ be an odd prime and let $f(x)=\sum_{i=1}^ka_ix^{p^{\alpha_i}+1}\in\Bbb F_{p^n}[x]$, where $0\le \alpha_1<...<\alpha_k$. We consider the exponential sum $S(f,n)=\sum_{x\in\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\pi…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\…

Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where…