Related papers: On Kloosterman sums with multiplicative coefficien…

We construct sequences $\{a_n\}_{n\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$ with small values of signed harmonic sums \[ \sum_{n\in\mathcal{A}\cap[1,N]}\frac{a_n}{n}, \] for any reasonably dense subsets $\mathcal{A}\subset\mathbb{N}.$ We…

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski.…

Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime…

In the present paper, we generalize some of the results on Kloosterman sums proven in \cite{BG} for prime moduli to general moduli. This requires to establish the corresponding additive properties of the reciprocal set $$…

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box $[-X,X]^4$ that satisfy the determinant equation $x_1x_2-x_3x_4=r$ for $r \neq 0$ with two of entries to be…

In this paper, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove a new Omega result in the region $\exp((\log q)^{\frac12+\delta})\le N\le\sqrt q$, where $q$ is the…

Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $\psi:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $\chi: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz…

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n), as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as…

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N…

In this paper, we improve on Fouvry and Radziwi{\l}{\l}'s results on unbalanced convolutions. In particular, we find that if $(\alpha_m)$ and $(\beta_n)$ are sequences supported on $m\sim M$ and $n\sim M$ where $\beta$ is equidistributed…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for…

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows…

We use some elementary arguments to obtain a new bound on bilinear sums with weighted Kloosterman sums which complements those recently obtained by E. Kowalski, P. Michel and W. Sawin (2020).

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower…

We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums.…

We prove a non-trivial bound for $\operatorname{Sp}(2n)$ Kloosterman sums of moduli not equal to a prime multiple of the identity. These sums are attached to Siegel modular forms on the group $\operatorname{Sp}(2n)$ and appear in the…

Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be the finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $\mathbb F_q$ and $a\in\mathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of…

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…