Related papers: On Kloosterman sums with multiplicative coefficien…

In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

Let $n_0$ be 1 or 3. If a multiplicative function $f$ satisfies $f(p+q-n_0) = f(p)+f(q)-f(n_0)$ for all primes $p$ and $q$, then $f$ is the identity function $f(n)=n$ or a constant function $f(n)=1$.

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$…

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(n\alpha)$ where $f$ is a $1$-bounded multiplicative function and $\alpha\in\mathbb R$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+…

Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g :…

Let $\mathcal{K}(a)$ denote the Kloosterman sum on the finite field of order $2^n$. We give a simple characterization of $\mathcal{K}(a)$ modulo 16, in terms of the trace of $a$ and one other function. We also give a characterization of…

We give generating functions for Gauss sums for finite general linear and unitary groups. For the general linear case only our method of proof is new, but we deduce a bound on Kloosterman sums which is sometimes sharper than Deligne's bound…

We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman…

We prove two results on Kloosterman sums over finite fields, using Stickelberger's theorem and the Gross-Koblitz formula. The first result concerns the minimal polynomial over Q of a Kloosterman sum, and the second result gives a…

We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs…

We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the…

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to…

Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(\alpha; N)=\sum_{n\le…

We establish a simple identity and using it we find a new proof of a result of Kloosterman.

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$…

We present a class of multiplicative functions $f:\mathbb{N}\to\mathbb{C}$ with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by $1$. The key feature is that they pretend to be the constant…

We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential…

As early as the 1930s, P\'al Erd\H{o}s conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to\{-1,1\}$, the partial sums $\sum_{n\leq x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider…

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$…