Related papers: Large character sums

We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials…

Let $G$ be a connected reductive group over $\kk$, an algebraic closure of a finite field. For an integer $r\ge 1$ let $G_r=G(\kk[\e]/(\e^r))$ viewed as an algebraic group of dimension $r\dim G$ over $\kk$. We show that the character of the…

For integer $q$, let $\chi$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}\Lambda(n)\chi(n+a),$$ where $\Lambda(n)$ is the von Mangoldt function. This…

Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations,…

Writing for a general mathematical audience, we provide elementary upper and lower bounds on the growth (as a function of N) of the sum \sum_{n=1}^N (-1)^{\floor{n x}} for various fixed x. For example, if x is a quadratic irrational, then…

We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.

Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written as the sum of a prime and a number with at most 395 prime factors. We also show, under assumption of the generalised Riemann hypothesis,…

In this paper, we exhibit upper and lower bounds with explicit constants for some objects related to entire $L$-functions in the critical strip, under the generalized Riemann hypothesis. The examples include the entire Dirichlet…

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of…

In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between…

The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach,…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

For a given Dirichlet character $\chi (n) = e^{i \theta_n}$, we prove central limit theorems for the series $\sum_{p'} \cos \theta_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$…

We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational…

A subfield $K$ of $\bar{\mathbb{Q}}$ is $large$ if every smooth curve $C$ over $K$ with a rational point has infinitely many rational points. A subfield $K$ of $\bar{\mathbb{Q}}$ is $big$ if for every positive integer $n$, $K$ contains a…

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.

We evaluate the character sum sigma_m sigma_n (m/n) where the two sums are of approximately the same length. The answer is surprising.

Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best…