Related papers: Discretized sum-product for large sets

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen…

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x +…

Let s_1,...,s_d be d positive integers and consider the multiple Hurwitz-zeta value zeta(s_1,...,s_d;-1/2,...,-1/2)/2^w where w=s_1+...+s_d is called the weight. For d<n+1, let T(2n,d) be the sum of all these values with even arguments…

In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset \mathbb R$, and for all convex functions $f$ which satisfy an additional technical condition.…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

A detailed, internal symmetry exists between individual terms $n^{-s}$, where $n \in P$ is less than a particular value $n_p$, and sums over conjugate regions consisting of adjoining steps $n$ greater than $n_p$. The boundaries of the…

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A…

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…

Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma;q)=\big\{\sum_{n=0}^\infty…

We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups…

Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. Let SF(G) denotes the set of all sum-free subets of $G$ and $\sigma(G)$ denotes…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote…

If $A$ and $B$ are subsets of an abelian group, their sumset is $A+B:=\{a+b:a\in A, b\in B\}$. We study sumsets in discrete abelian groups, where at least one summand has positive upper Banach density. Renling Jin proved that if $A$ and $B$…

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations…

Let q be a prime, A be a subset of a finite field $F=\Bbb Z/q\Bbb Z$, $|A|<\sqrt{|F|}$. We prove the estimate $\max(|A+A|,|A\cdot A|)\ge c|A|^{1+\epsilon}$ for some $\epsilon>0$ and c>0. This extends the result of J. Bourgain, N. Katz, and…

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\subset \mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominant tends to zero, in 2006…

Consider a positive integer $r$ and a graph $G=(V,E)$ with maximum degree $\Delta$ and without isolated edges. The least $k$ so that a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ exists such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$…