Related papers: Exponential sums over primes are unbounded

We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new…

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-\epsilon}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] +…

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $\lambda\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ for which the interval…

Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.

Hardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $\mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by…

We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…

In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool…

Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that $\lim_{n\rightarrow\infty} \sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to…

We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise…

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…

We calculate the triple correlations for the truncated divisor sum $\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to do. We also…

Let $\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \to +\infty$, $$ \sum_{n \leqslant x} f(n) \ll \sqrt{x} (\log_2 x)^{\frac{3}{4}+ \varepsilon}. $$

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for…

Given a positive integer $m$, let $\mathbb{Z}_m$ be the set of residue classes mod $m$. For $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\sigma_A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let…

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are…