Related papers: Chebyshev Estimates for Beurling Generalized Prime…

For first-order optimization of non-convex functions with Lipschitz continuous gradient and Hessian, the best known complexity for reaching an $\varepsilon$-approximation of a stationary point is $\mathcal{O}(\varepsilon^{-7/4})$. Existing…

There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane $\Bbb C$ has the…

This paper provides a detailed expository and computational account of the elementary methods developed by P. L. Chebyshev and J. J. Sylvester to establish explicit bounds on the prime counting function. The core of the method involves…

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X^\varepsilon_t(x))_{t\geqslant 0}$ with $\varepsilon$-small additive L\'evy noise and initial…

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system…

In this study, we discuss the convergence and divergence of generalized integrals,\int_{0}^{+\infty}\frac{sin^{b}x}{x^{a}}dx(a\epsilon R^{+},b\epsilon N^{+}), and use the transformation method, the partial integration method, the…

Let $X = [0,1]$, and let $T:X\to X$ be an expanding piecewise linear map sending each interval of linearity to $[0,1]$. For $\psi:\mathbb N\to\mathbb R_{\geq 0}$, $x\in X$, and $N\in\mathbb N$ we consider the recurrence counting function \[…

Let $(T,d)$ be a metric space and $\phi:\mathbb{R}_+\to \mathbb{R}$ an increasing, convex function with $\phi(0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,\phi$, that is,…

We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-\epsilon}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and…

We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A…

We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(\pi(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This…

Let $s: [1, \infty) \to \C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \C$ such that $$\lim_{t\to \infty} \tau(t) = A, \quad…

In this paper we collect several examples of convergence of functions of random processes to generalized functionals of those processes. We remark that the limit is always finitely absolutely continuous with respect to Wiener measure. We…

Chaundy and Jolliffe [4] proved that if $\{a_{n}\}$ is a non-increasing (monotonic) real sequence with $\lim\limits_{n\to \infty}a_{n}=0$, then a necessary and sufficient condition for the uniform convergence of the series…

In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindel\"of hypothesis (LH) for general sequences which coincides with the usual Lindel\"of hypothesis for the Riemann zeta function in the case of the sequence of positive…

We fix a gap in the proof of a result in our earlier paper "Generalised notions of amenability, II" (J. Funct. Anal. 254 (2008) 1776-1810), and so provide a new proof to a characterization of amenability for Beurling algebras. The result…

We generalize the Stein-Tomas [17] $L^2$-restricition theorem and the uniform Sobolev estimates of Kenig, Ruiz and the second author [11] by allowing critically singular potential. We also obtain Strichartz estimates for Schr\"odinger and…

Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…

We obtain some sufficient conditions for the Central Limit Theorem for the random processes (fields) with values in the separable part of Holder space in the modern terms of majorizing (minorizing) measures, belonging to X.Fernique and…

For a base $b\geq 2$ and a set of digits $\mathcal{A}\subset \{0,...,b-1\}$, let $\mathcal{P}$ denote the set of prime numbers with digits restricted to $\mathcal{A}$, when written in base-$b$. We prove that if $A\subset \mathbb{N}$ has…