Related papers: Explicit estimates for the Goldbach summatory func…

A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the…

We approach a new proof of the strong Goldbach's conjecture for sufficiently large even integers by applying the Dirichlet's series. Using the Perron formula and the Residue Theorem in complex variable integration, one could show that any…

This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach zeta function). As an application we…

We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we…

In this paper, we introduce a geometrical summation method that makes the original Riemann series converge over the critical strip. This method gives an analytical function, that coincides with z\^eta. This point of view allows us to…

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We…

We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}(…

We present a common ground for infinite sums, unordered sums, Riemann/Lebesgue integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…

Let $\psi_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $\psi^{(1)}_{\mathbb K}(x):=\int_{0}^{x}\psi_{\mathbb K}(t)\,d t$ and $\psi^{(2)}_{\mathbb K}(x):=2\int_{0}^{x}\psi^{(1)}_{\mathbb K}(t)\,d t$. We prove…

By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all…

We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some complex power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in…

Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates…

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several…

We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We…