A note on an average additive problem with prime numbers

In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.

The aim of this paper is to present a revised version of my proof of the Riemann Hypothesis in which a few more details and explanations have been added

A proposed solution to the Riemann Hypothesis

Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors an analogous result by Friedlander-Goldston.

The aim of this work is to illustrate a conditional result involving the exponential sums over primes in short intervals under the assumption that both the Generalized Riemann Hypothesis and the Density Hypothesis for Dirichlet $L$-functions are true.

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.

In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in small intervals.

We cosider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also estabish an asymptotic formula if the interval is not very short.

In this paper, we use the transference principle to investigate the representation of sufficiently large positive integers as the sum of prime powers and integer powers, where the primes are drawn from a positive density subset of the set of all primes , and the integer powers are drawn from a positive density subset of k-th powers.