Related papers: A note on an average additive problem with prime n…

Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math.…

Assume the Riemann hypothesis throughout. We obtain some new estimates for the size of the set of large values of the error term in the prime number theorem. Our argument is based on an analysis of the behavior of zeros of the Riemann zeta…

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences…

We prove explicit versions of Cram\'er's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

We provide an asymptotic expansion for the mean-value of the logarithm of the middle prime factor of an integer, defined according to multiplicity or not, thus generalising a recent study of McNew, Pollack, and Singha Roy. This yields an…

Fujii obtained a formula for the average number of Goldbach representations with lower order terms expressed as a sum over the zeros of the Riemann zeta-function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an…

In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.

In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…

This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently…

We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially…

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set…

Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable…

We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.

Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…

In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (1998).