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

Prime number theorem asserts that (at large $x$) the prime counting function $\pi(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N…

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…

The author sharpens a result of Jia (1996), showing that the interval $[n, n+n^{\frac{1}{21.5}+\varepsilon}]$ contains prime numbers for almost all $n$. Watt's mean value bound, a delicate sieve decomposition and more accurate estimates for…

We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of…

We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these…

We examine the representation of numbers as the sum of two squares in $\mathbb{Z}_n$ for a general positive integer $n$. Using this information we make some comments about the density of positive integers which can be represented as the sum…

Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erd\H{o}s--Kac theorem on the distribution of the number of prime factors of a random…

A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that…

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.

In this paper, we study the average size of the product of a number and its multiplicative inverse modulo a prime p. This turns out to be related to moments of L-functions and leads to a curious asymptotic formula for a certain triple…

When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

We improve some results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$ and $n=p^{\ell_1} + m^{\ell_2}$, where $\ell_1, \ell_2\ge 2$ are…

This paper studies power indices based on average representations of a weighted game. If restricted to account for the lack of power of dummy voters, average representations become coherent measures of voting power, with power distributions…

We give an asymptotic formula for the mean value of the number of representations of an integer as sum of two squares known as the Gauss circle problem.

In this article, we investigate conditional large values of quadratic Dirichlet character sums with multiplicative coefficients. We prove some Omega results under the assumption of the generalized Riemann hypothesis.

The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…

A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the…

We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an…

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to…

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.