Related papers: Short interval results for certain prime-independe…

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 paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable…

Using estimates on Hooley's $\Delta$-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also…

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.

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the…

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.

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression.…

We prove an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields.

Let $R_{m, \mathrm{sq-full}}(N)$ be a representation function for the sum of a prime and a square-full number. In this article, we prove an asymptotic formula for the sum of $R_{m, \mathrm{sq-full}}(N)$ over positive integers $N$ in a short…

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…

In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.

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…

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the…

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…

Given good knowledge on the even moments, we derive asymptotic formulas for $\lambda$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…

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.