相关论文: Simple proof of Gallagher's singular series sum es…

We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.

Exact summatory functions that count the number of prime $k$-tuples up to some cut-off integer are presented. Related $k$-tuple analogs of the first and second Chebyshev functions are then defined.

In this paper, we will give some estimation for the average error of the prime number theorem.

We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint…

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 use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a…

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Assuming a suitably uniform version of the prime $k$-tuples conjecture, we show that the number \begin{align*} \sum_{n=1}^\infty \frac{\omega(n)}{2^n} \end{align*} is…

In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.

We determine a formula for the average values of L-series associated to eigenforms at complex values.

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of…

We continue our recent work on averages for ternary additive problems with powers of prime numbers.

We obtain the general k-correlations for a short divisor sum related to primes.

We study the prime pair counting functions $\pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of…

Let $K\geq 2$ be a natural number and $a_i,b_i\in\mathbb{Z}$ for $i=1,\ldots,K-1$. We use the large sieve to derive explicit upper bounds for the number of prime $k$-tuplets, i.e., for the number of primes $p\leq x$ for which all $a_ip+b_i$…

We prove the numerical Hodge standard conjecture for the square of a simple abelian variety of prime dimension, and also in some related cases.

Consider a $d$-dimensional simplex whose vertices are random points chosen independently according to the standard Gaussian distribution on $\mathbb R^d$. We prove that the expected angle sum of this random simplex equals the angle sum of…

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 survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.