Related papers: On Weyl sums over primes in short intervals
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…
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg…
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.
We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial. The method we use is a blend of techniques coming from…
We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…
We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
Based on Euclid's algorithm, we find a kind of special sequences which play an interesting role in the study of primes. We call them W Sequences. They not only ties up the distribution of primes in short interval but also enables us to give…
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.
We obtain new estimates - both upper and lower bounds - on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures of C. Demeter and B. Langowski (2022), and improve some of…
A few elementary estimates of a basic character sum over the prime numbers are derived here. These estimates are nontrivial for character sums modulo large q. In addition, an omega result for character sums over the primes is also included.
For sufficiently large integers $K$, $x$, $y$, and $q$ satisfying $K \le y < x$, where $f(u) = \alpha u^n + \alpha_{n-1}u^{n-1} + \ldots + \alpha_1 u$ is a polynomial of degree $n$ with real coefficients, $n$ is a fixed positive integer,…
On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…
We show that there are short intervals $[x,x+y]$ containing $\gg y^{1/10}$ numbers expressible as the sum of two squares, which is many more than the average when $y=o( (\log{x})^{5/9})$. We obtain similar results for sums of two squares in…
In this paper we obtain some new estimates for multilinear exponential sums in prime fields with a more general class of weights than previously considered. Our techniques are based on some recent progress of Shkredov on multilinear sums…
We give some results on a priori estimates and on estimates of type sup+inf and sup*inf.
Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha…
We survey the classical results on the prime number theorem
A survey paper on some recent results on additive problems with prime powers.
We obtain the general k-correlations for a short divisor sum related to primes.