Related papers: On Weyl sums over primes in short intervals
In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.
In this paper, we develop interval estimation methods for means of bounded random variables based on a sequential procedure such that the sampling is continued until the sample sum is no less than a prescribed threshold.
This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two…
We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$…
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums.…
In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x…
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent…
This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n), as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as…
In recent years, maximizing G\'al sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of G\'al type, with respect to the $L^1$-norm. We also consider…
Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over…
This work consists of a heuristic study on the distribution of prime numbers in short intervals. We have modelled the occurrence of prime numbers such intervals as a counting experiment. As a result, we have provided an experimental…
Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling generalized primes.
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
In this paper we extend the well-known investigations of Montgomery and Goldston & Montgomery, concerning the pair-correlation function and its relations with the distribution of primes in short intervals, to a more general version of the…