Related papers: On an average ternary problem with prime powers

This is an article for a general mathematical audience on the author's work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. It is based on several one hour lectures, chiefly given at…

We present a new, elementary, dynamical proof of the prime number theorem.

In this paper, we state a conjecture on the prime factorization of numbers of the form $n!+1$, explore its implications, and compare it with empirical evidence and established results based on the $abc$ conjecture.

In the present paper we construct normal numbers in base $q$ by concatenating $q$-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.

In this paper, we provide formulas for partial sums of weighted averages over regular integers modulo $n$ of the $\gcd$-sum function with any arithmetic function. Many interesting applications of the results are also given.

We introduce extremely symmetric primes and provide some elementary properties of these.

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The…

This paper is a complement of our recent works on the semilinear Tricomi equations in [8] and[9].

We discuss various recent advances on weak forms of the Twin Prime Conjecture.

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We…

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to…

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

We use the Hardy-Littlewood circle method to study primes of the form $n_1^u + n_2^v + k$, on average.

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…

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for…

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the…

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.