Explicit formulae for averages of Goldbach representations

Abstract

We prove an explicit formula, analogous to the classical explicit formula for ψ(x), for the Cesàro-Riesz mean of any order k > 0 of the number of representations of n as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein. (2000): 11P32, 11N05 Mathematics Subject Classification

On an Average Goldbach Representation Formula of Fujii

“…The term E(N ) in Fujii's formula will give main terms ≫ N 3/2 from zeros with β ≥ 3/4. For weighted versions of Fujii's theorem there are formulas where the error term corresponding to E(N ) is explicitly given in terms of sums over zeros, see [LZ15] and [BKP19]. In principle one could use an explicit formula for Ψ(z) in Theorem 1 to obtain a complicated formula for E(N ) in terms of zeros, but we have not pursued this.…”

A note on Clebsch–Gordan integral, Fourier–Legendre expansions and closed form for hypergeometric series

“…It is important to emphasize again how these topics and techniques can be related to other, and sometimes unexpected, mathematical topics; for example, if it is quite natural to think about the classical Ramanujan-type series for 1/π (for a survey of this topic, see, for example, [3] and for formulas via hypergeometric transformations see [15]), the connection with additive number theory problems is probably less evident. Indeed, fractional operators applied to particular power series are involved in the study of explicit formulas for the so-called Cesaro average that counts the number of representations of an integer as sums of primes, prime powers, and squares of integers (for the interested reader, see [7,[11][12][13]20]); therefore, it is plausible to think that the techniques developed may also be of interest for these types of problems In this paper, we will focus on a results of Zhou [21] about a closed form for the generalized Clebsch-Gordan integral 1 −1 P μ (x) P ν (x) P ν (−x) dx, where P ν (x) , P μ (x) are the Legendre functions of arbitrary complex degree ν, μ ∈ C. We show that this result can be can be interpreted in terms of the FL theory and this point of view allows to evaluate series in which addends are powers of central binomials (and so, particular hypergeometric functions). Furthermore, we will show that from Zhou's results, we can obtain some formulas that recall the well-known Brafman's formula [6] and we can evaluate very easily some integral moments involving combinations of complete elliptic integrals of the first kind.…”

Some Identities Involving the Cesàro Average of the Goldbach Numbers

“…where N ≥ 2, 1 ≤ H ≤ N. k. In a very recent paper Brüdern, Kaczorowski and Perelli [2] were able to find an explicit formula which holds for all k > 0. Similar averages of arithmetical functions are common in literature, see, e.g., Chandrasekharan -Narasimhan [5] and Berndt [1] who built on earlier classical work.…”

“…This explicit formula extend, in some sense, the main result in [13] (since their formula holds for k > 1) and in [8] (since the authors assume the Riemann Hypothesis). Furthermore it provides different way to write the explicit formula of the Cesàro average of Goldbach representations in the case k = 1 respect to the main formula in [2]. The study of the double series…”