Some Identities Involving the Cesàro Average of the Goldbach Numbers

Abstract

Let Λ (n) be the von Mangoldt function and r G (n) := m1+m2=n Λ (m 1 ) Λ (m 2 ) be the counting function for the numbers that can be written as sum of two primes (that we will call "Goldbach numbers", for brevity) and let S (z) := n≥1 Λ (n) e −nz , with z ∈ C, Re (z) > 0. In this paper we will prove the identity

A Cesàro average for an additive problem with an arbitrary number of prime powers and squares

“…with Re (a) > 0 (see formulas (8) and (9) of [1]). We also need an integral representation of the Bessel J function with real argument u and complex order v: 8) on page 177 of [28]). Assume that k > 0.…”

“…Now we want to show that it is possible to exchange the integral with the series in the right side of (8).…”

“…The presence of a smooth weight allowed to obtain an asymptotic formula with terms of decreasing orders of magnitude and depending on the non-trivial zeros of the Riemann Zeta function; furthermore, the weights allowed to obtain results independent of the Riemann hypothesis. Since the Cesàro weights depend on a non-negative real parameter k and are equal to 1 if k = 0 (that is, for k = 0 we have a simple average without weights) it is important to obtain results with the smallest possible k. During the last few years there have been some improvements regarding the optimal k in the case of the Goldbach's problem: in [21] results hold for k > 1, in [14] (assuming Riemann hypothesis) and in [8] (unconditionally) for k = 1 and in [4] for k > 0. Unfortunately, for case k = 0, that is, without the Cesàro weights, it is not yet possible to obtain in the same form or with the same quantity of terms as in the other cases (see, for example, [23], [5] and [25]).…”

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.…”

Explicit formula for the average of Goldbach and prime tuples representations

“…see [7], [14] and [16]. Even if the technique of Languasco and Zaccagnini, developed to study (1), can be applied to various problems (see [5,6,17]) in all of these papers there are some limitations over the parameter k due to some convergence problems.…”