A Cesàro average for an additive problem with prime powers

Abstract

In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes, that appeared in [10] (see also Theorem 2.2 of [6]). Let 1 ≤ ℓ 1 ≤ ℓ 2 be two integers, Λ be the von Mangoldt function and r ℓ 1 , ℓ 2 (n) = m ℓ 1 1 +m ℓ 2 2 =n Λ(m 1 )Λ(m 2 ) be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let N ≥ 2 be an integer. We prove that the Cesàro average of weight k > 1 of r ℓ 1 , ℓ 2 over the interval [1, N ] has a development as a sum of terms depending explicitly on the zeros of the Riemann zeta-function.

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

“…It is important to underline that in some particular configurations of the parameters some terms of the asymptotic (but not the dominant term) could be incorporated in the error. Despite the apparently complicated form of the terms, it is not difficult to recognize the results obtained in the previous work on this topic, for example setting d = 2, h = 0 and r = (1, 1) (the Goldbach numbers case [22]) or r = ( 1 , 2 ) , 1 ≤ 1 ≤ 2 integers (the generalized Goldbach numbers case [23]). Furthermore, it is quite natural to conjecture that at least the main term of this asymptotic is valid for k ≥ 0 instead of k > d+h 2 as suggested by similar studies but with other techniques (see, e.g., the papers by the present authors [10] and with Languasco [8]).…”

“…Given the flexibility of the technique introduced in [22], the latter has been applied to other types of additive problems like the one studied in [23]. Similar averages of arithmetical functions are common in the literature: see, e.g., Berndt [1].…”

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

“…It is important to underline that in some particular configurations of the parameters some terms of the asymptotic (but not the dominant term) could be incorporated in the error. Despite the apparently complicated form of the terms, it is not difficult to recognize the results obtained in the previous work on this topic, for example setting d = 2, h = 0 and r = (1, 1) (the Goldbach numbers case [21]) or r = (ℓ 1 , ℓ 2 ) , 1 ≤ ℓ 1 ≤ ℓ 2 integers (the generalized Goldbach numbers case [19]). Furthermore, it is quite natural to conjecture that at least the main term of this asymptotic is valid for k ≥ 0 instead of k > d+h 2 as suggested by similar studies but with other techniques (see, e.g., [10][11]).…”