On an average ternary problem with prime powers

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

“…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]).…”

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

A note on an average additive problem with prime numbers

“…We pursue recent investigations by the present authors and Alessandro Languasco in [2] and [1]. In this short note we study general average additive problems: let k = (k 1 , k 2 , .…”

“…Proving the expected individual asymptotic formula for R(n; k) as n → ∞ along "admissible" residue classes (that is, avoiding those residue classes which can not contain values of the form p k 1 1 + • • • + p k r r because of the uneven distribution of prime powers in residue classes) is very difficult if either r or ρ is small. Our main goal is to give an asymptotic formula for the average value of R(n; k) for n ∈ [N + 1, N + H] where N → +∞ and H = H(N; k) is as small as possible.…”

“…Theorem 1.1 contains as special cases all results in [2] and [1], whereas Theorem 1.2 is occasionally slightly weaker because our basic combinatorial identity here, equation (2), is less efficient than the identities we used in the papers mentioned above.…”