The average number of solutions of the Diophantine equation U²+V²=W³and related arithmetic functions

Abstract

Abstract. For the number of integer solutions of the title equation, with W ≤ x ( x a large parameter), an asymptotics of the form Ax log x + Bx + O x 1/2 (log x) 3 (log log x) 2 is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.

Short interval asymptotics for a class of arithmetic functions

“…For general ζ K , the result has been established as a Lemma in [8]. (The extension to both signs in the exponent of ζ K (1 + it) is clear by a trivial modification of the last step in the proof in [8].) Lemma B.…”

Representations of integers by certain positive definite binary quadratic forms

“…Then using Kronecker's result, we prove a proposition relating genus characters to poles of the Rankin-Selberg convolution of L-functions. The first estimate is a recent result of Kühleitner and Nowak [13], namely Theorem 2.1. Let a(n) be an arithmetic function satisfying a(n) ≪ n ǫ for every ǫ > 0, with a Dirichlet series…”

Asymptotic formulas for certain arithmetic functions

“…For the details of the arguments the reader is referred to the authors' original articles [1], [2], and [6].…”

“…However, it is amazing how many interesting applications (cf. also [2] and [6] for a few more) are contained in the special case considered.…”