Asymptotic formulas for certain arithmetic functions

Abstract

ABSTRACT. This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere.

Asymptotics for a class of arithmetic functions

“…where Q 3 (u) is a polynomial in u of degree 3. The estimate O(x 3/5+ε ) was improved to O(x 1/2+ε ) by Wilson [34], and to O(x 1/2 (log x) 5 log log x) by K. Ramachandra and A. Sankaranarayanan [28], and by M. Z. Garaev, M. Kühleitner, F. Luca and W. G. Nowak [4]. Recently, Jia and Sankaranarayanan [14] showed that the error term in (6.3) can be improved to O(x 1/2 (log x) 5 ).…”

“…In [4], M. Z. Garaev, M. Kühleitner, F. Luca and W. G. Nowak gave an interesting and fairly general theorem which includes applications to sums like where d(n) is the Dirichlet divisor function. For papers in this direction, see also [14,5,6,21,28,32,34].…”