Short interval asymptotics for a class of arithmetic functions

Abstract

We provide a general asymptotic formula which permits applications to sums like x<n x+ywhere d(n) and r(n) are the usual arithmetic functions (number of divisors, sums of two squares), and y is small compared to x.

The mean square of the divisor function

“…Using the method similar to that in this paper, we can prove the following proposition. Combining the method of this paper with that of [2], we can prove the following proposition. Let r(n) be the number of representations of n as the sum of two squares.…”

“…If the result of Iwaniec [4] could be generalized to ζ K (s), then the error terms in (6.2) and (6.3) could be improved to O(x 1 2 (log x) 3 ). Furthermore, the sums studied in [2] x<n≤x+y r 2 (n), x<n≤x+y r(n 3 ), x<n≤x+y d(n)r(n) could also be improved correspondingly.…”

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.…”

Short interval results for certain arithmetic functions

“…In this example, F (n) is either τ (n) 2 , or τ (n 3 ). Improving on a result from[5], Zhai[13, Corollary 4] showed thatx<n x+y F (n) ∼ C F y(log x) 3 for y = o(x) with y x 1/2 log x → ∞ lorsque x → ∞ and where C F = 1 6ζ(2)…”