On the variance of sums of divisor functions in short intervals

Abstract

Given a positive integer n the k-fold divisor function d k (n) equals the number of ordered k-tuples of positive integers whose product equals n. In this article we study the variance of sums of d k (n) in short intervals and establish asymptotic formulas for the variance of sums of d k (n) in short intervals of certain lengths for k = 3 and for k ≥ 4 under the assumption of the Lindelöf hypothesis.

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

“…Our main concern is to understand its mean square. For relatively long intervals, Lester [29] proves an asymptotic (assuming RH for k > 3) similar to the result (1.3):…”

On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

“…Our proof follows the argument of Lester [18]. Most of the steps are analogous, but we present the details for the sake of completeness.…”

“…Jutila's method is not applicable here essentially for two reasons; first one being that trigonometric polynomials in the truncated GL(n)-Voronoi summation formula are more complicated than in the lower rank setting and the other one is that the error term in the relevant truncated Voronoi summation formula gives larger contribution than the expected main term. Instead, we follow Lester [18] who treats a similar problem for the error term of the Dirichlet divisor problem for the k-fold divisor function by combining Jutila's method with the one of Selberg [25]. Selberg's method can also be applied to other problems concerning automorphic forms, see e.g.…”

“…by Lester [18] under the Lindelöf hypothesis for the ζ-function and we follow his strategy. Many of the details are similar, but we present the whole argument for the sake of completeness as only a bound of the form A(m, 1, ..., 1) ≪ ε m ϑ+ε , for some fixed ϑ ≥ 0, is known for the Hecke eigenvalues.…”

The variance of divisor sums in arithmetic progressions

“…What is known rigorously over the integers in the short-interval setting of Conjecture 2 follows from using summation formulas related to the functional equation for the Riemann zeta function. In this way Lester [23] has evaluated the variance for c ∈ (k − 1, k). It is likely that a similar argument could be used to verify Conjecture 1 in this restricted range for all k (indeed, this is close to the strategy of [22] in the case k = 2).…”