On the variance of sums of divisor functions in short intervals (Q2827352)

We quote author's perfect summary: ``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 \(k\geq 4\) under the assumption of the Lindelöf hypothesis.''NEWLINENEWLINENEWLINENEWLINEAs the author says in the Introduction: \dots ``there is a connection between the variance of sums of \(d_k(n)\) and the \(2k\)th moment of the Riemann zeta-function. This connection becomes more pronounced when looking at short intervals.'' He should have quoted, for example, reviewer's paper [Publ. Inst. Math., Nouv. Sér. 88(102), 99--110 (2010; Zbl 1337.11056)].NEWLINENEWLINEActually, see the quoted reviewer's paper for terminology, the author is giving an asymptotic estimate for a kind of Selberg integral of the arithmetic function \(d_k(n)\), both for \(k=3\), see Theorem 1, unconditionally (but for inner short intervals with length \(h=h(x)\gg x^{7/12+\varepsilon}\)) and for \(k\geq 4\), see Theorem 2, under the Lindelöf hypothesis. In turn, this is done approximating the inner short sums over \(d_k(n)\) in terms of, as the author says, ``short trigonometric polynomials'' (a kind of Voronoï \thinspace formula for \(d_k\)).NEWLINENEWLINEAs the author says ``Our main innovation is to (essentially) combine Jutila's approach with Selberg's method.'' (See the paper for the details)