On the modified Selberg integral of the three-divisor function $d_3$

Abstract

We prove a non-trivial result for the,say,modified Selberg integral $\modSel_3(N,h)$, of the divisor function $d_3(n):={\sum }_a{\sum }_b{\sum }_{c,abc=n}1$; this integral is a slight modification of the corresponding Selberg integral, that gives the expected value of the function in short intervals. We get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a byproduct, we obtain indications on the way in which it may be proved the weak sixth moment of the Riemann zeta function.(This was OLD abstract)

arXiv commentThe square-root cancellation for the modified Selberg integral of d3 is now a Conjecture. In fact,our proof of the Proposition is wrong;actually, the Proposition is too strong to be proven with present methods

References in corpus (1)

• On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals, III