BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Nguyen (UCSB) DTSTART:20210305T230000Z DTEND:20210306T000000Z DTSTAMP:20260423T024739Z UID:UCSBsga/16 DESCRIPTION:Title: Distribution of the ternary divisor function in arithmetic progressions\nby David Nguyen (UCSB) as part of UCSB Seminar on Geometry and Arithme tic\n\n\nAbstract\nThe ternary divisor function\, denoted $\\tau_3(n)$\, c ounts the number of ways to write a natural number $n$ as an ordered produ ct of three positive integers. Thus\, $\\sum_{n=1}^\\infty \\tau_3(n) n^{- s} = \\zeta^3(s).$ Given two coprime positive integers $a$ and $q$\, we st udy the distribution of $\\tau_3$ in arithmetic progressions $n \\equiv a (\\text{mod } q).$ The distribution of $\\tau_3$ in arithmetic progression s has a rich history and has applications to the distribution of prime num bers and moments of Dirichlet $L$-functions. We show that $\\tau_3$ is equ idistributed on average for moduli $q$ up to $X^{2/3}$\, extending the ind ividual estimate of Friedlander and Iwaniec (1985). We will also discuss a n averaged variance of $\\tau_3$ in arithmetic progressions related to a r ecent conjecture of Rodgers and Soundararajan (2018) about asymptotic of t his variance. One of the key inputs to this asymptotic come from a modifie d additive correlation sum of $\\tau_3.$\n LOCATION:https://researchseminars.org/talk/UCSBsga/16/ END:VEVENT END:VCALENDAR