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SUMMARY:Short sums of the polynomial Mobius function [MPIM]
DTSTART:20260826T123000Z
DTEND:20260826T133000Z
DTSTAMP:20260822T032500Z
UID:indico-event-1540@math-events.uni-bonn.de
DESCRIPTION:Speakers: Igor Shparlinski (University of New South Wales/MPI)
 \n\nNumber theory lunch seminar\nA classical result of Stickelberger relat
 es the polynomial Mobius function modulo a prime $p$ to quadratic characte
 rs of polynomial discriminants. We explain how to use a recent stratificat
 ion result of J. Xu (2020) to estimate the corresponding character sums ov
 er polynomials of degree $n$ with coefficients in a cubic box of side leng
 th $H$. We obtain a nontrivial bound with power saving below the Pólya–
 Vinogradov range\, namely for $H \\ge p^{1/2-\\gamma_n}$\, where $\\gamma_
 n>0$ is an explicit constant. The main novelty of our approach is a “cus
 tom-made” modification of the Burgess shift\, which allows us to handle 
 multivariate character sums involving non-homogeneous polynomials (in the 
 homogeneous case\, the classical Burgess shift applies).\nMotivated by wor
 k of S. Ganguly and C. S. Rajan (2023)\, we also apply a similar idea to i
 nvestigate $2\\times 2$ integral matrices with entries in $[1\,H]$ and an 
 irreducible characteristic polynomial. Our results are nontrivial for $H \
 \ge p^{1/8+\\varepsilon}$.\nThis is joint work with E. Fouvry and P. Xi \
 n\nhttps://math-events.uni-bonn.de/event/1540/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/1540/
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