TY - JOUR
AB - In this paper, we investigate the distribution of the maximum of partial sums of families of m -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of ℓ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of m -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.
AU - Autissier, Pascal
AU - Bonolis, Dante
AU - Lamzouri, Youness
ID - 10711
IS - 7
JF - Compositio Mathematica
KW - Algebra and Number Theory
SN - 0010-437X
TI - The distribution of the maximum of partial sums of Kloosterman sums and other trace functions
VL - 157
ER -