Abstract
Let M(χ) denote the maximum of ∑n≤Nχ(n) for a given
nonprincipal Dirichlet character χ(modq), and let Nχ denote a
point at which the maximum is attained. In this article we study the
distribution of M(χ)/q√ as one varies over characters (modq),
where q is prime, and investigate the location of Nχ. We show that the
distribution of M(χ)/q√ converges weakly to a universal distribution
Φ, uniformly throughout most of the possible range, and get (doubly
exponential decay) estimates for Φ's tail. Almost all χ for which
M(χ) is large are odd characters that are 1pretentious. Now,
M(χ)≥∑n≤q/2χ(n)=2−χ(2)πq√L(1,χ), and one knows how often the latter expression is large, which has
been how earlier lower bounds on Φ were mostly proved. We show, though,
that for most χ with M(χ) large, Nχ is bounded away from q/2,
and the value of M(χ) is little bit larger than q√πL(1,χ).
Original language  English 

Pages (fromto)  17591818 
Number of pages  60 
Journal  Journal of the European Mathematical Society 
Volume  20 
Issue number  7 
Early online date  22 May 2018 
DOIs  
Publication status  Published  22 May 2018 
Keywords
 math.NT
 Primary: 11N60, Secondary: 11K41, 11L40
Fingerprint
Dive into the research topics of 'The frequency and the structure of large character sums'. Together they form a unique fingerprint.Profiles

Dr Jonathan W Bober
 School of Mathematics  Heilbronn Senior Research Fellow
 Heilbronn Institute for Mathematical Research
 Pure Mathematics
Person: Academic , Member