Character Sums to Smooth Moduli are Small

D ec 2 00 7 Character sums to smooth moduli are small

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...

ec 2 00 7 Character sums to smooth moduli are small

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...

On Certain Character Sums over Smooth Numbers

Character Sums Associated to Finite Coxeter

The main result of this paper is a character sum identity for Cox-eter arrangements over nite elds which is an analogue of Macdonald's conjecture 19] proved by Opdam 21]. 0. Introduction Throughout this paper F will denote a nite eld of characteristic p diierent from 2. We rst slightly reformulate Macdonald's conjecture in a form which has a direct analogue over F. Let G be a nite subgroup of G...

Large Character Sums

where χ is a non-principal Dirichlet character χ (mod q). It is easy to show that such character sums are always ≤ q in absolute value, while G. Pólya and I.M. Vinogradov (see [3]) improved this to≤ √q log q around 1919, and H.L. Montgomery and R.C. Vaughan [13] to √q log log q in 1977, assuming the Generalized Riemann Hypothesis (GRH). Up to the constant this is “best possible” since R.E.A.C. ...