Large character sums

Authors:
Andrew Granville and K. Soundararajan

Journal:
J. Amer. Math. Soc. **14** (2001), 365-397

MSC (2000):
Primary 11L40; Secondary 11N25

DOI:
https://doi.org/10.1090/S0894-0347-00-00357-X

Published electronically:
October 20, 2000

MathSciNet review:
1815216

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Abstract | References | Similar Articles | Additional Information

Abstract: We make conjectures and give estimates for how large character sums can be as we vary over all characters mod , and as we vary over real, quadratic characters. In particular we show that the largest sums seem to depend on the value of the character at ``smooth numbers''.

**1.**P.T. Bateman and S. Chowla,*Averages of character sums*, Proc. Amer. Math. Soc**1**(1950), 781-787. MR**13:113d****2.**D.A. Burgess,*The distribution of quadratic residues and non-residues*, Mathematika**4**(1957), 106-112. MR**20:28****3.**H. Davenport,*Multiplicative number theory*, Second Edition, Springer Verlag, New York, 1980. MR**82m:10001****4.**J.B. Friedlander and H. Iwaniec,*A note on character sums*, Contemp. Math. J**166**(1994), 295-299. MR**95f:11058****5.**S.W. Graham and C.J. Ringrose,*Lower bounds for least quadratic non-residues*, Prog. Math**85**(1990), 269-309. MR**92d:11108****6.**A. Granville and K. Soundararajan,*The spectrum of multiplicative functions*(to appear).**7.**A. Granville and K. Soundararajan,*The distribution of*(to appear).**8.**G.H. Hardy and S. Ramanujan,*The normal number of prime factors of a number*, Quart. J. Math**48**(1917), 76-92.**9.**A. Hildebrand,*A note on Burgess's character sum estimate*, C.R. Acad. Sci. Roy. Soc. Canada**8**(1986), 35-37. MR**87e:11095****10.**A. Hildebrand and G. Tenenbaum,*Integers without large prime factors*, J. Théorie des Nombres, Bordeaux**5**(1993), 411-484. MR**95d:11116****11.**H.L. Montgomery,*An exponential polynomial formed with the Legendre symbol*, Acta Arithm.**37**(1980), 375-380. MR**82a:10041****12.**H.L. Montgomery,*Ten lectures on the interface between analytic number theory and harmonic analysis*, vol. 84, CBMS Regional Conference Series in Mathematics, AMS, 1994. MR**96i:11002****13.**H.L. Montgomery and R.C. Vaughan,*Exponential sums with multiplicative coefficients*, Invent. Math**43**(1977), 69-82. MR**56:15579****14.**R.E.A.C. Paley,*A theorem on characters*, J. London Math. Soc**7**(1932), 28-32.**15.**C. Pomerance,*On the distribution of round numbers*, Number Theory (Proc. Ootacamund, India), Springer Lecture Notes No. 1122, 1984, pp. 173-200. MR**87b:11095****16.**G. Tenenbaum,*Cribler les entiers sans grand facteur premier*, Phil. Trans. Roy. Soc.**345**(1993), 377-384. MR**95d:11119**

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Additional Information

**Andrew Granville**

Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602

Email:
andrew@math.uga.edu

**K. Soundararajan**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Address at time of publication:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Email:
skannan@math.princeton.edu, ksound@ias.edu

DOI:
https://doi.org/10.1090/S0894-0347-00-00357-X

Received by editor(s):
March 29, 1999

Received by editor(s) in revised form:
September 8, 2000

Published electronically:
October 20, 2000

Additional Notes:
The first author is a Presidential Faculty Fellow. He is also supported, in part, by the National Science Foundation. The second author is partially supported by the American Institute of Mathematics (AIM)

Dedicated:
Dedicated to John Friedlander

Article copyright:
© Copyright 2000
American Mathematical Society