Large character sums

Authors:
Andrew Granville and K. Soundararajan

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

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

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**0042445**, 10.1090/S0002-9939-1950-0042445-6**2.**D. A. Burgess,*The distribution of quadratic residues and non-residues*, Mathematika**4**(1957), 106–112. MR**0093504****3.**Harold Davenport,*Multiplicative number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR**606931****4.**J. B. Friedlander and H. Iwaniec,*A note on character sums*, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 295–299. MR**1284069**, 10.1090/conm/166/01632**5.**S. W. Graham and C. J. Ringrose,*Lower bounds for least quadratic nonresidues*, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269–309. MR**1084186****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.**Adolf Hildebrand,*A note on Burgess’ character sum estimate*, C. R. Math. Rep. Acad. Sci. Canada**8**(1986), no. 1, 35–37. MR**827113****10.**Adolf Hildebrand and Gérald Tenenbaum,*Integers without large prime factors*, J. Théor. Nombres Bordeaux**5**(1993), no. 2, 411–484. MR**1265913****11.**Hugh L. Montgomery,*An exponential polynomial formed with the Legendre symbol*, Acta Arith.**37**(1980), 375–380. MR**598890****12.**Hugh L. Montgomery,*Ten lectures on the interface between analytic number theory and harmonic analysis*, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR**1297543****13.**H. L. Montgomery and R. C. Vaughan,*Exponential sums with multiplicative coefficients*, Invent. Math.**43**(1977), no. 1, 69–82. MR**0457371****14.**R.E.A.C. Paley,*A theorem on characters*, J. London Math. Soc**7**(1932), 28-32.**15.**Carl Pomerance,*On the distribution of round numbers*, Number theory (Ootacamund, 1984) Lecture Notes in Math., vol. 1122, Springer, Berlin, 1985, pp. 173–200. MR**797790**, 10.1007/BFb0075761**16.**G. Tenenbaum,*Cribler les entiers sans grand facteur premier*, Philos. Trans. Roy. Soc. London Ser. A**345**(1993), no. 1676, 377–384 (French, with English and French summaries). MR**1253499**, 10.1098/rsta.1993.0136

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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:
http://dx.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