|
Large character sums
Authors:
Andrew Granville and K. Soundararajan
Journal:
J. Amer. Math. Soc. 14 (2001), 365-397
MSC (2000):
Primary 11L40; Secondary 11N25
Posted:
October 20, 2000
MathSciNet review:
1815216
Full-text PDF Free Access
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
(13,113d), http://dx.doi.org/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
(20 #28)
- 3.
Harold
Davenport, Multiplicative number theory, 2nd ed., Graduate
Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980.
Revised by Hugh L. Montgomery. MR 606931
(82m:10001)
- 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
(95f:11058), http://dx.doi.org/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
(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.
Adolf
Hildebrand, A note on Burgess’ character sum estimate,
C. R. Math. Rep. Acad. Sci. Canada 8 (1986), no. 1,
35–37. MR
827113 (87e:11095)
- 10.
Adolf
Hildebrand and Gérald
Tenenbaum, Integers without large prime factors, J.
Théor. Nombres Bordeaux 5 (1993), no. 2,
411–484. MR 1265913
(95d:11116)
- 11.
Hugh
L. Montgomery, An exponential polynomial formed with the Legendre
symbol, Acta Arith. 37 (1980), 375–380. MR 598890
(82a:10041)
- 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, 1994. MR 1297543
(96i:11002)
- 13.
H.
L. Montgomery and R.
C. Vaughan, Exponential sums with multiplicative coefficients,
Invent. Math. 43 (1977), no. 1, 69–82. MR 0457371
(56 #15579)
- 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
(87b:11095), http://dx.doi.org/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 (95d:11119), http://dx.doi.org/10.1098/rsta.1993.0136
- 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
Similar Articles
Retrieve articles in Journal of the American Mathematical Society
with MSC (2000):
11L40,
11N25
Retrieve articles in all journals
with MSC (2000):
11L40,
11N25
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
PII:
S 0894-0347(00)00357-X
Received by editor(s):
March 29, 1999
Received by editor(s) in revised form:
September 8, 2000
Posted:
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
|
