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Erschienen in:
2006 | OriginalPaper | Buchkapitel
Character Sums
Erschienen in: Handbook of Number Theory I
Verlag: Springer Netherlands
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Excerpt
1)
For χ any nonprincipal character modulo p (prime) and any positive integer x
a)
G. Pólya. Über die Verteilung der quadratische Reste und Nichtreste. Göttingen Nachrichten, 1918, 21–29 and I.M. Vinogradov. On the distribution of residues and non-residues of powers. Journal of the Physico-Mathematical Society of Perm. 1 (1918), 94–96.
Remark. Actually, one can establish the above inequality with the constant c=1
b)
where x and r are arbitrary positive integers and N is any integer.
where x and r are arbitrary positive integers and N is any integer.
D.A. Burgess. On character sums and L-series. II. Proc. London Math. Soc. (3), 13 (1963), 524–536.
2)
Let χ denote a primitive character modulo k. Write
a)
If r=1 or 2 then, for every ɛ>0,
b)
For any integer r>0, if k has non-trivial cubic factor then the estimate from a) holds. sp ]D.A. Burgess. On character sums and L-series. II. Proc. London Math. Soc. (3), 13 (1963), 524–536.
c)
Let r=3. Let ɛ>0. For every prime power k=pα, of every prime p>3, the estimate from a) holds (with the implied constant independent of both p and α).
D.A. Burgess. Estimation of character sums modulo a power of a prime. Proc. London Math. Soc. (3), 52 (1986), 215–235.
d)
Let r=3. Then the estimate from a) holds.
D.A. Burgess. The character sum estimate with r=3. J. London Math. (2), 33 (1986), 219–226.
3)
Let χ be a nonprincipal primitive even character modulo k. Then
for even n, where B n is the Bernoulli polynomial, and
is a generalized Bernoulli number. (Generalization of the Pólya-Vinogradov inequality.)
S. Kanemitsu and K. Shiratani. An application of the Bernoulli functions to character sums. Mem. Fac. Sci. Kyushu Univ. Ser. A. 30 (1976), 65–73.