1 § XI.1 Pólya-Vinogradov inequality and a generalization. Character sums modulo a prime power. Burgess’ estimate
- 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.
D.A. Burgess. On character sums and L-series. II. Proc. London Math. Soc. (3), 13 (1963), 524–536.
- a)
- 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.
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- a)
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(2006). Character Sums. In: Handbook of Number Theory I. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3658-2_11
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