Character Sums

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.

• 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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