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In mathematics, a character sum is a sum : of values of a Dirichlet character χ ''modulo'' ''N'', taken over a given range of values of ''n''. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue ''modulo'' ''N''. Character sums are often closely linked to exponential sums by the Gauss sums (this is like a finite Mellin transform). Assume χ is a nonprincipal Dirichlet character to the modulus ''N''. ==Sums over ranges== The sum taken over all residue classes mod ''N'' is then zero. This means that the cases of interest will be sums over relatively short ranges, of length ''R'' < ''N'' say, : A fundamental improvement on the trivial estimate is the Pólya–Vinogradov inequality (George Pólya, I. M. Vinogradov, independently in 1918), stating in big O notation that : Assuming the generalized Riemann hypothesis, Hugh Montgomery and R. C. Vaughan have shown〔Montgomery and Vaughan (1977)〕 that there is the further improvement : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「character sum」の詳細全文を読む スポンサード リンク
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