Large values of character sums

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Abstract

The occurrence of large values for the sums S(χ, x) = Σnx χ(n), where χ is a primitive character (mod q), is investigated. It is shown that the Pólya-Vinogradov bound O(√qlog q) for S(χ, x) is attained only very rarely, and a more precise bound that depends on rational approximations to xq is given. Moreover, improved values for the constant in the Pólya-Vinogradov inequality are obtained.

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Supported by NSF Grant DMS 8640693.