The evaluation of the sum over arithmetic progressions for the coefficients of the Rankin-Selberg series

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  • ICHIHARA Yumiko
    DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF SCIENCE AND ENGINEERING WASEDA UNIVERSITY GRADUATE SCHOOL OF INFORMATION ENGINEERING HIROSHIMA UNIVERSITY

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  • evaluation of the sum over arithmetic progressions for the coefficients of the Rankin Selberg series
  • The Evaluation of the Sum over Arithmetic Progressions for the Coefficients of the Rankin-Selberg Series II

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We study where c n s are the coefficients of the Rankin-Selberg series, p is an odd prime, r is a natural number, and a is also a natural number satisfying (a, p) = 1. For any natural number d, we know the asymptotic formula for , where χ is a primitive Dirichlet character mod d. This is obtained by using the Voronoiformula of the Riesz-mean . In particular, in case d = p r,the fourth power of the Gauss sum appears in that Voronoiformula. We consider the sum over all characters mod pr, then the fourth power of the Gauss sum produces the hyper-Kloosterman sum. Hence, applying the results of Deligne and Weinstein, we can estimate the error term in the asymptotic formula for .

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