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On the 2m-th power mean of Dirichlet L-functions with the weight of trigonometric sums

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Abstract

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a 0 + a 1 x + ... + a k x k is a k-degree polynomial with integral coefficients such that (p, a 0, a 1, ..., a k ) = 1, for any integer m, we study the asymptotic property of

$$ \sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}} {p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } , $$

where e(y) = e2πiy. The main purpose is to use the analytic method to study the 2m-th power mean of Dirichlet L-functions with the weight of the general trigonometric sums and give an interesting asymptotic formula. This result is an extension of the previous results.

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References

  1. Carlitz L and Uchiyama S, Bounds for exponential sums, Duke Math. J. 24(1) (1957) 37–41

    Article  MATH  MathSciNet  Google Scholar 

  2. Davenport H, On certain exponential sums, J. Reinc u. Angew. Math. 169 (1933) 158–176

    Google Scholar 

  3. Harman Glyn, Wait Nigel and Wong Kam, A new mean-value result for Dirichlet L-functions and polynomials, Quart. J. Math. 55 (2004) 307–324

    Article  MATH  Google Scholar 

  4. Hua L K and Min S H, On a double exponential sum, Science Record 1 (1942) 23–25

    MathSciNet  Google Scholar 

  5. Hua L K, On an exponential sum, J. Chinese Math. Soc. 2 (1940) 301–312

    MathSciNet  Google Scholar 

  6. Hua L K, On exponential sums over an algebraic field, Canadian J. Math. 3 (1951) 44–51

    MATH  MathSciNet  Google Scholar 

  7. Min S H, On systems of algebraic equations and certain multiple exponential sums, Quart. J. Math. Oxford 18 (1947) 133–142

    Article  MathSciNet  Google Scholar 

  8. Mordell L J, On a sum analogous to a Gauss’s sum, Quart. J. Math. Oxford 3 (1932) 161–167

    Article  Google Scholar 

  9. Pan C D and Pan C B, Element of the Analytic Number Theory (Beijing: Science Press) (1991) (in Chinese)

    Google Scholar 

  10. Ren G, On the mean value of a generalized Dedekind sums with the weight of Hurwitz zeta-functions, Pure Appl. Math. 19(01) (2003) 23–25

    Google Scholar 

  11. Stankus E, Mean value of Dirichlet L-functions in the critical strip, Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys) vol. 31, no. 4, pp. 678–686 (October–December 1991)

    MATH  MathSciNet  Google Scholar 

  12. Tom MA, Introduction to Analytic Number Theory (New York: Springer-Verlag) (1976)

    MATH  Google Scholar 

  13. Weil A, On some exponential sums, Proc. Nat. Acad. Sci. USA 34 (1948) 204–207

    Article  MATH  MathSciNet  Google Scholar 

  14. Yi Y and Zhang WP, On the 2k-th power mean of Dirichlet L-functions with the weight of Gauss sums, Adv. Math. 31(6) 517–526 (2002)

    MathSciNet  Google Scholar 

  15. Zhang W P, Yi Y and He X L, On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, J. Number Theory 84 (2000) 199–213

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Science, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China

    Rong Ma & Junhuai Zhang

  2. The School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China

    Yulong Zhang

Authors
  1. Rong Ma
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  2. Junhuai Zhang
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  3. Yulong Zhang
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Correspondence to Rong Ma.

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Ma, R., Zhang, J. & Zhang, Y. On the 2m-th power mean of Dirichlet L-functions with the weight of trigonometric sums. Proc Math Sci 119, 411–421 (2009). https://doi.org/10.1007/s12044-009-0046-8

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  • DOI: https://doi.org/10.1007/s12044-009-0046-8

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