On the 2 k -th power mean of L ' L ( 1 , χ ) with the weight of Gauss sums

Dongmei Ren; Yuan Yi

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 781-789
  • ISSN: 0011-4642

Abstract

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The main purpose of this paper is to study the hybrid mean value of L ' L ( 1 , χ ) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value ∑ χ ≠ χ 0 | τ ( χ ) | | L ' L ( 1 , χ ) | 2 k of L ' L and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.

How to cite

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Ren, Dongmei, and Yi, Yuan. "On the $2k$-th power mean of $\frac{L^{\prime }}{L}(1,\chi )$ with the weight of Gauss sums." Czechoslovak Mathematical Journal 59.3 (2009): 781-789. <http://eudml.org/doc/37958>.

@article{Ren2009,
abstract = {The main purpose of this paper is to study the hybrid mean value of $\frac\{L^\{\prime \}\}\{L\}(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum _\{\chi \ne \chi _0\} |\tau (\chi )| |\frac\{L^\{\prime \}\}\{L\}(1,\chi )|^\{2k\}$ of $\frac\{L^\{\prime \}\}\{L\}$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.},
author = {Ren, Dongmei, Yi, Yuan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dirichlet L-function; Gauss sums; asymptotic formula; Dirichlet L-function; Gauss sum; asymptotic formula},
language = {eng},
number = {3},
pages = {781-789},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the $2k$-th power mean of $\frac\{L^\{\prime \}\}\{L\}(1,\chi )$ with the weight of Gauss sums},
url = {http://eudml.org/doc/37958},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Ren, Dongmei
AU - Yi, Yuan
TI - On the $2k$-th power mean of $\frac{L^{\prime }}{L}(1,\chi )$ with the weight of Gauss sums
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 781
EP - 789
AB - The main purpose of this paper is to study the hybrid mean value of $\frac{L^{\prime }}{L}(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum _{\chi \ne \chi _0} |\tau (\chi )| |\frac{L^{\prime }}{L}(1,\chi )|^{2k}$ of $\frac{L^{\prime }}{L}$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.
LA - eng
KW - Dirichlet L-function; Gauss sums; asymptotic formula; Dirichlet L-function; Gauss sum; asymptotic formula
UR - http://eudml.org/doc/37958
ER -

References

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  1. Apostol, Tom M., Introduction to Analytic Number Theory, Springer-Verlag, New York (1976), 160-162. (1976) Zbl0335.10001MR0434929
  2. Chengdong, Pan, Chengbiao, Pan, Element of the Analytic Number Theory, Science Press, Beijing (1991), 243-248. (1991) 
  3. Ireland, K., Rosen, M., A Classical Introduction to Modern Number Theory, Springer-Verlag, New York (1982), 88-91. (1982) Zbl0482.10001MR0661047
  4. Yuan, Yi, Wenpeng, Zhang, On the first power mean of Dirichlet L-functions with the weight of Gauss sums, Journal of Systems Science and Mathematical Sciences 20 (2000), 346-351. (2000) MR1790030
  5. Yuan, Yi, Wenpeng, Zhang, On the 2 k -th Power mean of Dirichlet L-function with the weight of Gauss sums, Advances in Mathematics 31 (2002), 517-526. (2002) MR1959549
  6. Davenport, H., Multiplicative Number Theory, Springer-Verlag, New York (1980). (1980) Zbl0453.10002MR0606931
  7. Wenpeng, Zhang, A new mean value formula of Dirichlet's L-function, Science in China (Series A) 35 (1992), 1173-1179. (1992) MR1223064
  8. Huaning, Liu, Xiaobeng, Zhang, On the mean value of L ' L ( 1 , χ ) , Journal of Mathematical Analysis and Applications 320 (2006), 562-577. (2006) MR2225976
  9. Siegel, C. L., 10.4064/aa-1-1-83-86, Acta. Arith. 1 (1935), 83-86. (1935) Zbl0011.00903DOI10.4064/aa-1-1-83-86
  10. Chengdong, Pan, Chengbiao, Pan, The Elementary Number Theory, Peking University Press, Beijing (2003). (2003) 

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