Averages of values of series
Author:
Emre Alkan
Journal:
Proc. Amer. Math. Soc. 141 (2013), 11611175
MSC (2010):
Primary 11M06, 11L05
Published electronically:
August 28, 2012
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We obtain an exact formula for the average of values of series over two independent odd characters. The average of any positive moment of values at is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of series at and the average size of positive moments of character sums related to the class number are deduced.
 1.
Emre
Alkan, On Dirichlet 𝐿functions with periodic coefficients
and Eisenstein series, Monatsh. Math. 163 (2011),
no. 3, 249–280. MR 2805873
(2012e:11151), http://dx.doi.org/10.1007/s0060501002112
 2.
Emre
Alkan, On the mean square average of special values of
𝐿functions, J. Number Theory 131 (2011),
no. 8, 1470–1485. MR
2793888, http://dx.doi.org/10.1016/j.jnt.2011.02.013
 3.
Emre
Alkan, Values of Dirichlet 𝐿functions, Gauss sums and
trigonometric sums, Ramanujan J. 26 (2011),
no. 3, 375–398. MR
2860694, http://dx.doi.org/10.1007/s1113901092928
 4.
Emre
Alkan, Florin
Stan, and Alexandru
Zaharescu, Lehmer 𝑘tuples, Proc. Amer. Math. Soc. 134 (2006), no. 10, 2807–2815 (electronic).
MR
2231602 (2007d:11090), http://dx.doi.org/10.1090/S000299390608484X
 5.
Bruce
C. Berndt and Alexandru
Zaharescu, Finite trigonometric sums and class numbers, Math.
Ann. 330 (2004), no. 3, 551–575. MR 2099193
(2005f:11173), http://dx.doi.org/10.1007/s0020800405595
 6.
Harold
Davenport, Multiplicative number theory, 3rd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000.
Revised and with a preface by Hugh L. Montgomery. MR 1790423
(2001f:11001)
 7.
Huaning
Liu and Wenpeng
Zhang, On the mean value of
𝐿(𝑚,𝜒)𝐿(𝑛,\overline𝜒) at
positive integers 𝑚,𝑛≥1, Acta Arith.
122 (2006), no. 1, 51–56. MR 2217323
(2007a:11122), http://dx.doi.org/10.4064/aa12215
 8.
Ming
Gao Qi, A class of mean square formulas for
𝐿functions, J. Tsinghua Univ. 31 (1991),
no. 3, 34–41 (Chinese, with English summary). MR 1168609
(93g:11090)
 9.
Zhefeng
Xu and Wenpeng
Zhang, Some identities involving the Dirichlet
𝐿function, Acta Arith. 130 (2007),
no. 2, 157–166. MR 2357653
(2008j:11108), http://dx.doi.org/10.4064/aa13025
 10.
Wen
Peng Zhang, A formula for quartic mean values of the
𝐿function, Kexue Tongbao (Chinese) 34
(1989), no. 9, 647–650 (Chinese). MR 1020426
(90j:11087)
 11.
Wen
Peng Zhang, On the fourth power mean of Dirichlet
𝐿functions, Lecture notes in contemporary mathematics, 1989,
Science Press, Beijing, 1990, pp. 173–179. MR 1180537
(93g:11091)
 12.
Wen
Peng Zhang, On the mean value of the 𝐿function, J.
Math. Res. Exposition 10 (1990), no. 3, 355–360
(Chinese, with English summary). MR 1072441
(91i:11108)
 13.
Wen
Peng Zhang, A note on a class of mean square values of
𝐿functions, J. Northwest Univ. 20 (1990),
no. 3, 9–12 (Chinese, with English summary). MR 1077163
(91j:11068)
 14.
Wen
Peng Zhang, On the general Dedekind sums and one kind identities of
Dirichlet 𝐿functions, Acta Math. Sinica (Chin. Ser.)
44 (2001), no. 2, 269–272 (Chinese, with
English and Chinese summaries). MR 1831528
(2002c:11106)
 1.
 E. Alkan, On Dirichlet functions with periodic coefficients and Eisenstein series, Monatsh. Math. 163 (2011), 249280. MR 2805873
 2.
 E. Alkan, On the mean square average of special values of functions, J. Number Theory 131 (2011), 14701485. MR 2793888
 3.
 E. Alkan, Values of Dirichlet functions, Gauss sums and trigonometric sums, Ramanujan J. 26, no. 3 (2011), 375398. MR 2860694
 4.
 E. Alkan, F. Stan, A. Zaharescu, Lehmer tuples, Proc. Amer. Math. Soc. 134 (2006), 28072815. MR 2231602 (2007d:11090)
 5.
 B. C. Berndt, A. Zaharescu, Finite trigonometric sums and class numbers, Math. Ann. 330 (2004), 551575. MR 2099193 (2005f:11173)
 6.
 H. Davenport, Multiplicative Number Theory, Third Edition, Graduate Texts in Mathematics 74, SpringerVerlag, 2000. MR 1790423 (2001f:11001)
 7.
 H. Liu, W. P. Zhang, On the mean value of at positive integers , Acta Arith. 122 (2006), 5156. MR 2217323 (2007a:11122)
 8.
 M. G. Qi, A class of mean square formulas for functions, J. Tsinghua Univ. 31 (1991), 3441. MR 1168609 (93g:11090)
 9.
 Z. Xu, W. P. Zhang, Some identities involving the Dirichlet function, Acta Arith. 130 (2007), 157166. MR 2357653 (2008j:11108)
 10.
 W. P. Zhang, A formula for quartic mean values of the function, Kexue Tongbao 34 (1989), 647650. MR 1020426 (90j:11087)
 11.
 W. P. Zhang, On the fourth power mean of Dirichlet functions, Lecture Notes in Contemporary Mathematics (1989), 173179, Science Press, Beijing, 1990. MR 1180537 (93g:11091)
 12.
 W. P. Zhang, On the mean value of the function, J. Math. Res. Exposition 10 (1990), 355360. MR 1072441 (91i:11108)
 13.
 W. P. Zhang, A note on a class of mean square values of functions, J. Northwest Univ. 20 (1990), 912. MR 1077163 (91j:11068)
 14.
 W. P. Zhang, On the general Dedekind sums and one kind identities of Dirichlet functions, Acta Math. Sinica (Chin. Ser.) 44 (2001), 269272. MR 1831528 (2002c:11106)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11M06,
11L05
Retrieve articles in all journals
with MSC (2010):
11M06,
11L05
Additional Information
Emre Alkan
Affiliation:
Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sarıyer, Istanbul, Turkey
Email:
ealkan@ku.edu.tr
DOI:
http://dx.doi.org/10.1090/S000299392012115060
PII:
S 00029939(2012)115060
Keywords:
$L$series,
special values,
Jordan’s totient function,
Euler’s totient function,
averages,
Gauss sums,
cotangent sums
Received by editor(s):
August 9, 2011
Received by editor(s) in revised form:
August 15, 2011
Published electronically:
August 28, 2012
Additional Notes:
The author is supported by the Distinguished Young Scholar Award, TübaGebip of Turkish Academy of Sciences
Dedicated:
Dedicated to the memory of Professor Cemal Koç
Communicated by:
Ken Ono
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
