Frontiers of Mathematics in China >
Multiplication formulas for Kubert functions
Received date: 21 Oct 2013
Accepted date: 25 Nov 2013
Published date: 01 Feb 2014
Copyright
The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of (generalized) Kubert functions. Second, we apply the special case of Carlitz’s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz’s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz (for sums) and Mordell (for integrals), both of which are generalizations of preceding results by Frasnel, Landau, Mikolás, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck’s lamma is the same as Carlitz’s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.
Hailong LI , Jing MA , Yuichi URAMATSU . Multiplication formulas for Kubert functions[J]. Frontiers of Mathematics in China, 2014 , 9(1) : 101 -109 . DOI: 10.1007/s11464-013-0348-0
| 1 |
Apostol T M. Mathematical Analysis. Reading: Addison-Wesley Publishing Company, Inc, 1957
|
| 2 |
Apostol T M, Vu T H. Identities for sums of Dedekind type. J Number Theory, 1982, 14: 391-396
|
| 3 |
Beck M. Dedekind cotangent sums. Acta Arith, 2003, 109: 109-130
|
| 4 |
Berndt B C. A generalization of a theorem of Gauss on sums involving [x]. Amer Math Monthly, 1973, 82: 44-51
|
| 5 |
Carlitz L. Some finite summation formulas of arithmetic character. Publ Math Debrecen, 1959, 6: 262-268
|
| 6 |
Espinosa O, Moll V. On some integrals involving the Hurwitz zeta-function: Part 1. Ramanujan J, 2002, 6: 159-188
|
| 7 |
Espinosa O, Moll V. On some integrals involving the Hurwitz zeta-function: Part 2. Ramanujan J, 2002, 6: 449-468
|
| 8 |
Franel J. Les suites de Farey et le probl`eme des nombres premiers. Nachr Math Ges Wiss Gtingen, 1924: 198-201
|
| 9 |
Fukuhara S, Yui N. A generating function for higher-dimensional Apostol-Zagier sums and their reciprocity law. J Number Theory, 2006, 117: 87-105
|
| 10 |
Hashimoto M, Kanemitsu S, Li H L. Examples of the Hurwitz transform. J Math Soc Japan, 2009, 61: 651-660
|
| 11 |
Kanemitsu S, Li H L, Ma J. Some results on Kubert functions (in preparation)
|
| 12 |
Kanmeitsu S, Ma J, Zhang W P. On the discrete mean value of the product of two Dirichlet L-functions. Abh Math Sem Univ Hamburg, 2009, 79: 149-164
|
| 13 |
Kanemitsu S, Tsukada H. Vistas of Special Functions. Singapore-London-New York: World Scientific, 2007
|
| 14 |
Katsurada M, Matsumoto K. The mean values of Dirichlet L-function at integer points and class numbers of cyclotomic fields. Nagoya Math J, 1994, 134: 151-172
|
| 15 |
Knopp M. Hecke operators and an identity for Dedekind sums. J Number Theory, 1980, 12: 2-9
|
| 16 |
Kozuka K. Dedekind type sums attached to Dirichlet characters. Kyushu J Math, 2004, 58: 1-24
|
| 17 |
Kozuka K. Some identities for multiple Dedekind sums attached to Dirichlet characters. Acta Arith, 2011, 146: 103-114
|
| 18 |
Landau E. Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel. Nachr Math Ges Wiss Göttingen, 1924: 202-206
|
| 19 |
Landau E. Vorlesungen über Zahlentheorie, B. II. Lepzig, 1927
|
| 20 |
Liu G D, Wang N L, Wang X H. The discrete mean square of Dirichlet L-function at integral arguments. Proc Japan Acad Ser A Math Sci, 2010, 86(9): 149-153
|
| 21 |
Maier H. Cyclotomic polynomials whose orders contain many prime factors. Period Math Hungar, 2001, 43: 155-164
|
| 22 |
Mikolás M. Mellinsche Transformation und Orthogonalität bei ζ(s, u). Verallgemeinrung der Riemannschen Funktionalgleichung von ζ(s). Acta Sci Math (Szeged), 1956, 17: 143-164
|
| 23 |
Mikolás M. Integral formulae of arithmetical characteristics relating to the zeta-function of Hurwitz. Publ Math Debrecen, 1957, 5: 44-53
|
| 24 |
Mikolás M. On certain sums generating the Dedekind sums. Pacific J Math, 1957, 7: 1167-1178
|
| 25 |
Milnor J. On polylogarithms, Hurwitz zeta-functions, and the Kubert identities Enseign Math, 1983, 29(2): 281-322
|
| 26 |
Mordell L J. Integral formulae of arithmetical character. J Lond Math Soc, 1958, 33: 371-375
|
| 27 |
Nagasaka C. Dedekind type sums and Hecke operators. Acta Arith, 1984, 44: 207-214
|
| 28 |
Nielsen N. Traitéélémentaire des nombres de Bernoulli. Paris: Gauther-Villars, 1923
|
| 29 |
Parson L A, Rosen K. Hecek operators and generalized Dedekind sum. Math Scand, 1981, 49: 5-14
|
| 30 |
Romanoff N P. Hilbert spaces and number theory II. Izv Akad Nauk SSSR, Ser Mat, 1951, 15: 131-152
|
| 31 |
Sun Z W. On covering equivalence. In: Jia Ch-H,Matsumoto K, eds. Analytic Number Theory. Dordrecht: Kluwer, 2002, 277-302
|
| 32 |
Walum H. Multiplication formulae for periodic functions. Pacific J Math, 1991, 149(2): 383-396
|
| 33 |
Widder D W. Advanced Calculus. 2nd ed. New York: Dover Publ, 1989
|
| 34 |
Zheng Z. The Petersson-Knopp identity for the homogenous Dedekind sums. J Number Theory, 1996, 57: 223-230
|
/
| 〈 |
|
〉 |