Abstract: The Dirichlet series are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by . We obtain a formula for the exponential generating function of , where is an arbitrary positive integer. In particular, for , say, , where is square-free and , we show that can be expressed as a linear combination of the four functions , where is a nonnegative integer not exceeding , and with being a constant depending on . Moreover, the Dirichlet series can be easily computed from the generating function formula for . Finally, we show that the main ingredient in the formula for has a combinatorial interpretation in terms of the -alternating augmented -signed permutations defined by Ehrenborg and Readdy. More precisely, when is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers . When is not square-free, say , the numbers can be written as a linear combination of the numbers of -alternating augmented -signed permutations with integer coefficients, where .
1.
D. André, Sur les permutations alternées, J. Math. Pures Appl. 7 (1881), 167-184.
6.G.
H. Hardy, Ramanujan: twelve lectures on subjects suggested by his
life and work., Chelsea Publishing Company, New York, 1959. MR 0106147
(21 #4881)
7.Michael
E. Hoffman, Derivative polynomials, Euler polynomials, and
associated integer sequences, Electron. J. Combin. 6
(1999), Research Paper 21, 13 pp. (electronic). MR 1685701
(2000c:11027)
V.I. Arnol'd, The calculus of snakes and the combinatorics of Bernoulli, Euler, and Springer numbers of Coxeter groups, Uspekhi Mat. Nauk. 47 (1992), 3-45 (Russian); Russian Math. Surveys 47 (1992), 1-51. MR 1171862 (93h:20042)
P.T. Bateman and R.A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367. MR 0148632 (26:6139)
N. Metropolis, G.-C. Rota, V. Strehl and N. White, Partitions into chains of a class of partially ordered sets, Proc. Amer. Math. Soc.71 (1978), 193-196. MR 0551483 (58:27667)
M. Purtill, André permutations, lexicographic shellability and the -index of a convex polytope, Trans. Amer. Math. Soc. 338 (1993), 77-104. MR 1094560 (93j:52017)
D. Shanks, On the conjecture of Hardy and Littlewood concerning the number of primes of the form , Math. Comp. 14 (1960), 321-332. MR 0120203 (22:10960)
William Y. C. Chen Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
chen@nankai.edu.cn
Neil J. Y. Fan Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
fjy@cfc.nankai.edu.cn
Jeffrey Y. T. Jia Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email:
jyt@cfc.nankai.edu.cn