Series of the form
(1) |
(2) |
(3) |
(4) |
Let or , where are distinct Odd Primes. Then there are three possible types of primitive -series with Real Coefficients. The requirement of Real Coefficients restricts the Character to for all and . The three type are then
(5) |
(6) |
(7) |
The first few primitive Negative -series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A003657), corresponding to the negated discriminants of imaginary quadratic fields. The first few primitive Positive -series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A046113).
The Kronecker Symbol is a Real Character modulo , and is
in fact essentially the only type of Real primitive Character
(Ayoub 1963). Therefore,
(8) | |||
(9) |
(10) | |||
(11) |
(12) | |||
(13) | |||
(14) | |||
(15) | |||
(16) | |||
(17) |
(18) |
(19) |
See also Dirichlet Beta Function, Dirichlet Eta Function
References
Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Buell, D. A. ``Small Class Numbers and Extreme Values of -Functions of Quadratic Fields.'' Math. Comput. 139, 786-796, 1977.
Ireland, K. and Rosen, M. ``Dirichlet -Functions.'' Ch. 16 in A Classical Introduction to Modern Number Theory, 2nd ed.
New York: Springer-Verlag, pp. 249-268, 1990.
Sloane, N. J. A. Sequences
A046113 and
A003657/M2332
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
Weisstein, E. W. ``Class Numbers.'' Mathematica notebook ClassNumbers.m.
Zucker, I. J. and Robertson, M. M. ``Some Properties of Dirichlet -Series.'' J. Phys. A: Math. Gen. 9, 1207-1214, 1976.
© 1996-9 Eric W. Weisstein