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

- 1. If (e.g., , 3, 5, ...) or (e.g., , 12, 20, ...), there is exactly one primitive -series.
- 2. If (e.g., , 24, ...), there are two primitive -series.
- 3. If , or where (e.g., , 6, 9, ...), there are no primitive -series

(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) |

where is the Kronecker Symbol. The functional equations for are

(10) | |||

(11) |

For a Positive Integer

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) | |||

(17) |

where and are Rational Numbers. can be expressed in terms of transcendentals by

(18) |

No general forms are known for and in terms of known transcendentals. For example,

(19) |

where is defined as Catalan's Constant.

**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

1999-05-24