“Dirichlet L-Series”的概念、定义、习题解题方法及翻译-数学辞典

单词

Dirichlet L-Series

释义

Dirichlet L-Series

Series of the form

(1)

where the Character (Number Theory)

is an Integer function with period

. These series appear innumber theory (they were used, for instance, to prove Dirichlet's Theorem) and can be written as sums ofLerch Transcendents with

a Power of

. The Dirichlet EtaFunction

(2)

(for

) and Dirichlet Beta Function

(3)

and Riemann Zeta Function

(4)

are Dirichlet series (Borwein and Borwein 1987, p. 289).

is called primitive if the Conductor

.Otherwise,

is imprimitive. A primitive

-series modulo

is then defined as one for which

is primitive. All imprimitive

-series can be expressed in terms of primitive

-series.

Let or , where are distinct Odd Primes. Then there are three possible types ofprimitive -series with Real Coefficients. The requirement ofReal Coefficients restricts the Characterto 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

(Zucker and Robertson 1976). All primitive

-series are algebraically independent and divide into two types according to

(5)

Primitive

-series of these types are denoted

. For a primitive

-series with RealCharacter (Number Theory), if

, then

(6)

, then

(7)

and if

, then there is a primitive function of each type (Zucker and Robertson 1976).

The first few primitive Negative -series are , , , , , , ,, , , , , , , , , , ,, , , , , , , , , , ,, , ... (Sloane's A003657), corresponding to the negated discriminants of imaginary quadratic fields. Thefirst few primitive Positive -series are , , , , , , ,, , , , , , , , , , ,, , , , , , , , , , ,, , ... (Sloane's A046113).

The Kronecker Symbol is a Real Character modulo , and isin 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)

where

is the Class Number and

is the Dirichlet Structure Constant. Some specific valuesof primitive

-series are

No general forms are known for

and

in terms of known transcendentals. For example,

(19)

where

is defined as Catalan's Constant.

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. SequencesA046113 andA003657/M2332in ``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.

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