“Tau Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Tau Function

释义

Tau Function

A function related to the Divisor Function , also sometimes called Ramanujan's TauFunction. It is given by the Generating Function

(1)

and the first few values are 1,

, 252,

, 4380, ... (Sloane's A000594).

is also given by

(2)

(3)

(4)

In Ore's Conjecture, the tau function appears as the number of Divisors of .Ramanujan conjectured and Mordell proved that if , then

(5)

Ramanujan

conjectured and Watson proved that

is divisible by 691 for almost all

. If

(6)

then

(7)

Values of

for which the first equation holds are

, 3, 5, 7, 23.

Ramanujan also studied

(8)

which has properties analogous to the Riemann Zeta Function. It satisfies

(9)

and Ramanujan's

Tau-Dirichlet Series conjecture alleges that all nontrivial zeros of

lie on the line

can be split up into

(10)

where

			(11)
			(12)

The Summatory tau function is given by

(13)

Here, the prime indicates that when

is an Integer, the last term

should be replaced by

Ramanujan's tau theta function is a Real function for Real and isanalogous to the Riemann-Siegel Function . The number of zeros in the critical stripfrom to is given by

(14)

where

is the Riemann Theta Function and

is the Tau-Dirichlet Series, defined by

(15)

Ramanujan

conjectured that the nontrivial zeros of the function are all real.

Ramanujan's function is defined by

(16)

where

is the Tau-Dirichlet Series.

See also Ore's Conjecture, Tau Conjecture, Tau-Dirichlet Series

References

Hardy, G. H. ``Ramanujan's Function .'' Ch. 10 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959.

Sloane, N. J. A. SequenceA000594/M5153in ``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.

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