## Tau Function

A function related to the Divisor Function , also sometimes called Ramanujan's Tau Function. 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 is analogous to the Riemann-Siegel Function . The number of zeros in the critical strip from 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.

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. Sequence A000594/M5153 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.