A function related to the Divisor Function , also sometimes called Ramanujan's Tau
Function. It is given by the Generating Function
(1) |
(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) |
(6) |
(7) |
Ramanujan also studied
(8) |
(9) |
(10) |
(11) | |||
(12) |
The Summatory tau function is given by
(13) |
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) |
(15) |
Ramanujan's function is defined by
(16) |
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. 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.
© 1996-9 Eric W. Weisstein