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) |

**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

1999-05-26