Tau Dirichlet Series
Ramanujan's Dirichlet L-series is defined as
where tau(n) is the tau function. Note that the notation F(s) is sometimes used instead of f(s) (Hardy 1999, p. 164).
f(s) has properties analogous to the Riemann zeta function, and is implemented as RamanujanTauL [s].
Ramanujan conjectured that all nontrivial zeros of f(s) lie on the line R[s]=6.
f(s) satisfies the functional equation
(Hardy 1999, p. 173) and has the Euler product representation
for sigma=R[s]>7 (since tau(n)=O(n^6)) (Apostol 1997, p. 137; Hardy 1999, p. 164).
f(s) can be split up into
| f(6+it)=z(t)e^(-itheta(t)), |
(4)
|
where
![画像:-1/2iln[(Gamma(6+it))/(Gamma(6-it))]-tln(2pi).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TauDirichletSeries/Inline16.svg){kind=link}
The functions theta(t), and z(t) are returned by the Wolfram Language commands RamanujanTauTheta [t] and RamanujanTauZ [t], respectively.
Ramanujan's tau Z-function z(t) is a real function for real t and is analogous to the Riemann-Siegel function Z(t). The number of zeros in the critical strip from t=0 to T is given by
![画像: N(t)=(Theta(T)+I{ln[f(6+iT)]})/pi,](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TauDirichletSeries/NumberedEquation5.svg){kind=link}
where Theta(z) is the Ramanujan theta function. Ramanujan conjectured that the nontrivial zeros of the function are all real.
Ramanujan's tau_z function is defined by
See also
Tau FunctionExplore with Wolfram|Alpha
More things to try:
References
Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, 1997.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Keiper, J. "On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip." Math. Comput. 65, 1613-1619, 1996.Spira, R. "Calculation of the Ramanujan Tau-Dirichlet Series." Math. Comput. 27, 379-385, 1973.Yoshida, H. "On Calculations of Zeros of L-Functions Related with Ramanujan's Discriminant Function on the Critical Line." J. Ramanujan Math. Soc. 3, 87-95, 1988.Referenced on Wolfram|Alpha
Tau Dirichlet SeriesCite this as:
Weisstein, Eric W. "Tau Dirichlet Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TauDirichletSeries.html