Tau Dirichlet Series

Ramanujan's Dirichlet L-series is defined as

(1)

where  is the tau function. Note that the notation  is sometimes used instead of  (Hardy 1999, p. 164).

 has properties analogous to the Riemann zeta function, and is implemented as RamanujanTauL[s].

Ramanujan conjectured that all nontrivial zeros of  lie on the line .

 satisfies the functional equation

(2)

(Hardy 1999, p. 173) and has the Euler product representation

(3)

for  (since ) (Apostol 1997, p. 137; Hardy 1999, p. 164).

 can be split up into

(4)

where

			(5)
			(6)

The functions , and  are returned by the Wolfram Language commands RamanujanTauTheta[t] and RamanujanTauZ[t], respectively.

Ramanujan's tau -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

(7)

where  is the Ramanujan theta function. Ramanujan conjectured that the nontrivial zeros of the function are all real.

Ramanujan's  function is defined by

(8)

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 -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.