Tau Function

A function $\tau(n)$ related to the Divisor Function $\sigma_k(n)$, also sometimes called Ramanujan's Tau Function. It is given by the Generating Function

$$\sum_{n=1}^\infty \tau(n)x^n = \prod_{n=1}^\infty (1-x^n)^{24},$$ (1)

and the first few values are 1, $-24$, 252, $-1472$, 4380, ... (Sloane's A000594). $\tau(n)$ is also given by

$$g(-x)=\sum_{n=1}^\infty (-1)^n\tau(n)x^n$$ (2)

$$g(x^2)=\sum_{n=1}^\infty \tau({\textstyle{1\over 2}}n)x^n$$ (3)

$$\sum_{n=1}^\infty \tau(n)x^n = x(1-3x+5x^3-7x^6+\ldots)^8.$$ (4)

In Ore's Conjecture, the tau function appears as the number of Divisors of $n$. Ramanujan conjectured and Mordell proved that if $(n,n')=1$, then

$$\tau(nn')=\tau(n)\tau(n').$$ (5)

Ramanujan conjectured and Watson proved that $\tau(n)$ is divisible by 691 for almost all $n$. If

$$\tau(p)\equiv 0\ \left({{\rm mod\ } {p}}\right),$$ (6)

then

$$\tau(pn)\equiv 0\ \left({{\rm mod\ } {p}}\right).$$ (7)

Values of $p$ for which the first equation holds are $p=2$, 3, 5, 7, 23.

Ramanujan also studied

$$f(x)\equiv\sum_{n=1}^\infty \tau(n)n^{-s},$$ (8)

which has properties analogous to the Riemann Zeta Function. It satisfies

$${f(s)\Gamma(s)\over (2\pi)^s} = {f(12-s)\over (2\pi)^{12-s}},$$ (9)

and Ramanujan's Tau-Dirichlet Series conjecture alleges that all nontrivial zeros of $f(s)$ lie on the line $\Re[s]=6$. $f$ can be split up into

$$f(6+it)=z(t)e^{-i\theta(t)},$$ (10)

where

$$z(t) = \Gamma(6+it)f(6+it)(2\pi)^{-it}\sqrt{\sinh(\pi t)\over \pi t(1+t^2)(4+t^2)(9+t^2)(16+t^2)(25+t^2)}$$ (11)

$$\theta(t) = -{\textstyle{1\over 2}}i\ln\left[{\Gamma(6+it)\over\Gamma(6-it)}\right]-t\ln(2\pi).$$ (12)

The Summatory tau function is given by

$$T(n)=\setbox0=\hbox{$\scriptstyle{n\leq x}$}\setbox2=\hbox{$... ...ern\dimen0\fi\fi \mathop{{\sum}'}_{\kern-\wd4 n\leq x}\tau(n).$$ (13)

Here, the prime indicates that when $x$ is an Integer, the last term $\tau(x)$ should be replaced by ${\textstyle{1\over 2}}\tau(x)$.

Ramanujan's tau theta function $Z(t)$ is a Real function for Real $t$ and is analogous to the Riemann-Siegel Function $Z$. The number of zeros in the critical strip from $t = 0$ to $T$ is given by

$$N(t) = {\Theta(T) + \Im\{\ln[\tau_{DS}(6+iT)]\}\over\pi},$$ (14)

where $\Theta$ is the Riemann Theta Function and $\tau_{DS}$ is the Tau-Dirichlet Series, defined by

$$\tau_{DS}(s) \equiv \sum_{n=1}^\infty {\tau(n)\over n^s}.$$ (15)

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

Ramanujan's $\tau_z$ function is defined by

$$\tau_z(t)={\Gamma(6+it)(2\pi)^{-it}\over \tau_{DS}(6+it)\sqrt{\sinh(\pi t)\over \pi t \prod_{k=1}^5 k^2 + t^2}},$$ (16)

where $\tau_{DS}(z)$ is the Tau-Dirichlet Series.

See also Ore's Conjecture, Tau Conjecture, Tau-Dirichlet Series

References

Hardy, G. H. ``Ramanujan's Function $\tau(n)$.'' 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.