With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in pure math almost never mention in books
Could you give a application of a special function on number theory or analysis?
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1Hmmm... don't you consider zeta:[$\zeta$](http://mathworld.wolfram.com/RiemannZetaFunction.html) as a special function? It is usually associated with [$\Gamma$](http://en.wikipedia.org/wiki/Gamma_function) and [$\theta$](http://en.wikipedia.org/wiki/Theta_function) – 2012-04-15
2 Answers
The zeta-function and the gamma-function, mentioned in the comments, are ubiquitous in analytic number theory. Bessel functions come in as well, e.g., if the partial quotients of the continued fraction of the real number $x$ form an arithmetic progression, then $x$ can be expressed in terms of Bessel functions. The theory of partitions gives us Dedekind's eta-function and serves as an introduction to modular forms and modular functions. Clausen functions, polylogarithms, hypergeometric functions ... number theory is full of special functions. They're out there --- just look a little harder!
Apart from the exceedingly ubiquitous $\zeta(s)$ in analytic number theory (as already mentioned by Gerry and others), a more mundane example is the logarithmic integral, $\mathrm{li}(x)=\mathrm{PV}\int_0^x\frac{\mathrm du}{\ln\,u}$, which turns up as one of the better estimates for describing the behavior of the prime counting function $\pi(x)$. It also turns up in Riemann's refined estimate
$R(x)=\sum_{k=1}^\infty \frac{\mu(k)}{k}\mathrm{li}(\sqrt[k]{x})$
There are a lot of fancier examples; just keep looking!