Could you give a application of a special function on number theory or analysis?

Question

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

Answer 1

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!

Answer 2

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!