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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

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The $\Gamma$ function helps with functional equations of $L$-functions. Off the top of my head. – anon Apr 15 '12 at 21:27
What functions do you class as special functions? – fretty Apr 15 '12 at 21:42
Not to mention differential equations use all sorts of special functions, hypergeometric functions (which basically encompass a large swath of functions) especially. Theta and elliptic and gamma and hyperbolic functions are probably studied in most complex analysis books, and when intersected with number theory we have modular forms. – anon Apr 15 '12 at 22:15
Hmmm... don't you consider zeta:$\zeta$ as a special function? It is usually associated with $\Gamma$ and $\theta$ – Raymond Manzoni Apr 15 '12 at 22:36
up vote 5 down vote accepted

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!

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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!

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