We have using Perron,s formula an integral for M(x) Mertens function:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int_{c-i\infty}^{c+i\infty}ds\frac{x^{s}}{\zeta(s) s}= 2i\pi M(x) [/tex]

Of course we can,t calculate this integral directly, let,s put x=exp(t) and s=c+iu then our integral becomes:

[tex] \int_{-\infty}^{\infty}du \frac{e^{iut+ict}}{\zeta(c+iu)(c+iu)} [/tex]

now we could writte the complex integrand involving \zeta(c+iu)(c+iu) as a sum of a real and complex part U(c,u)+iV(c,u), so we should make an approach for big t--->oo of the integral:

[tex] \int_{-\infty}^{\infty}duU(c,u)e^{iut+ict} [/tex]

so we can calculate M(e^{t}) of course RH is true iff for big t the factor:

[tex] M(e^{t})e^{-t(1/2+e)} [/tex] 0<e<<<<<<1 (epsilon)

tends to 0, so we could give a "proof" of Riemann Hypothesis.

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# A trick to proof RH using asymptotic Analysis

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