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August 24th, 2019, 12:01 PM   #1
Joined: Aug 2019
From: spain

Posts: 2
Thanks: 1

Generalization of Riemann-weil formula for arithmetic functions

$\displaystyle \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\gamma}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} </math>$

$\displaystyle \sum_{n=1}^{\infty} \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\gamma}\frac{h( \gamma)\zeta(2 \rho )}{\zeta '( \rho)}+ \frac{1}{\zeta (1/2)}\int_{-\infty}^{\infty}dx g(x)$

$\displaystyle \sum_{n=1}^{\infty} \frac{\varphi (n)}{\sqrt{n}}g(\log n)= \frac{6}{\pi ^2} \int_{-\infty}^{\infty}dx g(x)e^{3x/2}+ \sum_{\gamma}\frac{h( \gamma)\zeta(\rho -1 )}{\zeta '( \rho)}+ \frac{1}{2}\sum_{n=1}^{\infty} \frac{\zeta (-2n-1)}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dx g(x)e^{-x(2n+1/2)} $

think this could be interesting is based on a idea by myself could someone help me to get a Ph D thesis based on this ?
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