 My Math Forum Generalization of Riemann-weil formula for arithmetic functions
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 August 24th, 2019, 12:01 PM #1 Newbie   Joined: Aug 2019 From: spain Posts: 2 Thanks: 1 Generalization of Riemann-weil formula for arithmetic functions $\displaystyle \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\gamma}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} [/itex]$ $\displaystyle \sum_{n=1}^{\infty} \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\gamma}\frac{h( \gamma)\zeta(2 \rho )}{\zeta '( \rho)}+ \frac{1}{\zeta (1/2)}\int_{-\infty}^{\infty}dx g(x)$ $\displaystyle \sum_{n=1}^{\infty} \frac{\varphi (n)}{\sqrt{n}}g(\log n)= \frac{6}{\pi ^2} \int_{-\infty}^{\infty}dx g(x)e^{3x/2}+ \sum_{\gamma}\frac{h( \gamma)\zeta(\rho -1 )}{\zeta '( \rho)}+ \frac{1}{2}\sum_{n=1}^{\infty} \frac{\zeta (-2n-1)}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dx g(x)e^{-x(2n+1/2)}$ 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 ? Tags arithmetic, formula, functions, generalization, riemannweil Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post raul21 Number Theory 6 May 24th, 2014 09:50 AM raul21 Number Theory 2 May 23rd, 2014 10:04 AM mhss12345 Number Theory 1 November 2nd, 2012 11:59 AM miller Abstract Algebra 0 August 20th, 2010 12:11 AM Jalaska13 Number Theory 0 May 30th, 2010 03:30 PM

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