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We know that Wilson's theorem states the following : $x$ is a prime if $(\frac {\Gamma(x)+1}{x})$ is an integer . We can extend this to Twin primes as : $x$ and $x+2$ is prime if $(\frac {4(\Gamma( …

Cramér proved the following theorem (see the announcement in [1] and [2]): Consider the following function: $$V(z)=\sum_k e^{\rho_kz}$$ Where $\rho_k$ runs through non trivial zeta zeros with $Im(\r …

This is a straight forward question so apologies in advance Consider the following sums: $$\sum_k1_{\rho_k}$$ $$\sum_k{\rho_k}$$ (i.e. first sum counts non trivial zeros of Zeta function) I want …

Consider the analytic function $g(x)$ Define $$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$$ Note that: $$f(p)=g(p) \text{ for prime } p$$ And $f(n)=0$ fo …

Consider the following lacunary sum with parameter $x$: $$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$ As we can see for $x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\fra …

I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following : The Rie …

Consider the following line of thinking: $$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$ Here, $\operatorname{R}(x) = \sum …

Consider the analytic function $g(x)$ Now define $f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$ Such that $|f(x+it)|=o(e^{2πt})$ uniformly for every $x …