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There is no simple equation that given $n$ gives us $\gamma_n$, with sufficient approximation. Given a constant $c>0$, the error in both equations are at times greater than $c/\log n$ for any given c …

The function $$f(s):=\sin(\tfrac{\pi}{2}(s+1)) \zeta(s)+\sin(\tfrac{\pi s}{2}) L(\tfrac{s+1}{2})\tag{1}$$ where $L(s)$ is the second Dirichlet series $$L(s)=\sum\limits_{n=1}^{\infty } \frac{(-1)^{n-1 …

For the first question about $2\log\zeta(\sigma_0)|$, I think the reasoning is this: For $\sigma>2$ we have $$|L(s,\chi)-1|\le \sum_{n=2}^\infty\frac{1}{n^\sigma}=\zeta(\sigma)-1<1.$$ Hence $\log L(s, …

Assuming the Riemann hypothesis $\rho=\frac12+i\gamma$, then $$1-\frac{1}{\rho}=-\frac{\frac12-i\gamma}{\frac12+i\gamma}=e^{2i\theta},\qquad \theta=\arctan\frac{1}{2\gamma}.$$ $$\sum_\rho\Bigl[1-\Big …

You have also Guinand formula for $N(T)$, see, for example, in this answer https://mathoverflow.net/a/104570/7402

Your question is a little imprecise, because you use $\arg(\zeta(1/2+it))$ without explaining what value to take. I assume that you refer to $\pi S(t)$ as defined in the book of Titchmarsh (section 9. …

We know is that the difference between $\pi(x)$ y $\textrm{Li}(x)$ is less than $xe^{-c\sqrt{\log x}}$, and we have the asymptotic expansion $$\textrm{Li}(x)-\textrm{Li}((x+1)^2)+\textrm{Li}(x^2)=$ …

Assuming the Riemann Hypothesis Montgomery consider the function $$F(\alpha)=F(\alpha,T)=\Bigl(\frac{T}{2\pi}\log T\Bigr)^{-1} \sum_{0<\gamma, \gamma'\le T} T^{i\alpha(\gamma-\gamma')}w(\gamma-\gamma' …

In chapter 4, p. 15-18 of the book H. Iwaniec, Lectures on the Riemann Zeta Function, American Mathematical Society, University Lecture Series nº 62, 2014 there is an elementary proof of the prime …

In the paper: M. W. Coffey, "Asymptotic estimation of $\xi^{(2n)}(1/2)$: On a conjecture of Farmer and Rhoades", Mathematics of Computation, {\bf 78} (2009) 1147--1154 you may find the first terms o …

This has almost nothing to do with the zeros of zeta. To show it you can substitute $\frac{\Im(\rho_n)}{2\pi}$ in your program with an adequate approximation. Since the number of zeros of zeta $$N( …

If $\varphi$ is in the class of Schwartz we have $$\int_{-\infty}^{+\infty}\varphi(t)\zeta(\frac12+it)\,dt= \sum_{n=0}^\infty\Bigl\{ \frac{1}{\sqrt{n+1}}\widehat{\varphi}\Bigl(\frac{1}{2\pi}\log (n+1) …

I think it is false: Consider a simple series with a zero at $s=2$. For example $$E(s)=1-\frac{1}{2^s}-\frac{12}{4^s}=P(2^{-s}),\quad \text{with} \quad P(x)=1-x-12x^2.$$ We have $E(2)=0$ and $E(1)=- …

The Riemann hypothesis implies that the function $\Xi(z)=\xi(1/2+iz)$ is in the Laguerre-Pólya class. Therefore it is a limit, uniformly on compact sets, of a sequence of polynomials with real roots. …

The formula is given by Guinand as said in his answer by Matthew Watkins but it can be found in page 111 of the paper and reads: assuming Riemann Hypothesis and $T>0$ $$\frac12(N(T+0)+N(T-0))=\frac{T} …