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This is not a completely satisfactory answer. I would like a simpler one. Nevertheless still probably a good exercise in Complex variables. I will only sketch it. What we want to show is equivalent t …

The asymptotic expansion of Cipolla starts $$p_n=n\log n+n\log\log n-n+n\frac{\log\log n}{\log n}+O(n(\log\log n/\log n)^2)$$ So the given approximations have errors $$p_n=n\frac{\Gamma'(n)}{\Gamma(n …

I have posted in arXiv:1702.05442 the English translation of a paper about Fabius function that I published in Spanish in 1982 (we will refer to it as (A)). With the Theorems in this paper the ques …

We have Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that $$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$ Then for a …

The zeta function can be written $\zeta(1/2+it)=Z(t)e^{-i\vartheta(t)}$ where $Z(t)$ are real analytic and $\vartheta(t)$ is a simple function related to $\Gamma(s)$. The relation between $\zeta(1/2 …

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} …

This is well known (Riemann could have writen it) $$\zeta(s)=\frac{1}{2}\frac{\pi^{s/2}}{(s-1)\Gamma(1+s/2)}\prod_{\Im\rho > 0}\Bigl\{ \Bigl(1-\frac{s}{\rho}\Bigr)\Bigl(1-\frac{s}{\overline{\rho}}\Bi …

It is known that the number of zeros with $T-1 < Im(\rho)<T+1$ is $O(\log(T))$. Therefore the multiplicity of a zero $\beta+\gamma i$ will be less than $C \log|\gamma|$ for some absolute constant $C …

In fact for odd $n\ge3$ we have $$\Bigl.\frac{d^{n}}{ds^n}\log\zeta(s)\Bigr|_{s=\frac12}= \frac{(n-1)!}{2}\Bigl(2^n L(n,\chi)+(2^n-1)\zeta(n)\Bigr)$$ The proof (due to Voros) is the following: It …

The series $\sum_\rho \rho^{-1}$ over the non-trivial zeros is not absolutely convergent, this is proved in Davenport p. 80. But as Davenport says and proves in page 81-82 the series converges con …

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) …

Assuming the Riemann hypothesis $$|p_n -\text{li}^{-1}(n) |\le \pi^{-1} \sqrt{n} (\log n)^{5/2}, \qquad n \ge 11$$ This follows from known bounds for $\pi(x)$ due to Schoenfeld. You may see the detai …

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. …

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)=$ …

The figure of egg is not an elliptic curve. Since the curve pass through the point $(-2,0)$ and $(1,0)$ the equation will be of type $$y^2=(ax+b)(x+2)(x-1)$$ We find the numbers $y_0$ and $y_1$ wh …