Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
@Helfgott I computed the integral aproximately with Mathematica for $2\times 10^{-6}<|t|<2\times 10^6$ obtaining $0.81492827$ and then my sum oscillates, but the mean of the sums to 300000 give me the value $0.814346$. This is not too good, it is what I can say.
Making some transformations from the integral, transformations that I will be ashamed to confess, I get to this expression $$-\frac12\sum_{n=1}^\infty \frac1n\Bigl(\sum_{ab=n}\mu(a)\mu(b)|\log(a/b)|\Bigr).$$ Numerically it appear to have some sense.
Your first formula is not true, take for example a sequence $ s_n=\rho_n$ the sequence of non trivial zeros of zeta, then you get $\log(-1)/\rho_n\to -\log p_n$? But if you put $\lim_{s\to+\infty}$ instead, the equation appear to be true.
@Wane Riesel and Gohl never use $\sum_{n=1}^\infty \mu(n)=-2$. Only say that to make the bound of the error it is preferable to use some $N$ such that $\sum_{n=1}^N \mu(n)=-2$. Such $N$ exist because the partial sums oscillates between positive and negative values. But Riesel and Gohl objective was how to compute $\pi(x)$ with a bounded error using Riemann's formula.
@Wane Landau proves the main formula of Riemann (you write it, except for a summand $-\log 2$). All the other formulas (when correctly written, I have not checked yours) follow relatively easy from this. As Steven Clark says, the formulas can be written in several equivalent forms, depending on how you treat the trivial zeros of zeta. Riesel and Gohl have a valuable simplification of this formulas. I translated to Spanish the paper of Landau in part to clarify the bad typing of the original. Understanding Landau and writing correctly his formulas in Latex.
@Wane Not, I have a copy from many years ago. But the French journals are usually in NUMDAM numdam.org . Landau have also a German version of his paper. Neuer Beweis der Riemannschen Primzahlformel, Sitzunberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1908, p. 737-745. But this I have not seen.
@Wane The exact reference to the paper of Landau is: E. Landau, Nouvelle demonstration pour la formule de Riemann sur le nombre des nombres premiers inférieurs a une limite donnée et demonstration d'une formule plus générale pour le cas des nombres premiers d'une progression arithmétique, Ann. Ec. Norm. (3) 25, (1908) 399-442.
Landau has a paper published in Ann. Ec. Norm., (3) 25, (1908) 399-442 with title something as "New proof of Riemann's formula for the number of prime numbers less than a limit", but I am translating this from my Spanish translation of the paper. In this paper Landau (with all the care to the detail that we all know) proves this formula. Showing in particular the convergence of the series of $\text{Li}(x^\rho)$ adequately ordered.
$\Im f(1/5)$ appear to be a solution of the equation $81x^4-45x^2+5=0$ and the imaginary part of the expression with exponential appear to be a root of $81x^4-585x^2+5=0$. All this as given in Mathematica and recognising the algebraic numbers.
@Cohen With Mathematica only the real parts coincide. This is a summary of all my previous comments. Usually there is a cut on the real axis for the implementation of the hypergeometric function.
The paper of Connes and Consani refer explicitly to zeros on the critical line. For example see Theorem 6.4 on page 52. The book by H. Iwaniec Lectures on the Riemann zeta function, have a Part with title The critical zeros After Levinson, and uses critical zeros to denote zeros on the critical line.
If you are interested in probability you may assume that the measure is complete (each subset of a set of measure zero is measurable). In many cases $\Omega$ will contain a set of measure zero with cardinal the continuum. Then you may modify the random variable in this measure zero set so that the range is $\Bbb R$.
