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... then you are making the $\arg\Gamma(1/4+it/2)$ discontinuous at certain points, but not the zeros of zeta. Nevertheless the two functions will be equal modulo $\pi$.
@OneTwoOne $\frac{x}{2}\log\pi-\arg\Gamma(1/4+it/2)$ is a very simple function, continuous and indefinitely differentiable. $\arg\zeta(1/2+ix)$ is continuous except at zeros of zeta (to simplify I assume the Riemann Hypothesis here) the function has jumps at the zeros of zeta. The equality you write is only true $\mod \pi$. But if you consider the function of user 64494
@OneTwoOne What you have done have nothing to do with the Riemann Hypothesis. $\arg\zeta(1/2+it)$ contains many information about the zeros. $\Gamma(1/4+it/2)$ almost nothing.
@OneTwoOne It is not so easy to explain. It is defined in the book by Titchmarsh Section 9.3. $\arg\zeta(1/2+iT)$ is obtained by continuous variation along the straight lines joining 2, $2+iT$, $1/2+iT$, starting with the value $0$. It is a discontinuous function. While $\arg\Gamma(1/4+it/2)$ is usually meaning as the continuous argument. So the relation you writes between then is not true.
@OneTwoOne No. This not disproof the Riemann hypothesis. when one uses $\arg\zeta(1/2+ix)$ one refers to the "discontinuous" arg defined in a certain specific way. Your formula for $\arg\zeta(1/2+it)$ is not correct. It is only an equality $\bmod \pi$
@user64494 Plot it from 0 to 20, the problem is that the argument of gamma can be defined as a continuous function that sometimes cross pi, 3pi and so on. Of course you can consider the integral with the discontinuous argument. But usually when one speaks about the argument of Gamma one uses the continuous one.
Elsevier notified Yves Meyer that the copies of his book "Algebraic numbers and Harmonic analysis" was to be destroyed. Because they do not want to store them. Some months ago an old copy was in Amazon for 1000$, now there are none. How can they ask for rights of copyrights?
@Carlo Beenakker. Sometime, somewhere we will find a place where copyright laws are dictated to defend the authors. Mathematicians are all glad to see his works read. Mathematicians do not get any from all the paywalls we find in our papers.
@Greg Martin I will suggest that once translated the paper should be posted in arXiv. Translation of some (some 60 papers) of Euler's papers are posted in arXiv. I have translations of several German number theory papers, but to Spanish, so I have not posted them in arXiv.
