@HenriCohen in page 218 there are conditions for the expansion to be valid. Forgotten mathematics

In Nörlund book Vorlesungen über Differenzenrechnung p. 203 speak about Stirlingsche Interpolationsformel, under which appear to be examples the two forms for the cos. In page 211 there is an expansion for sin pi x / x, but it is not the same as yours. Maybe there is more in the book. At least this say what to search.

@Salvo The figures of the partial sums $\sum_{k=1}^n e^{ik+i\gamma_k}$ are very similar to the ones in the book by Montgomery, I have done it for $1\le n\le 100000$. Impressive.

@Salvo I confirm your plot. There are other jumps, the next one between $10000<n<15000$ and other between $[50000,60000]$. Perhaps $\sum_{k=1}^n e^{ikx+i\gamma_k}$ behaves as the sum considered by H. Montgomery in \emph{Ten Lectures on the Interface between Number Theory and Harmonic Analysis} p. 45--60.

@Jesse Elliot My main point is that you have the same plots with $\tilde\gamma$ and the proof of the equivalence with $\tilde\gamma$ is easier. My proof is long, but not difficult really. Besides, the equivalence with Odlyzko's formulation is not so interesting as with $\tilde\gamma$

@Amdeberham When $s >5$ say, $\zeta(s)$ and $L(s)$ are practically 1. So the plot is indistinguishable from that of $$\sin(\pi(s+1)/2)+\sin(\pi s/2)=\sqrt{2}\sin(\pi s/2+ \pi/4)$$

@Amdeberham it was interesting the Plot of the complex function $t \mapsto e^{-\pi t/2} \zeta(1/2+it). The zeros of this function are not on the critical line. (in general).

@Amdeberham I am not sure I know how to do that. L[s_] := Sum[(-1)^(n - 1)/(2 n - 1)^(2 s - 1), {n, 1, Infinity}]; f[s_] := Sin[(Pi (s + 1))/2] Zeta[s] + Sin[(Pi (s))/2] L[(s + 1)/2]; Table[f[s], {s, 2, 8}]

@Clark my $L(s)$ is defined as $$L(s)=\sum_{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)^{2s-1}}.$$