Cotangent sums related to the Riemann hypothesis for various shifts of the argument

Cotangent sums of the form
$$ c_{0}(r)\,=\,\sum\limits_{m=1}^{b-1}\frac{m}{b}\cot{\Bigl(\frac{\pi m}{b}\Bigr)} $$
play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. M.Th. Rassias and the speaker in several joint papers and M.Th. Rassias in his thesis have investigated moments as well as the distribution of these cotangent sums for several variables of the arguments: for variable $r$ and fixed large $b$, for $r$ varying over prime numbers and $b$ being a fixed large prime. They found a close connection to the sum
$$ g(\alpha)\,:=\,\sum\limits_{l=1}^{+\infty}\frac{1-2\{l\alpha\}}{l},\quad \alpha\in (0,1), $$
where ${u} := u -[u]$, $u \in \mathbb{R}$. The speaker will give a short overview on these results but focus on the joint distribution of
$$ c_{0}\Bigl(\frac{r+a_{l}}{q}\Bigr),\quad 1\leqslant l\leqslant L $$
with $a_{1}, \ldots , a_{L}$ being distinct non-negative integers. The main tools are results on Exponential Sums in Finite Fields due to Weil as well as Fouvry and Michel.

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