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Consider the constants $c(s_1,s_2,...)=\sum_{n=1}^\infty \frac{1}{n^{s_n}}$ where the $s_i$'s are fixed positive integers and the number of $i$ with $s_i=1$ is finite (so that the previous sum converges).

For example, when $s_i$ is equal to a fixed positive integer $s\geq 2$ for every $i,$ this constant is the value of the zeta function at $s.$

My first question is if these numbers have ever been considered so far. The second is if in the literature there exists at least one proof of the transcendence or algebraicity of $c$ for values of the $s_i$ not all equal to an even positive integer.

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  • $\begingroup$ Did you mean $n^{s_n}$ in the sum? $\endgroup$
    – Dan Piponi
    Commented Jul 8, 2022 at 16:56
  • $\begingroup$ Yes, thank. Edited. $\endgroup$
    – Nick Belane
    Commented Jul 8, 2022 at 17:00
  • $\begingroup$ If the $s_i$ alternate between two values $t$ and $u$ then this should be expressible as a simple linear combination of $\zeta(t)$ and $\zeta(u)$. $\endgroup$
    – Gerry Myerson
    Commented Jul 9, 2022 at 1:30
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    $\begingroup$ If the $s_n$ grow rapidly enough (i.e., the sum converges rapidly enough), there are some results which give the irrationality or transcendence of the sum. In the amount of generality your question is posed, I don't think you'll find much in the literature. Even the case $s_n = n$ is wide open in terms of just the irrationality of the sum. $\endgroup$
    – Joshua Stucky
    Commented Jul 9, 2022 at 6:33
  • $\begingroup$ @JoshuaStucky could you provide references for the case where $s_n$ grow fast? $\endgroup$
    – Nick Belane
    Commented Jul 9, 2022 at 6:51

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