What is the value of $$\sum_{i=0}^\infty \frac{1}{(3i+1)^2} ?$$
Methods for other $a\pmod p$ would be helpful, i.e., the value of
$$\sum_{i=0}^\infty \frac{1}{(pi+a)^2} .$$
Thanks in advance Herman :+)
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You can express $\sum_{i=0}^{\infty} \frac{1}{(pi+a)^2}$ as a linear combination of Dirichlet $L$-values $L(\chi,2) = \sum_{n=1}^{\infty} \chi(n)/n^2$ where the $\chi$'s are Dirichlet characters modulo $p$. If $\chi$ is even, there are standard formulas giving $L(\chi,2)$ as an algebraic multiple of $\pi^2$, but if $\chi$ is odd, there is none. In fact, thanks to a deep theorem of Borel, $L$-values at $2$ of odd Dirichlet characters are related to $K$-theory, more precisely to a regulator defined on $K_3$ of the abelian extension cut out by the Dirichlet character. Such regulators are expected to be algebraically independent from $\pi$, but this conjecture is out of reach. I think it's still not known whether $L(\chi_3,2)$ and $L(\chi_4,2)$ (which is also known as Catalan's constant) are irrational. See Rivoal, T.; Zudilin, W. Diophantine properties of numbers related to Catalan's constant. Math. Ann. 326 (2003), no. 4, 705--721. |
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In the few seconds before closure: Maple gives a closed form in terms of the polygamma function $$\sum_{k=0}^\infty\frac{1}{(pk+c)^2}=\frac{1}{p^2}\Psi\Big(1,\frac{c}{p}\Big)\,$$ that should not be difficult to find in the literature or prove directly. |
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There is a paper on the arxvi claiming that Dirichlet series L(χm, 2) for a nonprincipal character mod m are irrational numbers, for example, L(χ 3 ,2) and L(χ 4 ,2). |
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The reference website of the paper is http://arxiv.org/abs/1105.2042. |
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