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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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What do you mean by "methods for other a(mod p)"? –  Ricky Demer Feb 2 '12 at 17:10
    
this question is not appropriate for this site. See the FAQ for an explanation and alternative suggestions –  Yemon Choi Feb 2 '12 at 18:12
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@Ralph I misread the original form of the question, and thought it had an easy expression in terms of even zeta values. I do still think that the questions is unmotivated, and would need more context to really be appropriate for MO (research-related, etc) –  Yemon Choi Feb 2 '12 at 18:40
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I think this question is probably MO appropriate. I don't see that it needs much motivation, it's interesting simply by analogy to $\sum 1/k^2=\pi^2/6$. I'm going to vote to re-open and edit to make it clearer and add a number theory tag. Only reason maybe not MO appropriate is that one can use characters in a standard way to express it in terms of Dirichlet $L$-series (I think), so maybe not really research level. But still, I think many on MO would find it interesting to see a solution. –  Joe Silverman Feb 2 '12 at 20:33
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@Yemon Choi: Sure, it's a good question, I have no problem with someone asking about a specific case of something that's easy to generalize. More generally, one might ask about the value of $\sum_{n=1}^\infty f(n)^{-1}$ for a polynomial $f\in\mathbb{Z}[x]$ that takes positive values on positive integers. And there are obvious multivariable generalizations of this generalization that have been much studied. But it's very reasonable to start by asking about a specific non-trivial example. –  Joe Silverman Feb 3 '12 at 3:10

4 Answers 4

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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Easy enough to prove directly: $\Phi(1, 1/3)$ is defined to be something like $\sum_n 1/(n+1/3)^2$. See the section "Series representation" in the Wikipedia article on the Polygamma function. –  David Speyer Feb 2 '12 at 18:37
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Is there a closed expression for $\Psi(1,1/3)$ like $\pi^2/6$ for $\Gamma(2)$ ? –  Ralph Feb 2 '12 at 18:46
    
No, the one with closed-formm expression is the sum from $-\infty$ to $\infty$. –  Gerald Edgar Feb 2 '12 at 19:02
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@Ralph : No, there isn't, because the non-trivial Dirichlet $\chi$ mod 3 is so and there is a formula for $L(\chi,m)$ iff $m$ is odd. On the other hand, there is an expression in terms of the dilogarithm function evaluated at cube roots of unity, see for example maths.dur.ac.uk/~dma0hg/dilog.pdf on page 17. –  François Brunault Feb 2 '12 at 19:03
    
Sorry, this should read : $\chi$ mod 3 is odd, and there is... –  François Brunault Feb 2 '12 at 19:04

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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Reference?${}{}$ –  Gerry Myerson Feb 3 '12 at 22:51

The reference website of the paper is http://arxiv.org/abs/1105.2042.

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