MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 :+)

share|cite|improve this question
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
@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
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
@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

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.

share|cite|improve this answer

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.

share|cite|improve this answer
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
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
@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 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).

share|cite|improve this answer
Reference?${}{}$ – Gerry Myerson Feb 3 '12 at 22:51

The reference website of the paper is

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.