Sum of reciprocals of squares of integers congruent to 1 mod 3 ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:55:43Z http://mathoverflow.net/feeds/question/87348 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3 Sum of reciprocals of squares of integers congruent to 1 mod 3 ? Herman 2012-02-02T17:05:17Z 2012-02-10T18:36:13Z <p>What is the value of <code>$$\sum_{i=0}^\infty \frac{1}{(3i+1)^2} ?$$</code> </p> <p>Methods for other $a\pmod p$ would be helpful, i.e., the value of <code>$$\sum_{i=0}^\infty \frac{1}{(pi+a)^2} .$$</code> </p> <p>Thanks in advance Herman :+)</p> http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3/87357#87357 Answer by Pietro Majer for Sum of reciprocals of squares of integers congruent to 1 mod 3 ? Pietro Majer 2012-02-02T18:33:54Z 2012-02-02T18:33:54Z <p>In the few seconds before closure: Maple gives a closed form in terms of the <a href="http://en.wikipedia.org/wiki/Polygamma" rel="nofollow">polygamma function</a> $$\sum_{k=0}^\infty\frac{1}{(pk+c)^2}=\frac{1}{p^2}\Psi\Big(1,\frac{c}{p}\Big)\,$$ that should not be difficult to find in the literature or prove directly.</p> http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3/87373#87373 Answer by François Brunault for Sum of reciprocals of squares of integers congruent to 1 mod 3 ? François Brunault 2012-02-02T21:18:48Z 2012-02-02T21:18:48Z <p>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.</p> <p>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.</p> <p>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</p> <p>Rivoal, T.; Zudilin, W. Diophantine properties of numbers related to Catalan's constant. Math. Ann. 326 (2003), no. 4, 705--721.</p> http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3/87468#87468 Answer by W-S-Park for Sum of reciprocals of squares of integers congruent to 1 mod 3 ? W-S-Park 2012-02-03T19:17:17Z 2012-02-03T19:17:17Z <p>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).</p> http://mathoverflow.net/questions/87348/sum-of-reciprocals-of-squares-of-integers-congruent-to-1-mod-3/88127#88127 Answer by W-S-Park for Sum of reciprocals of squares of integers congruent to 1 mod 3 ? W-S-Park 2012-02-10T18:36:13Z 2012-02-10T18:36:13Z <p>The reference website of the paper is <a href="http://arxiv.org/abs/1105.2042" rel="nofollow">http://arxiv.org/abs/1105.2042</a>.</p>