Is special value of Epstein zeta function in 3 variables a period?

Question

Kontsevich-Zagier's article "Periods" contains the following question

Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?

($\sum'$ means we do not sum over $(x,y,z) = (0,0,0)$.)

Here period is loosely interpreted as "integral of algebraic differential forms (on an smooth projective algebraic variety) over algebraic subvariety". Extended period is a period up to a factor of integer powers of $\pi$.

(I am intentionally making this vague since I don't think even Kontsevich-Zagier tried to pin this down very precisely, but hopefully you know a period when you see one.)

My question: Is this now known? (It has been 15 years since Kontsevich-Zagier first published the article) More generally in this setup, the question would be

Let $Q(x_1,\cdots,x_n)$ be a positive definite quadratic form in an odd number ($\ge 3)$ of variables with coefficients in $\mathbb{Q}$. Is $\displaystyle \sum_{x_1,\cdots,x_n \in \mathbb{Z}} \frac{1}{Q(x_1,\cdots,x_n)^s}$ an extended period for $s > n/2$?

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Some background There is a beautiful theorem due to Beilinson and Deninger-Scholl,

Theorem Let $f$ be a modular form of weight $k \ge 2$ defined over $\overline{\mathbb{Q}}$. Then $L(f,m)$ are extended periods for all integers $m \ge k$.

(When $m < k$ this is still true; here I am merely distinguishing $m$ by whether it is a critical value or not for $L(f,s)$.)

The question above for quadratic forms with even number of variables is then solved by this theorem, since it is the $L$-function for the theta series $\sum'_{x_1,\cdots,x_n} q^{Q(x_1,\cdots,x_n)}$ of weight $n/2$ if $n$ is even.

I tried to look up generalization of this theorem to half-integral weight modular forms with no success.

Answer 1

Nice question. I think this is what's going on:

If $Q$ is a (positive definite) quadratic form, then its Epstein zeta function $Z_Q(s)$ can be expressed as a linear combination of L-functions of cusp forms and Eisenstein series.

In the case of an even number of variables $2k$, the Eisenstein series involved are certain modular forms of weight $k$, which by the work of Hecke can be expressed in terms of Dirichlet L-functions. See for example:

• Keiju Sono, Higher moments of the Epstein zeta functions (2013)

As you mention, thanks to the result of Beilinson-Deninger-Scholl, modular forms evaluated at the integers are periods. $Z_Q(n)$ is then a linear combination of periods, and therefore a period.

The problem in the odd case seems to be that the Eisenstein series are not a combination of Dirichlet L-functions, and I don't think it is known whether their values at the integers are periods or not. It might even be the case that they aren't. Remember that Epstein zeta functions are automorphic forms, and their special values might very well be trascendental.

In fact, in another paper on periods, Maxim Kontsevich, co-author of the 2001 paper you mention says:

It is not clear to me whether the analogous statement holds for quadratic forms in odd number of variables.

• M. Kontsevich, Periods (october 2006) pp. 7