# A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives

The question below is the follow-up of this question on MathOverflow.

Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are related to surfaces with Picard rank 20(see the paper of Elkies and Schuett) in the Dwork family $$x_1^4+x_2^4+x_3^4+x_4^4=4\lambda x_1x_2x_3x_4.$$ Jesus Guillera found a few Ramanujan-type formulas for $1/\pi^2$(which can be found in W. Zudilin's paper), three of which(formula (92)(93)(94) in Zudilin's paper) are related to the Dwork family $$x_1^6+x_2^6+x_3^6+x_4^6+x_5^6+x_6^6=6\lambda x_1x_2x_3x_4x_5x_6$$through Picard-Fuchs equation. It is reasonable to conjecture that the Guillera's formulas are related to "singular" members in Dwork family.

The end of this paper suggests that the (Hasse-Weil)L-functions of those "singular" members behave differently from those "ordinary" members. It seems that "hypergeometric motive" package(developed by M. Watkins, based on the work of N. Katz et al.) in MAGMA offers a possible way to investigate those L-function numerically(although the L-functions are different from Hasse-Weil L-function).

Experiment: M. Watkins and David Roberts tried to find out imprimitive L-function attached to hypergeometric motives in this document(p.29, Table 15), where the L-function attached to the motive can be factorized to the product of two L-functions. One can immediately recognize the numbers corresponding to Guillera formula(formulas (86)(87)(88) in Zudilin's Paper). It is amazing that one can find out that EVERY L-function associated to the Guillera formula is imprimitive.

Example: Guillera conjectured that

$$\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n(\frac{1}{8})_n(\frac{3}{8})_n(\frac{5}{8})_n(\frac{7}{8})_n}{(n!)^5}(1920n^2+304n+15)\frac{1}{7^{4n}}=\frac{56\sqrt{7}}{\pi^2}.$$

The HGM package can evaluate Frobenius trace and Euler factors of the corresponding hypergeometric motive $$H([1/2,1/8,3/8,5/8,7/8],[0,0,0,0,0],\tilde{t}),$$ where $$1/\tilde{t}=t=\frac{1}{7^4}.$$ Almost each Euler factor $L_p(T)$ are quintic polynomials, and the absolute values of roots of Euler factors are $p^2.$ Adapting the primitivity test of M. Watkins(p. 25), if one assumes the truth of Selberg's conjecture for Selberg class, then(for a primitive L-function) the second moment of (normalized) Frobenius trace $a_p$ $$\frac{1}{\pi(X)}\sum_{p<X}\left(\frac{a_p}{p^2}\right)^2$$ should have limit 1.

A calculation with second moment of normalized Frobenius trace of the example above(up to $p\approx 246500$) is $\approx 2.018$, suggesting the mean value is $2$. The same test is performed for each hypergeometric motive attached to Guillera's formulas(formula (86)-(95), up to $p\approx 40000$), and all mean values are close to $2.$ The same phenomena are also observed for identities discovered by Ramanujan, B. Gourevich and J. Cullen(formulas on p. 33 of this paper). I also tried to find imprimitive L-function other than those associated to Guillera's formulas, but without any success.

Question: I am really amazed by the fact that each L-function associated to the Guillera formulas is likely to be able to be factorized(and one of the factors seems to be either Riemann zeta function or Dirichlet L-function[e.g. formula(90)(92)(95) and formula of B. Gourevich](tested for Euler factor for each Guillera formula up to $p\approx 1000$ and $p\approx 100$ for higher order examples of B. Gourevich and J. Cullen)), while the L-function of a randomly chosen hypergeometric motive cannot be further factorized(which is suggested by calculation with MAGMA).

• Is there any interpretation that how the imprimitivity of L-function attached to hypergeometric motives leads to period relation of hypergeometric functions?
• Is it possible to find new Ramanujan-type formula with this imprimitivity test?
• What are the p-adic counterparts of these identities?

Addendum: The same phenomena can also be observed when one worked with these formulas discovered by Guillera.

Edit: M. Watkins has already verified the observation for Guillera's formula (96) here(p. 17). Again the L-function attached to the motive has Riemann zeta function as a factor.

EUREKA!: If one uses the technique in Guillera's paper and the 3rd/last row of Table 15 of M. Watkins, PSLQ suggests that

$$\sum_{n=1}^{\infty}\frac{(1)_n^5}{(1/2)_n^3(1/3)_n(2/3)_n}\frac{28n^2-18n+3}{n^5}\left(-\frac{1}{27}\right)^n=-14\zeta(3)$$

and

$$\sum_{n=1}^{\infty}\frac{(1)_n^5}{(1/2)_n(1/3)_n(2/3)_n(1/4)_n(3/4)_n}\frac{172n^2-75n+9}{n^5}\left(-\frac{16}{27}\right)^n=-1792\zeta(3)!$$

Also $$\sum_{n=1}^{\infty}\frac{(1)_n^5}{(1/2)_n^5}\frac{10n^2-6n+1}{n^5}\left(-\frac{1}{4}\right)^n=-28\zeta(3).$$

[Remark: The three formulas above are already known to Guillera(see this paper).]

And more

$$\sum_{n=1}^{\infty}\frac{92n^3-84n^2+27n-3}{n^7}\frac{(1)_n^7}{(\frac{1}{2})_n^5(\frac{1}{3})_n(\frac{2}{3})_n}\left(\frac{4}{27}\right)^n=8\pi^4,$$

$$\sum_{n=1}^{\infty}\frac{84n^3-88n^2+32n-4}{n^7}\frac{(1)_n^7}{(\frac{1}{2})_n^7}\left(\frac{1}{64}\right)^n=\pi^4/2.$$

(Edit: The second formula has been discovered by Guillera, see this paper, p. 509)

And more!!!

$$\sum_{n=0}^{\infty}\frac{(1/2)_n^5(1/3)_n(2/3)_n(1/4)_n(3/4)_n}{(1)_n^9}(4528n^4+3180n^3+972n^2+147n+9)\left(-\frac{27}{256}\right)^n=\frac{768}{\pi^4},$$

\begin{align}\sum_{n=1}^{\infty}\frac{5532n^4-5600n^3+2275n^2-425n+30} {n^9}\cdot & \\ \frac{(1)_n^9}{(\frac{1}{2})_n^5(\frac{1}{5})_n(\frac{2}{5})_n(\frac{3}{5})_n(\frac{4}{5})_n}& \left(-\frac{1024}{3125}\right)^n =-380928\zeta(5)\end{align}

Update: I tested a few hypergeometric motives with degree=weight+1, weight>=10. None of them can pass the test. I am not sure if one can find more similar formulas for higher reciprocal powers of $\pi$.

Update2: A recent preprint of David P. Roberts and Fernando Rodriguez Villegas is provided here. It seems that the splitting of underlying hypergeometric motives does not only provide period relations of (generalized) hypergeometric functions at algebraic arguments, but also supercongruences of truncated hypergeometric series.

• In the latest Magma Handbook, Watkins also tested Guillera's (96) (over ${\bf Q}(\sqrt 5)$) numerically, again finding it to seem to have a $\zeta$-factor in the $L$-function. Zudilin would be the best person to ask about the HGM connection. – post.as.a.guest Sep 13 '17 at 22:34
• Is the ! in $... = -1792\zeta(3)!$ intended or a typo? – ypercubeᵀᴹ Sep 24 '17 at 8:41
• @zy_: Are you sure those three formulas for $\zeta(3)$ are in the paper cited? I am unable to find it. – Tito Piezas III Jan 23 '18 at 17:43
• @Tito Piezas: I sent an e-mail to Dr. Guillera on these discoveries. He pointed out to me that the counterpart of these three formulas has already been in his paper(formula (34)(36)(37)). He knows the upside-down version of these formulas as well. But the formulas for $1/\pi^4$,$\zeta(4)$ and $\zeta(5)$ are new to him. – Y. Zhao Jan 24 '18 at 16:25
• @zy_: Ah, I see it now. Yes, they are beautiful and the $\zeta(5)$ is unexpected. I noticed its argument is $-\frac{1024}{3125} = -\frac{4^5}{5^5}$ so tried to generalize it for $\zeta(7)$ using $-\frac{6^7}{7^7}$. Unfortunately, an integer relations algorithm couldn't find anything. – Tito Piezas III Jan 24 '18 at 16:42

## 1 Answer

It should follow from Grothendieck's period conjecture that whenever a period of an irreducible motive $V$ is also the period of a motive $W$, then $V$ is a summand of $W$ in the category of motives, and hence the L-function of $V$ is a factor of the L-function of $W$.

This predicts that in your cases you should see that the normalized average value of $a_p$ is one, except in the first case where you should average $a_p$ against the primitive quadratic Dirichlet character mod $28$ (= character of the field $\mathbb Q(\sqrt{7})$)

Conversely, it should be true that if the L-function splits, the motive splits, and so the period splits. However, there is no reason to believe that the motive splits into motives whose periods are given by some kind of closed-form formula.

To make this work in full detail one has to understand how the rational functions of $n$ you're throwing in affect the motive. In particular by figuring out what happens to the weight it should tell you what power of $\pi$ to expect and when to expect $\zeta(3)$.

• Thank you for your answer. I am wondering when a hypergeometric motive can be written as a summand of motives. The numerical calculation suggests that these exceptional cases rarely occur. – Y. Zhao Sep 21 '17 at 14:55
• @zy_ One should be able to prove from results of Cadoret and Tamagawa that for any hypergeometric function with large monodromy, the associated hypergeometric motive does not split except for finitely many rational values of the parameter. – Will Sawin Sep 21 '17 at 15:58