What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision anybody tried but still remain unproved?
I am also interested in values of special functions at certain points that have conjectural representations in terms of simpler functions (e.g. special values of hypergeometric functions, Meijer G-function or Fox H-function that representable in terms of elementary functions and well-known constants like $\pi$, $e$, Catalan, Euler–Mascheroni, Glaisher–Kinkelin or Khinchin).
To give some examples:
- Gourevitch conjecture mentioned at MathOverflow:
$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7\stackrel{?}{=}\frac{32}{\pi^3}.$$ - Riemann hypothesis (in an unusual form, also mentioned at MathOverflow)
$$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt\stackrel{?}{=}\frac{\pi(3-\gamma)}{32}.$$ - Another conjecture from MathOverflow attributed to J. M. Borwein, D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century:
$$\frac{\displaystyle\int_{\pi/3}^{\pi/2}\log\left|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}\right|dt}{\displaystyle\sum_{n=1}^\infty\left(\frac n7\right)\frac{1}{n^2}}\stackrel{?}{=}\frac{7\sqrt{7}}{24},$$
where $(\frac{n}{7})$ denotes the Legendre symbol.
Best Answer
Some conjectural formulas for $\pi$ are given in:
For example (I expressed an infinite sum given in the paper in terms of hypergeometric functions): $$224 \, _5F_4\left(\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3 };1,1,1,1;8235 \sqrt{5}-18414\right)\\-100 \sqrt{5} \, _5F_4\left(\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3 };1,1,1,1;8235 \sqrt{5}-18414\right)\\-1655540 \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };1,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\+740380 \sqrt{5} \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };1,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\-1237563 \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };2,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\\+553455 \sqrt{5} \, _5F_4\left(\frac{4}{3},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{5}{3 };2,2,2,2;\frac{27}{8} \left(5 \sqrt{5}-11\right)^3\right)\stackrel{?}{=}\frac{4}{\pi^2}$$
You can also look at:
And one more from J. Guillera homepage: $$\sum_{n=0}^\infty\frac{(5418n^2+693n+29)(6n)!}{(-23887872000)^n n!^6}\stackrel?=\frac{128\sqrt5}{\pi^2}.$$