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## Abstract:

Ramanujan's formulas for $1/\pi$ and their generalizations remain an amazing topic, with many mathematical challenges.

Recently it was observed that the formulas possess spectacular `supercongruence' counterparts.

For example, truncating the sum in Ramanujan's formula

$$

\sum_{k=0}^\infty\frac{\binom{4k}{2k}{\binom{2k}{k}}^2}{2^{8k}3^{2k}}\,(8k+1)=\frac{2\sqrt{3}}{\pi}

$$

to the first $p$ terms correspond to the congruence

$$

\sum_{k=0}^{p-1}\frac{\binom{4k}{2k}{\binom{2k}{k}}^2}{2^{8k}3^{2k}}\,(8k+1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3}

$$

valid for any prime $p>3$.

Some supercongruences were shown earlier through a tricky use of classical hypergeometric identities or the Wilf-Zeilberger method of creative telescoping.

The particular example displayed above (and many other entries) were resistant to such techniques.

In joint work with Victor Guo we develop a new method of `creative microscoping' that simultaneously proves both the underlying Ramanujan's formula and its finite supercongruence counterparts.

The main ingredient is an asymptotic analysis of $q$-analogues of Ramanujan's formulas at roots of unity.