## Experimental Mathematics

- Experiment. Math.
- Volume 14, Issue 3 (2005), 343-357.

### Extremal modular lattices, McKay Thompson series, quadratic iterations, and new series for $\pi$

#### Abstract

We give 20 new Ramanujan-type formulae and 20 quadratic approximations to {\small $\pi$}, parameterized by extremal modular lattices of minimal square norm 2 and 4, at the ten special levels corresponding to square-free orders of the Mathieu group {\small $M_{23}$}. An algorithm for uncovering rational relations between two given power series that we used to discover some of the explicit relations is also given. Explicit relations parameterized by modular lattices between McKay Thompson series for the group {\small $\Gamma_0(\ell)^+$} and {\small $\Gamma_0(2\ell)+p_j$}, where {\small $p_j$} range over all odd primes dividing the special level {\small $\ell$}, are uncovered.

#### Article information

**Source**

Experiment. Math., Volume 14, Issue 3 (2005), 343-357.

**Dates**

First available in Project Euclid: 3 October 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1128371759

**Mathematical Reviews number (MathSciNet)**

MR2172712

**Zentralblatt MATH identifier**

1084.11016

**Subjects**

Primary: 11F03: Modular and automorphic functions 11H31: Lattice packing and covering [See also 05B40, 52C15, 52C17] 11Y60: Evaluation of constants

**Keywords**

Modular lattices quadratic iterations McKay Thompson series $M_{23}$ Hauptmodul $\pi$

#### Citation

Chua, Kok Seng. Extremal modular lattices, McKay Thompson series, quadratic iterations, and new series for $\pi$. Experiment. Math. 14 (2005), no. 3, 343--357. https://projecteuclid.org/euclid.em/1128371759