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13 - Ramanujan Graphs and Digraphs
- By Ori Parzanchevski
Edited by Matthias Keller, Universität Potsdam, Germany, Daniel Lenz, Universität Potsdam, Germany, Radoslaw K. Wojciechowski
Book:

Analysis and Geometry on Graphs and Manifolds

Published online:

14 August 2020

Print publication:

20 August 2020, pp 344-367
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Summary

Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown to be of special interest, as they are extreme in the sense of an Alon-Boppana theorem, and they have remarkable combinatorial features, such as small diameter, Chernoff bound for sampling, optimal covering time and sharp cutoff. Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alon's conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan digraphs.

RAMANUJAN SERIES UPSIDE-DOWN
Part of
JESUS GUILLERA, MATHEW ROGERS
Journal:

Journal of the Australian Mathematical Society / Volume 97 / Issue 1 / August 2014

Published online by Cambridge University Press:

07 July 2014, pp. 78-106

Print publication:

August 2014
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We prove that there is a correspondence between Ramanujan-type formulas for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet $L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

ON η3(aτ)η3(bτ) WITH a+b=8
Part of
- Discontinuous groups and automorphic forms
HENG HUAT CHAN, SHAUN COOPER, WEN-CHIN LIAW
Journal:

Journal of the Australian Mathematical Society / Volume 84 / Issue 3 / June 2008

Published online by Cambridge University Press:

01 June 2008, pp. 301-313

Print publication:

June 2008
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We prove an observation associated with η3(τ)η3(7τ) which is found on page 54 of Ramanujan’s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988)). We then study functions of the type η3(aτ)η3(bτ) with a+b=8.