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The Ramanujan conjecture and its applications.

Wen-Ching Winnie Li1

  • 1Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|December 10, 2019
PubMed
Summary

This review explores the Ramanujan conjecture and its computer science applications, including Ramanujan graphs and quantum computing. It highlights their connection to zeta functions and the Riemann hypothesis.

Keywords:
Ramanujan complexesRamanujan conjectureRamanujan graphsRiemann hypothesiszeta functions

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Area of Science:

  • Number Theory
  • Computer Science
  • Quantum Computing

Background:

  • The Ramanujan conjecture is a fundamental concept in number theory with deep implications.
  • Its connections to combinatorial objects and computational problems are increasingly recognized.

Purpose of the Study:

  • To provide a comprehensive review of the Ramanujan conjecture in both classical and modern contexts.
  • To elucidate its diverse applications within computer science and related fields.

Main Methods:

  • Review of existing literature on the Ramanujan conjecture.
  • Explanation of its theoretical underpinnings and practical implementations.
  • Discussion of connections to spectral graph theory and quantum information.

Main Results:

  • Detailed exposition of Ramanujan graphs and complexes as spectrally extremal combinatorial objects.
  • Explanation of uniformly distributed points on spheres and Golden-Gate Sets in quantum computing.
  • Exploration of the link between Ramanujan graphs/complexes, their zeta functions, and the Riemann hypothesis.

Conclusions:

  • The Ramanujan conjecture offers powerful tools for constructing efficient algorithms and structures in computer science.
  • Its interdisciplinary nature bridges number theory, graph theory, and quantum computing.
  • Further research into these connections promises significant advancements.