scholarly journals The Ramanujan conjecture and its applications

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180441
Author(s):  
Wen-Ching Winnie Li
Keyword(s):  
Quantum Computing ◽  
Computer Science ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Golden Gate ◽  
Ramanujan Graphs ◽  
Combinatorial Objects ◽  
Explicit Constructions ◽  
Ramanujan Conjecture

In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

scholarly journals From Ramanujan graphs to Ramanujan complexes

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180445 ◽  
Author(s):  
Alexander Lubotzky ◽  
Ori Parzanchevski
Keyword(s):  
Random Walks ◽  
Computer Science ◽  
Quantum Computation ◽  
High Dimensional ◽  
Dimensional Theory ◽  
Ramanujan Graphs ◽  
Ramanujan Conjecture ◽  
High Dimensional Theory ◽  
Euclidean Spheres

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high-dimensional theory has emerged. In this paper, these developments are surveyed. After explaining their connection to the Ramanujan conjecture, we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to ‘golden gates’ which are of importance in quantum computation. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

scholarly journals Global Publications Output in Quantum Computing Research: A Scientometric Assessment during 2007-16

2018 ◽  
Vol 2 (4) ◽  
Author(s):  
S. M. Dhawan ◽  
B.M. Gupta ◽  
Sudhanshu Bhusan
Keyword(s):  
Quantum Computing ◽  
Computer Science ◽  
Citation Index ◽  
National Level ◽  
The Usa ◽  
The Subject ◽  
Scopus Database ◽  
Using Data ◽  
Scientometric Assessment ◽  
Highly Cited

The paper maps quantum computing research on various publication and citation indicators, using data from Scopus database covering 10-year period 2007-16. Quantum computing research cumulated 4703 publications in 10 years, registered a slow 3.39% growth per annum, and averaged 14.30 citations per paper during the period. Top 10 countries dominate the field with 93.15% global publications share. The USA accounted for the highest 29.98% during the period. Australia tops in relative citation index (2.0).  International collaboration has been a major driver of research in the subject; 14.10% to 62.64% of national level output of top 10 countries appeared as international collaborative publications. Computer Science is one of the most popular areas of research in quantum computing research. The study identifies top 30 most productive organizations and authors, top 20 journals reporting quantum computing research, and 124 highly cited papers with 100+ citations per paper.

THE PRIME GEODESIC THEOREM AND QUANTUM MECHANICS ON FINITE VOLUME GRAPHS: A REVIEW

2001 ◽  
Vol 13 (12) ◽  
pp. 1459-1503 ◽  
Author(s):  
NORMAN E. HURT
Keyword(s):  
Finite Volume ◽  
Riemannian Manifolds ◽  
Riemann Surfaces ◽  
Three Dimensional ◽  
Hyperbolic Spaces ◽  
Quantum Graphs ◽  
Selberg Zeta Function ◽  
Ramanujan Graphs ◽  
Prime Geodesic Theorem ◽  
Ramanujan Conjecture

The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the Selberg zeta function and the Selberg trace formula is outlined. Relationships to work on quantum graphs are surveyed. Extensions to compact Riemannian manifolds, in particular to three-dimensional hyperbolic spaces, are noted. Interconnections to the Selberg eigenvalue conjecture, the Ramanujan conjecture and Ramanujan graphs are developed.

scholarly journals Higher-rank zeta functions for elliptic curves

2020 ◽  
Vol 117 (9) ◽  
pp. 4546-4558 ◽  
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Vector Bundles ◽  
Zeta Functions ◽  
Smooth Curve ◽  
Isomorphism Classes ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier work by L.W., a nonabelian zeta function was defined for any smooth curve X over a finite fieldFqand any integern≥1bywhere the sum is over isomorphism classes ofFq-rational semistable vector bundles V of rank n on X with degree divisible by n. This function, which agrees with the usual Artin zeta function ofX/Fqifn=1, is a rational function ofq−swith denominator(1−q−ns)(1−qn−ns)and conjecturally satisfies the Riemann hypothesis. In this paper we study the case of genus 1 curves in detail. We show that in that case the Dirichlet serieswhere the sum is now over isomorphism classes ofFq-rational semistable vector bundles V of degree 0 on X, is equal to∏k=1∞ζX/Fq(s+k),and use this fact to prove the Riemann hypothesis forζX,n(s)for all n.

Integral moments of automorphic L-functions

2016 ◽  
Vol 12 (07) ◽  
pp. 1827-1843
Author(s):  
Hengcai Tang ◽  
Xuanxuan Xiao
Keyword(s):  
Real Number ◽  
Riemann Hypothesis ◽  
Cuspidal Representation ◽  
Generalized Riemann Hypothesis ◽  
Number Formula ◽  
Ramanujan Conjecture

Let [Formula: see text] be a self-contragredient irreducible unitary cuspidal representation of [Formula: see text] with [Formula: see text], and [Formula: see text] be the automorphic [Formula: see text]-function attached to [Formula: see text]. Assume that [Formula: see text] is self-contragredient. Under the Generalized Ramanujan Conjecture and Generalized Riemann Hypothesis for [Formula: see text], the estimate [Formula: see text] holds for all real number [Formula: see text] as [Formula: see text].

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