Riemann Hypothesis and strongly Ramanujan complexes from GL

Kang, Ming-Hsuan

doi:10.1016/j.jnt.2015.09.002

Cited by 11 publications

(15 citation statements)

References 7 publications

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“…We should mention that Kang [22] (and the other references mentioned above) obtained more detailed information about the location of the poles, by carefully examining the representations appearing in these cases.…”

Section: Introductionmentioning

confidence: 97%

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Random walks on Ramanujan complexes and digraphs

Lubetzky,

Lubotzky,

Parzanchevski

2017

Preprint

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The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group G over a local field F . We show that if T is any k-regular G-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the n-vertex Ramanujan complex has cutoff at time log k n. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of G. Via these, we show that operators T as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property (r-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group G, previously known for groups of type An and C2.

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Section: Introductionmentioning

confidence: 97%

“…Following W. Li, various authors have defined zeta functions associated with Ramanujan complexes, and verified the R.H. for them over groups of type A n and C 2 (cf. [15,[22][23][24][25]). These can all be described in our notation as Z Y T,X (u) for suitable collision-free branching operators T on B (see §6).…”

Section: Introductionmentioning

confidence: 99%

Random walks on Ramanujan complexes and digraphs

Lubetzky,

Lubotzky,

Parzanchevski

2017

Preprint

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“…In [41] Kang showed that if X Γ is strongly Ramanujan, then all Z r (X Γ , u) satisfy the Riemann hypothesis. The converse is not always true.…”

Section: (D) Strongly Ramanujan and The Riemann Hypothesismentioning

confidence: 99%

“…Given a finite quotient X Γ = Γ \B n,F and 1 ≤ r ≤ n − 1, the zeta function Z r (X Γ , u) is said to satisfy the Riemann hypothesis if all non-trivial poles have specifically prescribed absolute values, as given in [41].…”

Section: (D) Strongly Ramanujan and The Riemann Hypothesismentioning

confidence: 99%

“…In the final section, §6, we introduce a stronger concept, called strongly Ramanujan complex [40], in terms of the eigenvalues of an Iwahori-Hecke operator, or equivalently, the adjacency operator on the directed chambers of the complex, and discuss the relationship between Ramanujan complexes, strongly Ramanujan complexes and the Riemann hypothesis (cf. [41]).…”

Section: Introductionmentioning

confidence: 99%

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The Ramanujan conjecture and its applications

2019

Phil. Trans. R. Soc. A.

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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’.

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Ramanujan Complexes and Golden Gates in PU(3)

Evra

Parzanchevski

2022

Geom. Funct. Anal.

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The Clifford+T gate set is a topological generating set for P U (2), which has been wellstudied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of P U (2) has led to a fruitful interaction between quantum computation and number theory, leading in particular to a proof that words in these gates cover P U (2) in an almost-optimal manner.In this paper we study an analogue gate set for P U (3) called Clifford+D. We show that this set generates a full S-arithmetic subgroup of P U (3), and satisfies a slightly weaker almost-optimal covering property. Our proofs are different from those for P U (2): while both gate sets act naturally on a (Bruhat-Tits) tree, in P U (2) the generated group acts transitively on the vertices of the tree, and this is a main ingredient in proving both arithmeticity and efficiency. In the P U (3) (Clifford+D) case the action on the tree is far from being transitive. This makes the proof of arithmeticity considerably harder, and the study of covering rate by automorphic representation theory becomes more involved and results in a slower covering rate.

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Riemann Hypothesis and strongly Ramanujan complexes from GL

Cited by 11 publications

References 7 publications

Random walks on Ramanujan complexes and digraphs

Random walks on Ramanujan complexes and digraphs

The Ramanujan conjecture and its applications

Ramanujan Complexes and Golden Gates in PU(3)

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