The Riemann hypothesis and the zero distribution of angular lattice sums

McPhedran, R. C.; Botten, L. C.; Williamson, Dominic J.; Nicorovici, N. A.

doi:10.1098/rspa.2010.0566

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…For a square lattice in d = 2 dimensions, a simple approach for generating the HSEM operator coefficients is available that avoids derivatives of Epstein zeta functions and uses efficient summation formulas that have been found in the analysis of the Riemann hypothesis in higher dimensions [31]. For a two-dimensional square lattice, it has been shown that [41, equation (9)]…”

Section: A1 Hsem Operator Coefficientsmentioning

confidence: 99%

Singular Euler–Maclaurin expansion on multidimensional lattices

Buchheit

Keßler

2022

Nonlinearity

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We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for a precise and fast evaluation of singular sums that appear in multidimensional long-range interacting systems. We find that the approximation error decays exponentially with the expansion order for band-limited functions and that the runtime is independent of the number of particles. First, the EM summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler–Maclaurin expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite two-dimensional lattices, which are of relevance in condensed matter, statistical, and quantum physics. An implementation in mathematica is provided online along with this article.

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Section: A1 Hsem Operator Coefficientsmentioning

confidence: 99%

Singular Euler–Maclaurin expansion on multidimensional lattices

Buchheit

Keßler

2022

Nonlinearity

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“…The hypothesis states that all zeros of the Riemann zeta function ζ(s) must lie on the critical line ℜs = σ = 1/2, where ℑs = t. The approach taken here is to investigate the proof of the Riemann hypothesis by studying the properties of a quotient function, which is composed of a numerator known to have all its zeros on the critical line, and a denominator equal to ζ(2s − 1/2), together with a balancing function, introduced for reasons to be discussed in Section 2. ThIs approach has previously been used in McPhedran, Williamson, Botten & Nicorovici (2011; hereafter referred to as I) to show that, if the Riemann hypothesis holds for any one of a class of angular lattice sums denoted C(1, 4m; s), then it holds for all of them, where the lowest member of the class C(0, 1; s) = 4ζ(s)L −4 (s) is analytically known (Lorenz, 1871;Hardy, 1920). Here the notation L −4 (s) refers to a particular Dirichlet L or beta function, (Zucker and Robertson, 1976).…”

Section: Introductionmentioning

confidence: 99%

Constructing a Proof of the Riemann Hypothesis

McPhedran¹

2013

Preprint

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This paper compares the distribution of zeros of the Riemann zeta function ζ(s) with those of a symmetric combination of zeta functions, denoted T + (s), known to have all its zeros located on the critical line ℜ(s) = 1/2. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that ζ(s) must also have all its zeros on the critical line: the celebrated Riemann hypothesis. While the argument put forward is not at the level of rigour required to constitute a full proof of the Riemann hypothesis, it should convince non-specialists that it must hold.

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Lattice Sums Then and Now

et al. 2013

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The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

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The Riemann hypothesis and the zero distribution of angular lattice sums

Cited by 3 publications

References 21 publications

Singular Euler–Maclaurin expansion on multidimensional lattices

Singular Euler–Maclaurin expansion on multidimensional lattices

Constructing a Proof of the Riemann Hypothesis

Lattice Sums Then and Now

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