2017
DOI: 10.1093/imamat/hxx033
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Riemann’s zeta function and the broadband structure of pure harmonics

Abstract: Let a ∈ (0, 1) and let F s (a) be the periodized zeta function that is defined as F s (a) = n −s exp(2πina) for s > 1, and extended to the complex plane via analytic continuation. Let s n = σ n + it n , t n > 0, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, the Cesàro means of the sequence F sn (a) converge to the first harmonic exp(2πia) in the sense of periodic dis… Show more

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Cited by 4 publications
(7 citation statements)
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“…The latter shows how it manifests itself in two dimensions and how that translates into the language of quantum theory. It is likely that neither one of these two theorems is optimal in its present form as, indeed, based on conjectures discussed in [24] one may expect stronger statements to be true. Finally, we point out that at this stage too little is known to be sure whether the broadband redundancy is only a mathematical concept applicable to engineered systems or, indeed, a phenomenon that occurs in nature which ought to be considered in the fundamental quantum theory.…”
Section: Introductionmentioning
confidence: 94%
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“…The latter shows how it manifests itself in two dimensions and how that translates into the language of quantum theory. It is likely that neither one of these two theorems is optimal in its present form as, indeed, based on conjectures discussed in [24] one may expect stronger statements to be true. Finally, we point out that at this stage too little is known to be sure whether the broadband redundancy is only a mathematical concept applicable to engineered systems or, indeed, a phenomenon that occurs in nature which ought to be considered in the fundamental quantum theory.…”
Section: Introductionmentioning
confidence: 94%
“…It was observed in [24] that a pure tone, say sin x, can be well approximated by averages of certain special broadband signals (i.e. periodic functions whose Fourier coefficients decay relatively slowly).…”
Section: The Periodized Zeta Function and The Broadband Redundancymentioning
confidence: 99%
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“…First, D-matrices are used as effective signal processing tools [14], [16], and arise naturally in the analysis of nonlinear oscillators [15]. Second, they are the basic tool in applications of the newly discovered phenomenon of broadband redundancy [19], in particular in its quantummechanical applications [20]. Third, their finite-dimensional reductions turn out to furnish the universal building blocks of generic matrices [17]-an observation that one may hope to generalize to infinite dimensions once the topological properties of infinite D-matrices are well understood.…”
mentioning
confidence: 99%
“…However, an unconditional proof of this result was reported in[27] 4. The statement (35) about the Fourier coefficients of the periodized zeta function is trivial for σ > 1, but requires a nontrivial argument when 1 ≥ σ > 0, see[24]. In this article we will only consider the former case.…”
mentioning
confidence: 99%