Farey series and the Riemann hypothesis

Kanemitsu, Shigeru; Yoshimoto, Masami

doi:10.4064/aa-75-4-351-374

Cited by 30 publications

(24 citation statements)

References 6 publications

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“…The real part of (1.7) is which is another example of Riemann discussed by Casorati and Prym in 1865 11. We also note the result [8, p. 442, Corollary] which connects (1.8) to the Riemann Hypothesis. It fascinates us to imagine then that this contrast between the real and imaginary parts of (1.7) might have braked Riemann to study the real part (just denoting it by A ) in his posthumous fragment 13 (cf.…”

Section: Introductionmentioning

confidence: 60%

Arithmetical identities and zeta‐functions

Kanemitsu

Tanigawa

2010

Mathematische Nachrichten

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Dedicated to Professor Andrzej Shinzel on his seventieth birthday, sto lat, sto lat, neich zyje nam.In this paper we establish a class of arithmetical Fourier series as a manifestation of the intermediate modular relation, which is equivalent to the functional equation of the relevant zeta-functions. One of the examples is the one given by Riemann as an example of a continuous non-differentiable function.The novel interest lies in the relationship between important arithmetical functions and the associated Fourier series. E.g., the saw-tooth Fourier series is equivalent to the corresponding arithmetical Fourier series with the Möbius function. Further, if we squeeze out the modular relation, we are led to an interesting relation between the singular value of the discontinuous integral and the modification summand of the first periodic Bernoulli polynomial.

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Section: Introductionmentioning

confidence: 60%

Arithmetical identities and zeta‐functions

Kanemitsu

Tanigawa

2010

Mathematische Nachrichten

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“…Let r 1 , r 2 , ..., r A(x) denote the terms of the Farey sequence of order x and let h(ξ) denote the number of r v less than or equal to ξ. Kanemitsu and Yoshimoto [4] proved that each of the estimates ∑ r v ≤1/3 (r v − h(1/3)…”

Section: Shorter Intervals Of Farey Pointsmentioning

confidence: 99%

Farey Sequences and the Riemann Hypothesis

Cox¹

2016

Preprint

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Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev's second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function -the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

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“…The main results of [6] were two order of magnitude estimates which do not appear to follow from Theorem 2. In the notation of this paper, they are [7], who have many interesting results with weights of the form w(a/q), where w(x) is a function on the real line.…”

Section: Introductionmentioning

confidence: 99%