Publisher's Note: “Relation between primes and nontrivial zeros in the Riemann hypothesis; Legendre polynomials, modified zeta function and Schrödinger equation” [J. Math. Phys. 53, 122108 (2012)]

We study the question addressed by Barut and Dilley ( Barut & Dilley 1963 J. Math. Phys . 4 , 1401–1408) of counting the number of Regge poles for a radial Schrödinger equation. Using the asymptotics of Rudolph Langer, we acquire estimates for the free solutions at infinity for large generalized complex angular momentum | λ |. These estimates allow us to calculate the Wronskian of two particular solutions, which is the function whose zeros are the Regge poles, for large | λ | in the right-half λ -plane. These angular momentum asymptotics are rigorously related to the large-radius asymptotics by generalizing Marianna Shubova’s idea of formulating an integral equation for the solution at infinity. This leads to the proof that for integrable potentials there are only finitely many Regge poles. This should be compared with the ideas of Barut and Dilley, who require that the potential be analytic in the right-half plane with r 2 V ( r ) remaining bounded.

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The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽

10.3390/sym13122410 ◽

2021 ◽

Vol 13 (12) ◽

pp. 2410

Author(s):

Janyarak Tongsomporn ◽

Saeree Wananiyakul ◽

Jörn Steuding

Keyword(s):

Asymptotic Formula ◽

Zeta Function ◽

Riemann Zeta Function ◽

Real Line ◽

Riemann Hypothesis ◽

Critical Line ◽

Nontrivial Zeros ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v5 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

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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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v3 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

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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Farey Sequences and the Riemann Hypothesis

10.20944/preprints201607.0003.v1 ◽

2016 ◽

Author(s):

Darrell Cox

Keyword(s):

Zeta Function ◽

Riemann Hypothesis ◽

Analytic Number Theory ◽

Square Root ◽

Nontrivial Zeros ◽

Landau Theorem ◽

Farey Sequences ◽

The Riemann Zeta Function ◽

Riemann Zeta ◽

Function Zeros

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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