(RETRACTED) A note on prime zeta function and Riemann zeta function

RETRACTION NOTICE

As of June 2021, the paper “A note on prime zeta function and Riemann zeta function” by Mladen Vassilev–Missana, published in Notes on Number Theory and Discrete Mathematics, Vol. 22, 2016, No. 4, 12–15, has been retracted.

The reasons for retraction are identified errors, as pointed out by Richard P. Brent in the paper “On two theorems of Vassilev-Missana”, submitted on 23 March 2021 and immediately published in the June issue of Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 2, 49–50, DOI: 10.7546/nntdm. 2021.27.2.49-50. http://nntdm.net/volume-27-2021/number-2/49-50/

Correction entitled “A note on prime zeta function and Riemann zeta function. Corrigendum” was provided by Mladen Vassilev–Missana on 21 April 2021 and published in the same issue of Notes on Number Theory and Discrete Mathematics, Vol. 27, 2021, No. 2, 51–53, DOI: 10.7546/nntdm.2021.27.2.51-53. http://nntdm.net/volume-27-2021/number-2/51-53/

The electronic copy of the retracted paper is retained on the Journal website in order to maintain the scientific record, with the additional “Retracted Paper” watermark and the accompanying Retraction Notice, which as of June 2021 must be considered an integral part of the publication.

The Publisher apologizes for any inconvenience caused!

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Mladen Vassilev–Missana
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 12–15
Full paper (PDF, 135 Kb) (RETRACTED)

Details

Authors and affiliations

Mladen Vassilev–Missana
5 Victor Hugo Str, Ap. 3
1124 Sofia, Bulgaria

Abstract

In the present paper, we first deduce a new recurrent formula, that connects P(s), P(2s) and ζ(s), where P(s) is the prime zeta function and ζ(s) is Riemann zeta function. After that, with the help of this recurrent formula, we find a new formula for P(s) expressing P(s) as infinite nested radicals (roots), depending on the values of ζ(2^ks) for k = 0, 1, 2, 3, … .

Keywords

• Prime zeta function
• Riemman zeta function
• Prime numbers

AMS Classification

• 11A25
• 11M06

References

1. Glaisher, J. W. L. (1891) On the Sums of Inverse Powers of the Prime Numbers. Quart. J. Math., 25, 347–362.
2. Froberg, C.-E. (1968) On the Prime Zeta Function. BIT 8, 187–202.
3. Cohen, H. (2000) Advanced Topics in Computational Number Theory. New York: Springer-Verlag.