Erratum to “The Symmetric Series of Multiples of Primes” [Advances in Pure Mathematics, 12 (2022) 160-177]

Abstract

The original online version of this article (Doroszlai, P. and Keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://doi.org/10.4236/apm.2022.123014) unfortunately contains some small mistakes. The author discovered the possible problem in Equations (1.6), (1.7) and in the last equation of the proof of lemma 2.3 and the numbering of the chapters 3 and 4 should be changed. To fix the problem, the author wishes to change these equations and titles.

Keywords

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Doroszlai, P. and Keller, H. (2022) Erratum to “The Symmetric Series of Multiples of Primes” [Advances in Pure Mathematics, 12 (2022) 160-177]. Advances in Pure Mathematics, 12, 742-743. doi: 10.4236/apm.2022.1212056.

1) Page 163 Equation (6) should be revised to the following:

${\tau }_{appr}\left(c\right)\approx \underset{n=1}{\overset{\sqrt{c}}{\sum }}\frac{{\delta }_{2}}{\mathrm{ln}\left(c\right)\mathrm{ln}\left(n\right)\mathrm{ln}\left(2c-n\right)}$ (1.6)

2) Page 163 Equation (7) should be revised to the following:

${\tau }_{appr}\left(c\right)=\underset{j=1}{\overset{\sqrt{c}}{\sum }}\left(\underset{n=\left(j-1\right)\sqrt{c}}{\overset{j\sqrt{c}}{\sum }}\frac{{\delta }_{2}}{\mathrm{ln}\left(c\right)\mathrm{ln}\left(n\right)\mathrm{ln}\left(2c-n\right)}\right)\approx \underset{j=1}{\overset{\sqrt{c}}{\sum }}\frac{{\delta }_{2}\sqrt{c}}{\mathrm{ln}\left(c\right)\mathrm{ln}\left(j\sqrt{c}\right)\mathrm{ln}\left(2c-j\sqrt{c}\right)}$ (1.7)

3) Page 166 The numbering of the title should be changed to:

2. The low limit of the number of triads

4) Page 168 The numbering of the title should be changed to:

3. Consequences of the double density of occupation

5) Page 168 The last equation in lemma 2.3 should be changed to the following:

$\begin{array}{c}{\left[{P}_{\left(R\left(c\right)+1\right)}\right]}^{2}={\left[{P}_{\left(R\left(c\right)\right)}+2\right]}^{2}\\ ={\left[{P}_{\left(R\left(c\right)\right)}\right]}^{2}+4{P}_{\left(R\left(c\right)\right)}+4\ge {\left(\sqrt{c}-1\right)}^{2}+4\left(\sqrt{c}-1\right)+4\\ =c+2\sqrt{c}+1>c+\sqrt{c}\end{array}$

Conflicts of Interest

The authors declare no conflicts of interest.