On the Primorial Counting Function

Abstract

Let N_k denote the kth primorial and let K(x) be the primorial counting function, the function that counts the number of primorials not exceeding x. In this note, we examine the sign of the functions $K(x)-\pi (\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x)$ and $K(x)-\text{li}(\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x)$ and find a new necessary and sufficient criterion for the truth of the Riemann hypothesis.

MSC:

• 11A25
• 11A41

Acknowledgment

The author would like to express his great appreciation to the anonymous reviewer for the useful comments and suggestions to improve the quality of this paper. Moreover, the author would also like to thank the two beautiful souls Raul and Oskar for the never ending inspiration.

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 They wrote $K(x)\le {\text{li}}^{-1}(\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x)+0.12\sqrt{\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x}$ but meant $K(x)\le \text{li}(\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x)+0.12\sqrt{\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}x}$.