The prime index function

DOI10.1007/S13226-020-0458-9zbMATH Open1471.11016arXiv1905.03112OpenAlexW3091903236MaRDI QIDQ2221092

Theophilus Agama

Publication date: 26 January 2021

Published in: Indian Journal of Pure \& Applied Mathematics (Search for Journal in Brave)

Abstract: In this paper we introduce the prime index function �egin{align}iota(n)=(-1)^{pi(n)},

onumber end{align} where $pi(n)$ is the prime counting function. We study some elementary properties and theories associated with the partial sums of this function given by�egin{align}xi(x):=sum limits_{nleq x}iota(n). onumber end{align}We show that a prime $p>2$ is a twin prime if and only if $xi(p)=xi(p+2)$. We also relate the prime index function to Cramer's conjecture by showing that �egin{align}|xi(p_{n+1})-xi(p_n)|+2=p_{n+1}-p_n. onumber end{align}That is, Cramer's conjecture can be stated as �egin{align}xi(p_{n+1})-xi(p_n)ll (log p_n)^2.

onumber end{align}This reduces the problem to obtaining very good estimates of the second prime index function.

Full work available at URL: https://arxiv.org/abs/1905.03112

zbMATH Keywords

oscillationindexperiodprime

Mathematics Subject Classification ID

Primes (11A41)

Cites Work

• Title not available (Why is that?)
• Title not available (Why is that?)

Related Items (1)

Amalgams of prime index

This page was built for publication: The prime index function

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2221092)