Harmonic sets and the harmonic prime number theorem

Publisher： Cambridge University Press

E-ISSN： 1755-1633|71|1|127-137

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.71, Iss.1, 2005-02, pp. : 127-137

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.

Related content

The Gauss 2F1(1)-summation theorem and harmonic number identities

By Chen Xiaojing  

Integral Transforms and Special Functions, Vol. 20, Iss. 12, 2009-12 ,pp. : 925-935 (11)

Taylor & Francis Ltd

Access to resources Recommend Favorite

A Theorem of Liouville Type for Harmonic Morphisms

By Choi G.   Yun G.  

Geometriae Dedicata, Vol. 84, Iss. 1-3, 2001-02 ,pp. : 179-182 (4)

Springer Publishing Company

Access to resources Recommend Favorite

A Discrete Approach to the Prime Number Theorem

By Trigiante G.   Trigiante D.  

Journal of Difference Equations and Applications, Vol. 8, Iss. 1, 2002-01 ,pp. : 93-100 (8)

Taylor & Francis Ltd

Access to resources Recommend Favorite

Sets of Harmonicity for Finely Harmonic Functions

By Gardiner Stephen J.  

Potential Analysis, Vol. 21, Iss. 1, 2004-08 ,pp. : 1-6 (6)

Springer Publishing Company

Access to resources Recommend Favorite

Telescoping method, derivative operators and harmonic number identities

By Wei Chuanan   Gong Dianxuan   Yan Qinglun  

Integral Transforms and Special Functions, Vol. 25, Iss. 3, 2014-03 ,pp. : 203-214 (12)

Taylor & Francis Ltd

Access to resources Recommend Favorite