, Volume 29, Issue 1, pp 55-70
Date: 04 May 2007

The prime-counting function and its analytic approximations

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Abstract

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}} {{\log \,t}}} }\) , \({\text{li}}{\left( x \right)} - \frac{1} {2}{\text{li}}{\left( {{\sqrt x }} \right)}\) , and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0em} k}} } \right)}} k}} \right. \kern-0em} k} }\) , where μ is the Möbius function. The results show that π(x)x) for 2≤x≤1014, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}\) and \( - \frac{2} {5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.

Communicated by L. Reichel