The prime-counting function and its analytic approximations

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