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.
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References
Bays, C., Hudson, R.: A new bound for the smallest x with π(x)>li(x). Math. Comp. 69, 1285–1296 (2000)
Berndt, B.C.: Ramanujan’s Notebooks, Part IV. pp. 126–131. Springer, New York (1994)
Brent, R.P.: Irregularities in the distribution of primes and twin primes. Math. Comp. 29, 43–56 (1975)
de la Vallée Poussin, C.J.: Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mem. Cour. Acad. Roy. Belg. 59, 1 (1899)
Ford, K.: Vinogradov’s integral and bounds for the Riemann zeta function. Proc. London Math. Soc. 85, 565–633 (2002)
Gauss, C.F.: Werke, Vol. II. Königliche Gesellschaft der Wissenschaften zu Göttingen, pp. 444–447 (1863)
Gram, J.P.: Undersøgelser angående Mængden af Primtal under en given Grænse. Kong. Dansk. Videnskab. Selsk. Skr. (VI) 2, 183–308 (1884)
Ingham, A.E.: The distribution of prime numbers, pp. 105–106. Cambridge University Press, New York (1932)
Korobov, N.M.: Estimates of trigonometric sums and their applications. Usp. Mat. Nauk. 13, 185–192 (1958) (in Russian)
Legendre, A.M.: Essai sur la théorie des nombres, 2ème dition, p.394. Courcier, Paris (1808)
Lehman, R.S.: On the difference π(x)−li(x). Acta Arith. 11, 397–410 (1966)
Littlewood, J.E.: Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris 158, 1869–1872 (1914)
Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsber. Preuss. Akad. Wiss. 1859 671–680 (1859)
Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64–94 (1962)
Skewes, S.: On the difference π(x)−li(x). II. Proc. London Math. Soc. (3) 5, 48–70 (1955)
te Riele, H.J.J.: On the sign of the difference π(x)−li(x). Math. Comp. 48, 323–328 (1987)
Vinogradov, I.M.: A new estimate for ζ(1+it) [in Russian]. Izv. Akad. Nauk SSSR, Ser. Mat. 22, 161–164 (1958)
von Koch, H.: Sur la distribution des nombres premiers. Acta Math. 24, 159–182 (1901)
von Mangoldt, H.: Zu Riemann’s Abhandlung ‘Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse’. J. Reine Angew. Math. 114, 255–305 (1895)
Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie, pp. 175–188. VEB Deutscher Verlag, Berlin (1963)
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Kotnik, T. The prime-counting function and its analytic approximations. Adv Comput Math 29, 55–70 (2008). https://doi.org/10.1007/s10444-007-9039-2
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DOI: https://doi.org/10.1007/s10444-007-9039-2