Prime counting function π

Jan Górowski, Adam Łomnicki

Abstract


The aim of this paper is to derive new explicit formulas for thefunction π, where π(x) denotes the number of primes not exceeding x. Some justifications and generalisations of the formulas obtained by Willans (1964),Minac (1991) and Kaddoura and Abdul-Nabi (2012) are also obtained.

Keywords


prime number, prime counting function, congruence

References


Górowski, J., Łomnicki, A.: 2013, Around the Wilson’s theorem, Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia V, 51-56.

Kaddoura, J., Abdul-Nabi, S.: 2012, On formula to compute primes and the n th prime, Applied Math. Sciences 6(76), 3751-3757.

Lagarias, J. C., Miller, V. S., Odlyzko, A. M.: 1985, Computing π(x): the Meissel-Lehmer method, Math. Comp. 44(170), 537-560.

Oliveira e Silva, T.: 2006, Computing π(x): the combinatorial method, Revista do Detua 4(6), 759-768.

Ribenboim, P.: 1991, The little book of big primes, Springer Verlag, New York.

Sierpiński, W.: 1962, Co wiemy a czego nie wiemy o liczbach pierwszych, PZWS, Warszawa.

Willans, C. P.: 1964, On formulae for the n-th prime, Math. Gaz. 48, 413-415.


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