Despite outstanding results using sieve theory to obtain upper bounds on the number of primes or prime types (e.g., twin primes) up to a given number, it has proven to be somewhat disappointing in obtaining lower bounds. Some of the leading causes of the obstacles are reduced in this paper by using an alternative reading of the sieve  of Eratosthenes. This allows us, to avoid the difficulties caused by the Möbius function, twist the Legendre formula, bypass the parity problem and obtain a lower bound of the number of primes between  and n. In addition, we reprove Euclid’s theorem - that there are infinitely many primes - by arguably using only sieve theoretic means.

sieve theory, lower bound, prime numbers, prime counting function.

Received: March 1, 2021; Revised: April 26, 2021; Accepted: June 23, 2021; Published: July 3, 2021

How to cite this article: Madieyna Diouf, A lower bound on the number of primes between  and n, JP Journal of Algebra, Number Theory and Applications 51(2) (2021), 183-211. DOI: 10.17654/NT051020183

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] Carl Friedrich Gauss, Disquisitiones arithemeticae, Translated into English, by Arthur A. Clarke, 2nd, Corrected edition, Springer, New York, 1986.
[2] Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), 1121-1174.
[3] Daniel A. Goldston, Janos Pintz and Cem. Y. Yildirim, Primes in tuples. I, Ann. of Math. (2) 170 (2009), 819-862.
[4] V. Brun, Le crible d’Eratosthne et le thorme de Goldbach, Skr. Norske Vid.-Akad. Kristiania I, No. 3, 1920.
[5] J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157-176.
[6] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math. 78 (1874), 46-62 (in German).
[7] Nicomachus, of Gerasa, M. L. D’Ooge, F. E. Robbins and L. C. Karpinsk, Introduction to Arithmetic, Macmillan, New York, 1926.
[8] J. Hadamard, Un travail de jean merlin sur les nombres premiers, Bull. Sci. Math. 50 (1915), 121-136.
[9] A. Selberg, Sieve methods, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., Vol. 20, 1971, pp. 311-351.
[10] H. Iwaniec, Sieve methods, Proceedings of the International Congress of Mathematicians, Vol. 1, 1978, pp. 357-364.
[11] James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), 383-413.
[12] H. Halberstam and E. H. Richert, Sieve Methods, Academic Press, London, 1974.
[13] John Friedlander and Henryk Iwaniec, Asymptotic sieve for primes, Ann. of Math. (2) 148(3) (1998), 1041-1065. arXiv:math/9811186.
[14] Terence Tao, The parity problem in sieve theory,
https://terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sievetheory/
[15] Wikipedia contributors, Parity problem (sieve theory), Wikipedia, The Free Encyclopedia, 14 December 2020.
[16] Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167 (2008), 481-547.
[17] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2006.
[18] Wikipedia contributors, Prime omega function, Wikipedia, The Free Encyclopedia, 20 January 2021.
[19] Pierre Dusart, Estimates of some functions over primes without RH, 2010. arXiv preprint arXiv:1002.0442.
[20] M. Nair, On Chebyshev-type inequalities for primes, Amer. Math. Monthly 89(2) (1982), 126-129. doi:10.2307/2320934.
[21] G. H. Hardy, Note on a theorem of Mertens, J. London Math. Soc. 2 (1927), 70-72.
[22] Wikipedia contributors, Divergence of the sum of the reciprocals of the primes, Wikipedia, The Free Encyclopedia, 13 January 2021.