Inequalities concerning the function π(x): Applications
Laurenţiu Panaitopol
Acta Arithmetica, Tome 92 (2000), p. 373-381 / Harvested from The Polish Digital Mathematics Library
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Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while θ(x)=∑p≤xlogp. The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for x>e3/2. The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe and Schoenfeld announced that the first 3 500 000 zeros of ξ(s) lie on the critical line [9]. This result was followed by two papers [7], [10]; some of the inequalities they include will be used in order to obtain inequalities (11) and (12) below. In [6] it is proved that π(x)   x/(log x - 1). Here we will refine this expression by giving upper and lower bounds for π(x) which both behave as x/(log x - 1) as x → ∞.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:207436 
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Laurenţiu Panaitopol. Inequalities concerning the function π(x): Applications. Acta Arithmetica, Tome 92 (2000) pp. 373-381. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav94i4p373bwm/

[00000] [1] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1981, p. 16.

[00001] [2] D. K. Hensley and I. Richards, On the incompatibility of two conjectures concerning primes, in: Proc. Sympos. Pure Math. 24, H. G. Diamond (ed.), Amer. Math. Soc., 1974, 123-127. | Zbl 0264.10031

[00002] [3] C. Karanikolov, On some properties of the function π(x), Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 1971, 357-380.

[00003] [4] D. H. Lehmer, On the roots of the Riemann zeta-functions, Acta Math. 95 (1956), 291-298. | Zbl 0071.06603

[00004] [5] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134. | Zbl 0296.10023

[00005] [6] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. | Zbl 0122.05001

[00006] [7] J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), Math. Comp. 129 (1975), 243-269. | Zbl 0295.10036

[00007] [8] J. B. Rosser and L. Schoenfeld, Abstract of scientific communications, in: Intern. Congr. Math. Moscow, Section 3: Theory of Numbers, 1966.

[00008] [9] J. B. Rosser, J. M. Yohe and L. Schoenfeld, Rigorous computation and the zeros of the Riemann zeta functions, in: Proc. IFIP Edinburgh, Vol. I: Mathematics Software, North-Holland, Amsterdam, 1969, 70-76. | Zbl 0191.44504

[00009] [10] L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II, Math. Comp. 134 (1976), 337-360. | Zbl 0326.10037

[00010] [11] V. Udrescu, Some remarks concerning the conjecture π(x+y) < π(x)+π(y), Rev. Roumaine Math. Pures Appl. 20 (1975), 1201-1208. | Zbl 0319.10009