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Acta Arithmetica

2000 | 94 | 4 | 373-381
Tytuł artykułu

Inequalities concerning the function π(x): Applications

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EN
Abstrakty
EN
Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while $θ(x) = ∑_{p≤x} log p$.
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 $x > e^{3/2}$.
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 → ∞.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
373-381
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-04-26
poprawiono
2000-02-01
Twórcy
autor
• Faculty of Mathematics, University of Bucharest, 14 Academiei St., 70109 Bucureşti, Romania
Bibliografia
• [1] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1981, p. 16.
• [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.
• [3] C. Karanikolov, On some properties of the function π(x), Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 1971, 357-380.
• [4] D. H. Lehmer, On the roots of the Riemann zeta-functions, Acta Math. 95 (1956), 291-298.
• [5] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134.
• [6] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.
• [7] J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), Math. Comp. 129 (1975), 243-269.
• [8] J. B. Rosser and L. Schoenfeld, Abstract of scientific communications, in: Intern. Congr. Math. Moscow, Section 3: Theory of Numbers, 1966.
• [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.
• [10] L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II, Math. Comp. 134 (1976), 337-360.
• [11] V. Udrescu, Some remarks concerning the conjecture π(x+y) < π(x)+π(y), Rev. Roumaine Math. Pures Appl. 20 (1975), 1201-1208.
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Bibliografia
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