Inequalities concerning the function π(x): Applications

Laurenţiu Panaitopol

Acta Arithmetica (2000)

  • Volume: 94, Issue: 4, page 373-381
  • ISSN: 0065-1036

Abstract

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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 ≤ x l o g 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 → ∞.

How to cite

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Laurenţiu Panaitopol. "Inequalities concerning the function π(x): Applications." Acta Arithmetica 94.4 (2000): 373-381. <http://eudml.org/doc/207436>.

@article{LaurenţiuPanaitopol2000,
abstract = {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 → ∞.},
author = {Laurenţiu Panaitopol},
journal = {Acta Arithmetica},
keywords = {number of prime numbers; inequality; bounds; Chebyshev functions},
language = {eng},
number = {4},
pages = {373-381},
title = {Inequalities concerning the function π(x): Applications},
url = {http://eudml.org/doc/207436},
volume = {94},
year = {2000},
}

TY - JOUR
AU - Laurenţiu Panaitopol
TI - Inequalities concerning the function π(x): Applications
JO - Acta Arithmetica
PY - 2000
VL - 94
IS - 4
SP - 373
EP - 381
AB - 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 → ∞.
LA - eng
KW - number of prime numbers; inequality; bounds; Chebyshev functions
UR - http://eudml.org/doc/207436
ER -

References

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  1. [1] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1981, p. 16. 
  2. [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. Zbl0264.10031
  3. [3] C. Karanikolov, On some properties of the function π(x), Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. 1971, 357-380. 
  4. [4] D. H. Lehmer, On the roots of the Riemann zeta-functions, Acta Math. 95 (1956), 291-298. Zbl0071.06603
  5. [5] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134. Zbl0296.10023
  6. [6] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. Zbl0122.05001
  7. [7] J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), Math. Comp. 129 (1975), 243-269. Zbl0295.10036
  8. [8] J. B. Rosser and L. Schoenfeld, Abstract of scientific communications, in: Intern. Congr. Math. Moscow, Section 3: Theory of Numbers, 1966. 
  9. [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. Zbl0191.44504
  10. [10] L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II, Math. Comp. 134 (1976), 337-360. Zbl0326.10037
  11. [11] V. Udrescu, Some remarks concerning the conjecture π(x+y) < π(x)+π(y), Rev. Roumaine Math. Pures Appl. 20 (1975), 1201-1208. Zbl0319.10009

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