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The \(k\)th prime is greater than \(k(\ln k+\ln\ln k-1)\) for \(k\geq 2\). (English) Zbl 0913.11039

The prime number theorem shows that the \(k\)-th prime \(p_k\) has the asymptotic value \(k\log k\) as \(k\to\infty\). Applying the theorem with an error term, M. Cipolla [Napoli Rend (3) 8, 132–166 (1902; JFM 33.0214.04)] obtained an asymptotic formula for \(p_k\) with leading terms \(k(\log k+\log\log k-1+\cdots)\) and an error term \(O\big(k(\log\log k/\log k)^3\big)\).
Good explicit bounds for \(p_k\) can be obtained from estimates for the Chebyshev functions and calculations of the zeros of the Riemann-zeta function. Thus, J. B. Rosser [Proc. Lond. Math. Soc. (2) 45, 21–44 (1939; Zbl 0019.39401)] first found that \(p_k>k\log k\) for \(k>1\), and together with L. Schoenfeld [Math. Comput. 29, 243–269 (1975; Zbl 0295.10036)] extended the result to \(p_k>k(\log k+\log\log k-c)\), with \(c=3/2\). Later, G. Robin [Acta Arith. 42, 367–389 (1983; Zbl 0525.10024)] improved this to \(c=1.0072629\), and together with J.-P. Massias [J. Théor. Nombres Bordx. 8, 215–242 (1996; Zbl 0856.11043)] found that \(c=1\) is admissible for \(1<k\leq\exp(598)\) and \(k\geq\exp(1800)\).
The author shows that the estimate \[ | \psi(x)-x| \leq 0.905{\times}10^{-7}x \] holds for \(x\geq\exp(50)\), thereby deducing that the above result with \(c=1\) is valid for all \(k>1\). It is also shown that \[ k(\log p_k-2)<p_k<k\min(\log p_k,\log k+\log\log k)\quad\text{when } k\geq 6. \]

MSC:

11N05 Distribution of primes
11A41 Primes
11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
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[1] R. P. Brent, J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. II, Math. Comp. 39 (1982), no. 160, 681 – 688. , https://doi.org/10.1090/S0025-5718-1982-0669660-1 Herman J. J. te Riele, Corrigenda: ”On the zeros of the Riemann zeta function in the critical strip. II” [Math. Comp. 39 (1982), no. 160, 681 – 688; MR0669660 (83m:10067)] by R. P. Brent, J. van de Lune, te Riele and D. T. Winter, Math. Comp. 46 (1986), no. 174, 771. · Zbl 0486.10028
[2] M. CIPOLLA, La determinazione assintotica dell’\(n^{imo}\) numero primo, Matematiche Napoli 3 (1902), 132-166. · JFM 33.0214.04
[3] J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667 – 681. · Zbl 0585.10023
[4] Jean-Pierre Massias and Guy Robin, Bornes effectives pour certaines fonctions concernant les nombres premiers, J. Théor. Nombres Bordeaux 8 (1996), no. 1, 215 – 242 (French, with French summary). · Zbl 0856.11043
[5] Guy Robin, Estimation de la fonction de Tchebychef \? sur le \?-ième nombre premier et grandes valeurs de la fonction \?(\?) nombre de diviseurs premiers de \?, Acta Arith. 42 (1983), no. 4, 367 – 389 (French). · Zbl 0475.10034
[6] J. B. ROSSER, The \(n\)-th prime is greater than \(n\log n\), Proc. London Math. Soc. (2) 45 (1939), 21-44. · JFM 64.0100.04
[7] J. B. ROSSER, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211-232. · JFM 67.0129.03
[8] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64 – 94. · Zbl 0122.05001
[9] J. B. ROSSER & L. SCHOENFELD, Sharper Bounds for the Chebyshev Functions \(\theta(x)\) and \(\psi(x)\), Math. Of Computation 29 Number 129 (January 1975), 243-269. · Zbl 0295.10036
[10] L. SCHOENFELD, Sharper Bounds for the Chebyshev Functions \(\theta(x)\) and \(\psi(x)\).II, Math. Of Computation 30 Number 134 (April 1976), 337-360. · Zbl 0326.10037
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