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On the means of the values of prime counting function. (English) Zbl 1468.11190

Let \(A(\pi)\) denote the integral mean of the prime counting function \(\pi(x)\) on the interval \([2, b+2]\), with \(b>5\), and let \(p_n\) denote the largest prime not exceeding \(b+2\). It is proved that, as \(n\) tends to \(\infty\), one has \(A(\pi)= n/2 + O((\log n)/n)\). Defining the geometric and harmonic means in the classical manner as \(G(f)= \exp(A(\log f))\) and \(H(f)= 1/A(1/f))\), similar results for \(G(\pi)\) and \(H(\pi)\) are also proved.

MSC:

11N05 Distribution of primes
26E60 Means
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References:

[1] M. Cipolla, La determinazione assintotica dell’nimo numero primo (asymptotic determination of then-th prime),Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche, Series 3,8, (1902), 132-166. · JFM 33.0214.04
[2] M. Hassani, On the ratio of the arithmetic and geometric means of the prime numbers and the number e,International Journal of Number Theory,9(6), (2013), 1593-1603. · Zbl 1301.11066
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