Hassani, M. On the means of the values of prime counting function. (English) Zbl 1468.11190 Iran. J. Math. Sci. Inform. 13, No. 1, 15-22 (2018). 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. Reviewer: József Sándor (Cluj-Napoca) MSC: 11N05 Distribution of primes 26E60 Means Keywords:prime counting function; integral mean of a function PDFBibTeX XMLCite \textit{M. Hassani}, Iran. J. Math. Sci. Inform. 13, No. 1, 15--22 (2018; Zbl 1468.11190) Full Text: Link 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.