On the ratio of the arithmetic and geometric means of the prime numbers and the number \(e\) (Q2850710)

The author proves bounds for the arithmetic mean \(A_n\) and geometric mean \(G_n\) of the first \(n\) primes. For example, for all \(n\geq 2\) one has \(e/2-14.951/\log n< A_n/G_n> e/2+ 9.514/\log n\). He is not aware of the papers by \textit{L. Panaitopol} [Notes Number Theory Discrete Math. 5, No. 2, 52--54 (1999)], as well as the reviewer [ibid. 18, No. 1, 1--5 (2012; Zbl 1286.11013)], where many related bounds are proved. For example, Panaitopol proved that \(G_n<(1/e)\,p_n\) for all \(n\geq 10\), which is Conjecture 4.3 in the paper under review.