On an extremal problem of Selberg

We consider the minimum value mp of the product under the conditions that and . This problem stems from estimation of the Selberg integral appeared in the study of integral-valued entire functions [A. Selberg, Über einen Satz von A. Gelfond, Arch. Math. Naturvid. 44 (1941) 159–170]. We give upper and lower estimates of mp : . This, in particular, slightly improves the earlier lower estimate given by Selberg: logmp>p·0.31654924….