Hardy–Littlewood zeta-function conjectures
In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function.
Let be the total number of real zeros, be the total number of zeros of odd order of the function , lying on the interval .
Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of and on the density of zeros of on intervals for sufficiently great , and with as less as possible value of , where is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.
1. For any there exists such that for and the interval contains a zero of odd order of the function .
2. For any there exist and , such that for and the inequality is true.
In 1942 Atle Selberg studied the problem 2 and proved that for any there exists such and , such that for and the inequality is true.
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- Hardy, G.H.; Littlewood, J.E. (1921). "The zeros of Riemann's zeta-function on the critical line". Math. Zeits. 10: 283–317.
- Selberg, A. (1942). "On the zeros of Riemann's zeta-function". Shr. Norske Vid. Akad. Oslo 10: 1–59.
- Karatsuba, A. A. (1984). "On the zeros of the function ζ(s) on short intervals of the critical line". Izv. Akad. Nauk SSSR, Ser. Mat. 48 (3): 569–584.