Hardy–Littlewood zeta-function conjectures

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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.

Conjectures[edit]

In 1914 Godfrey Harold Hardy proved[1] that the Riemann zeta function has infinitely many real zeros.

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[2] 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.

Status[edit]

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.

In his turn, Selberg claim his conjecture[3] that it's possible to decrease the value of the exponent for which was proved forty-two years later by A.A. Karatsuba.[4]

References[edit]

  1. ^ Hardy, G.H. (1914). "Sur les zeros de la fonction ". Comp. Rend. Acad. Sci. 158: 1012–1014. 
  2. ^ Hardy, G.H.; Littlewood, J.E. (1921). "The zeros of Riemann's zeta-function on the critical line". Math. Zeits. 10: 283–317. 
  3. ^ Selberg, A. (1942). "On the zeros of Riemann's zeta-function". Shr. Norske Vid. Akad. Oslo. 10: 1–59. 
  4. ^ 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.