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.

Conjectures

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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

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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 made his conjecture[3] that it's possible to decrease the value of the exponent   for   which was proved 42 years later by A.A. Karatsuba.[4]

References

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  1. ^ Hardy, G.H. (1914). "Sur les zeros de la fonction  ". Compt. 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. Z. 10 (3–4): 283–317. doi:10.1007/bf01211614. S2CID 126338046.
  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.