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

In 1914, Godfrey Harold Hardy proved[1] that the Riemann zeta function ζ(12+it) has infinitely many real zeros.

Let N(T) be the total number of real zeros, N0(T) be the total number of zeros of odd order of the function ζ(12+it), lying on the interval (0,T].

Hardy and Littlewood claimed[2] two conjectures. These conjectures – on the distance between real zeros of ζ(12+it) and on the density of zeros of ζ(12+it) on intervals (T,T+H] for sufficiently great T>0, H=Ta+ε and with as less as possible value of a>0, where ε>0 is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.

1\. For any ε>0 there exists such T0=T0(ε)>0 that for T≥T0 and H=T0.25+ε the interval (T,T+H] contains a zero of odd order of the function ζ(12+it).

2\. For any ε>0 there exist T0=T0(ε)>0 and c=c(ε)>0, such that for T≥T0 and H=T0.5+ε the inequality N0(T+H)−N0(T)≥cH is true.

## Status

In 1942, Atle Selberg studied the problem 2 and proved that for any ε>0 there exists such T0=T0(ε)>0 and c=c(ε)>0, such that for T≥T0 and H=T0.5+ε the inequality N(T+H)−N(T)≥cHlog⁡T is true.

In his turn, Selberg made his conjecture[3] that it's possible to decrease the value of the exponent a=0.5 for H=T0.5+ε which was proved 42 years later by A.A. Karatsuba.[4]

## References

1. ↑ Hardy, G.H. (1914). "Sur les zeros de la fonction ζ(s)". 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. https://zenodo.org/record/1447415.
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.

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