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 [math]\displaystyle{ \zeta\bigl(\tfrac{1}{2}+it\bigr) }[/math] has infinitely many real zeros.

Let [math]\displaystyle{ N(T) }[/math] be the total number of real zeros, [math]\displaystyle{ N_0(T) }[/math] be the total number of zeros of odd order of the function [math]\displaystyle{ \zeta\bigl(\tfrac{1}{2}+it\bigr) }[/math], lying on the interval [math]\displaystyle{ (0,T] }[/math].

Hardy and Littlewood claimed[2] two conjectures. These conjectures – on the distance between real zeros of [math]\displaystyle{ \zeta\bigl(\tfrac{1}{2}+it\bigr) }[/math] and on the density of zeros of [math]\displaystyle{ \zeta\bigl(\tfrac{1}{2}+it\bigr) }[/math] on intervals [math]\displaystyle{ (T,T+H] }[/math] for sufficiently great [math]\displaystyle{ T \gt 0 }[/math], [math]\displaystyle{ H = T^{a + \varepsilon} }[/math] and with as less as possible value of [math]\displaystyle{ a \gt 0 }[/math], where [math]\displaystyle{ \varepsilon \gt 0 }[/math] is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.

1\. For any [math]\displaystyle{ \varepsilon \gt 0 }[/math] there exists such [math]\displaystyle{ T_0 = T_0(\varepsilon) \gt 0 }[/math] that for [math]\displaystyle{ T \geq T_0 }[/math] and [math]\displaystyle{ H=T^{0.25+\varepsilon} }[/math] the interval [math]\displaystyle{ (T,T+H] }[/math] contains a zero of odd order of the function [math]\displaystyle{ \zeta\bigl(\tfrac{1}{2}+it\bigr) }[/math].

2\. For any [math]\displaystyle{ \varepsilon \gt 0 }[/math] there exist [math]\displaystyle{ T_0 = T_0(\varepsilon) \gt 0 }[/math] and [math]\displaystyle{ c = c(\varepsilon) \gt 0 }[/math], such that for [math]\displaystyle{ T \geq T_0 }[/math] and [math]\displaystyle{ H=T^{0.5+\varepsilon} }[/math] the inequality [math]\displaystyle{ N_0(T+H)-N_0(T) \geq cH }[/math] is true.

## Status

In 1942, Atle Selberg studied the problem 2 and proved that for any [math]\displaystyle{ \varepsilon \gt 0 }[/math] there exists such [math]\displaystyle{ T_0 = T_0(\varepsilon) \gt 0 }[/math] and [math]\displaystyle{ c = c(\varepsilon) \gt 0 }[/math], such that for [math]\displaystyle{ T \geq T_0 }[/math] and [math]\displaystyle{ H=T^{0.5+\varepsilon} }[/math] the inequality [math]\displaystyle{ N(T+H)-N(T) \geq cH\log T }[/math] is true.

In his turn, Selberg made his conjecture[3] that it's possible to decrease the value of the exponent [math]\displaystyle{ a = 0.5 }[/math] for [math]\displaystyle{ H=T^{0.5+\varepsilon} }[/math] which was proved 42 years later by A.A. Karatsuba.[4]

## References

1. ↑ Hardy, G.H. (1914). "Sur les zeros de la fonction [math]\displaystyle{ \zeta(s) }[/math]". 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Hardy–Littlewood zeta-function conjectures. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Hardy–Littlewood_zeta-function_conjectures&oldid=2232668"