# 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 ${\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}}$ has infinitely many real zeros.

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

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

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

2. For any ${\displaystyle \varepsilon >0}$ there exist ${\displaystyle T_{0}=T_{0}(\varepsilon )>0}$ and ${\displaystyle c=c(\varepsilon )>0}$, such that for ${\displaystyle T\geq T_{0}}$ and ${\displaystyle H=T^{0.5+\varepsilon }}$ the inequality ${\displaystyle N_{0}(T+H)-N_{0}(T)\geq cH}$ is true.

## Status

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

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

## References

1. ^ Hardy, G.H. (1914). "Sur les zeros de la fonction ${\displaystyle \zeta (s)}$". 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.