Hardy–Littlewood 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.

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 made 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 42 years later by A.A. Karatsuba.[4]