Hardy–Littlewood zeta function conjectures

Conjectures

In 1914, Godfrey Harold Hardy proved that the Riemann zeta function ${\displaystyle \zeta (\frac{1}{2}+it)}$ 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 ${\displaystyle \zeta (\frac{1}{2}+it)}$, lying on the interval ${\displaystyle (0,T]}$.

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

1. For any ${\displaystyle \epsilon >0}$ there exists such ${\displaystyle T_0=T_0(\epsilon )>0}$ that for ${\displaystyle T\ge T_0}$ and ${\displaystyle H=T^{0.25+\epsilon }}$ the interval ${\displaystyle (T,T+H]}$ contains a zero of odd order of the function ${\displaystyle \zeta (\frac{1}{2}+it)}$.

2. For any ${\displaystyle \epsilon >0}$ there exist ${\displaystyle T_0=T_0(\epsilon )>0}$ and ${\displaystyle c=c(\epsilon )>0}$, such that for ${\displaystyle T\ge T_0}$ and ${\displaystyle H=T^{0.5+\epsilon }}$ the inequality ${\displaystyle N_0(T+H)-N_0(T)\ge cH}$ is true.