Selberg's zeta function conjecture

In 1942 Atle Selberg conjectured that for a fixed $\varepsilon$ satisfying the condition $0<\varepsilon <0.001$, a sufficiently large $T$ and $H=T^{a+\varepsilon }$, $a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}$, the interval $(T,T+H)$ contains at least $cH\ln T$ real zeros of the Riemann zeta function $\zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )}$. Atle Selberg proved it for the case $H\geq T^{1/2+\varepsilon }$.

In 1984 Anatolii Alexeevitch Karatsuba proved the Selberg conjecture.

In 1992 A.A. Karatsuba proved, that an analog of the Selberg conjecture holds for «almost all» intervals $(T,T+H]$, $H=T^{\varepsilon }$, where $\varepsilon$ is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals $(T,T+H]$, the length $H$ of which grows slower than any, even arbitrarily small degree $T$. In particular, he proved that for any given numbers $\varepsilon$, $\varepsilon _{1}$ satisfying the conditions $0<\varepsilon ,\varepsilon _{1}<1$ almost all intervals $(T,T+H]$ for $H\geq \exp {\{(\ln T)^{\varepsilon }\}}$ contain at least $H(\ln T)^{1-\varepsilon _{1}}$ zeros of the function $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$. This estimate is quite close to the one that follows from the Riemann hypothesis.