Selberg's zeta function conjecture

The Atle Selberg conjecture is about the density of zeros of the Riemann zeta function $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$.

Two Hardy-Littlewood conjectures

In 1914 Godfrey Harold Hardy proved that $\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}$ has infinitely many real zeros.

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

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

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

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

The Selberg conjecture

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

In his turn, Selberg claim a conjecture that it's possible to decrease the value of the exponent $a=0.5$ for $H=T^{0.5+\varepsilon }$.

In 1984 Anatolii Alexeevitch Karatsuba proved 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 $\zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )}$ and and therefore confirmed the Selberg conjecture.

The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as $T\to +\infty$.

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.