Selberg's zeta function conjecture

In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that 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

${\displaystyle N(T+H)-N(T)\geq cH\log T}$

holds true.

In his turn, Selberg stated a conjecture relating to shorter intervals,[1] namely that it is possible to decrease the value of the exponent a = 0.5 in

${\displaystyle H=T^{0.5+\varepsilon }.}$

In 1984 Anatolii Karatsuba proved[2][3][4] that for a fixed ${\displaystyle \varepsilon }$ satisfying the condition

${\displaystyle 0<\varepsilon <0.001,}$

a sufficiently large T and

${\displaystyle H=T^{a+\varepsilon },}$ ${\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}},}$

the interval in the ordinate t (T, T + H) contains at least cH ln T real zeros of the Riemann zeta function

${\displaystyle \zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )};}$

and thereby confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba cannot be improved in respect of the order of growth as T → +∞.

In 1992 Karatsuba proved[5] that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = T^ε, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits one to investigate zeroes 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 ε, ε₁ satisfying the conditions 0 < ε, ε₁< 1 almost all intervals (T, T + H] for H ≥ exp[(ln T)^ε] contain at least H (ln T)^1 −ε₁ zeros of the function ζ(1/2 + it). This estimate is quite close to the conditional result that follows from the Riemann hypothesis.