Selberg's zeta function conjecture

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.