Riemann hypothesis
Selberg's zeta function conjecture

Atle Selberg (1942) investigated the problem of Hardy–Littlewood 2 and proved that for any ε > 0 there exists such T 0 = T 0 (ε) > 0 {\displaystyle T_{0}=T_{0}(\varepsilon)>0} and c = c (ε) > 0, such that for T ≥ T 0 {\displaystyle T\geq T_{0}} and H = T 0.5 + ε {\displaystyle H=T^{0.5+\varepsilon }} the inequality N (T + H) − N (T) ≥ c H log T {\displaystyle N(T+H)-N(T)\geq cH\log T} is true. Selberg conjectured that this could be tightened to H = T 0.5 {\displaystyle H=T^{0.5}}. A. A. Karatsuba (1984a, 1984b, 1985) proved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large T and H = T a + ε {\displaystyle H=T^{a+\varepsilon }}, a = 27 82 = 1 3 − 1 246 {\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}}, the interval (T, T + H) contains at least cH log(T) real zeros of the Riemann zeta function ζ (1 2 + i t) {\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)} and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as T → ∞.

Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals (T, T + H ], H = T ε {\displaystyle H=T^{\varepsilon }}, where ε 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 ε, ε 1 {\displaystyle \varepsilon _{1}} satisfying the conditions 0 < ε, ε 1 < 1 {\displaystyle 0<\varepsilon,\varepsilon _{1}<1} almost all intervals (T, T + H ] for H ≥ exp { (log T) ε } {\displaystyle H\geq \exp {\{(\log T)^{\varepsilon }\}}} contain at least H (log T) 1 − ε 1 {\displaystyle H(\log T)^{1-\varepsilon _{1}}} zeros of the function ζ (1 2 + i t) {\displaystyle \zeta \left({\tfrac {1}{2}}+it\right)}. This estimate is quite close to the one that follows from the Riemann hypothesis.
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