Selberg's zeta function conjecture

In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ. It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of N, the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T.

Background

In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that for any
there exist
and
such that for
and
the inequality
holds true.
In his turn, Selberg stated a conjecture relating to shorter intervals, namely that it is possible to decrease the value of the exponent a = 0.5 in

Proof of the conjecture

In 1984 Anatolii Karatsuba proved that for a fixed satisfying the condition
a sufficiently large T and
the interval in the ordinate t contains at least cH ln T real zeros of the Riemann zeta function
and thereby confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba cannot be improved in respect of the order of growth as T → +∞.

Further work

In 1992 Karatsuba proved that an analog of the Selberg conjecture holds for "almost all" intervals ^ε] contain at least H ^{1 −ε₁} zeros of the function ζ. This estimate is quite close to the conditional result that follows from the Riemann hypothesis.