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(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T. 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,[1] namely that it is possible to decrease the value of the exponent a = 0.5 in In 1984 Anatolii Karatsuba proved[2][3][4] that for a fixed
satisfying the condition a sufficiently large T and the interval in the ordinate t (T, T + H) contains at least cH ln T real zeros of the Riemann zeta function and thereby confirmed the Selberg conjecture.
In 1992 Karatsuba proved[5] that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = Tε, where ε is an arbitrarily small fixed positive number.