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In 1914, Godfrey Harold Hardy proved that the Riemann zeta function has infinitely many real zeros.
Let be the total number of real zeros, be the total number of zeros of odd order of the function , lying on the interval
.
Hardy and Littlewood claimed two conjectures. These conjectures – on the distance between real zeros of and on the density of zeros of on intervals for sufficiently great , and with as less as possible value of , where is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.
1. For any there exists such that for and the interval contains a zero of odd order of the function .
2. For any there exist and , such that for and the inequality is true.
Status
In 1942, Atle Selberg studied the problem 2 and proved that for any there exists such and , such that for and the inequality is true.
In his turn, Selberg made his conjecture that it's possible to decrease the value of the exponent for which was proved 42 years later by A.A. Karatsuba.
References
Hardy, G.H. (1914). "Sur les zeros de la fonction ". Compt. Rend. Acad. Sci. 158: 1012–1014.