In mathematics, the Hardy–Littlewood zeta function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function.
In 1914, Godfrey Harold Hardy proved[1] that the Riemann zeta function
ζ
has infinitely many real zeros.
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[2] two conjectures.
These conjectures – on the distance between real zeros of
and on the density of zeros of
for sufficiently great
a + ε
ε > 0
is an arbitrarily small number – open two new directions in the investigation of the Riemann zeta function.
ε > 0
( ε ) > 0
0.25 + ε
contains a zero of odd order of the function
ε > 0
( ε ) > 0
c = c ( ε ) > 0
0.5 + ε
the inequality
{\displaystyle N_{0}(T+H)-N_{0}(T)\geq cH}
In 1942, Atle Selberg studied the problem 2 and proved that for any
ε > 0
( ε ) > 0
In his turn, Selberg made his conjecture[3] that it's possible to decrease the value of the exponent
which was proved 42 years later by A.A.
Karatsuba.