Gram points

E825421
mathematical concept sequence of real numbers

Gram points are specific real values of the argument on the critical line of the Riemann zeta function where the Hardy Z-function takes real values with alternating sign, playing a key role in studying the distribution of its zeros.

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Gram points canonical 1

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Predicate Object
instanceOf mathematical concept ⓘ
sequence of real numbers ⓘ
appearsIn computational studies of the Riemann hypothesis ⓘ
literature on zero spacing statistics of the Riemann zeta function ⓘ
asymptoticBehavior g_n grows roughly like 2πn / log(n) for large n (up to lower-order terms) ⓘ
coordinateSystem imaginary part t of s = 1/2 + it ⓘ
definedVia Riemann–Siegel theta function ⓘ
equation θ(t) = nπ ⓘ
domain real line ⓘ
field analytic number theory ⓘ
hasDefinition Gram interval is the interval [g_n, g_{n+1}] between consecutive Gram points ⓘ
real numbers g_n such that θ(g_n) = nπ, where θ is the Riemann–Siegel theta function ⓘ
hasProperty Gram’s law fails infinitely often ⓘ
Gram’s law states that zeros of the zeta function on the critical line usually lie between consecutive Gram points ⓘ
are close to successive zeros of the Riemann zeta function on the critical line ⓘ
are defined for nonnegative integers n ⓘ
are not themselves generally zeros of the zeta function ⓘ
are ordered as an increasing sequence g_0 < g_1 < g_2 < ... ⓘ
behavior is connected to the fine structure of the zeta function on the critical line ⓘ
between many consecutive Gram points there is typically exactly one zero of the zeta function on the critical line ⓘ
density increases with t but spacing decreases slowly as t grows ⓘ
distribution reflects oscillatory nature of the Riemann–Siegel theta function ⓘ
for many n, Z(g_n) and Z(g_{n+1}) have opposite signs ⓘ
form a discrete subset of the real line ⓘ
lie on the critical line s = 1/2 + it of the Riemann zeta function ⓘ
often exhibit alternating signs of the Hardy Z-function values Z(g_n) ⓘ
sign changes of Z(t) between Gram points indicate zeros of Z(t) ⓘ
some Gram intervals contain more than one zero or no zeros at all ⓘ
the Hardy Z-function Z(t) is real for real t, including at Gram points ⓘ
namedAfter Jørgen Pedersen Gram ⓘ
relatedTo Gram intervals ⓘ
Gram’s law ⓘ
Hardy Z-function ⓘ
Riemann zeta function ⓘ
Riemann–Siegel formula ⓘ
critical line of the Riemann zeta function ⓘ
zeros of the Riemann zeta function ⓘ
studiedBy Atle Selberg ⓘ
G. H. Hardy ⓘ
J. E. Littlewood ⓘ
linked to: John Edensor Littlewood
usedFor numerical verification of the Riemann hypothesis ⓘ
partitioning the critical line into intervals for zero counting ⓘ
studying the distribution of zeros of the Riemann zeta function ⓘ
usedIn high-precision computations of ζ(1/2 + it) ⓘ
locating zeros of the Hardy Z-function ⓘ
tabulation of zeros of the Riemann zeta function ⓘ

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Hardy Z-function → relatedTo → Gram points ⓘ