Gram points
E825421
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.
Statements (46)
| 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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| relatedTo |
Gram intervals
ⓘ
Gram’s law NERFINISHED ⓘ Hardy Z-function NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ Riemann–Siegel formula NERFINISHED ⓘ critical line of the Riemann zeta function ⓘ zeros of the Riemann zeta function ⓘ |
| studiedBy |
Atle Selberg
NERFINISHED
ⓘ
G. H. Hardy NERFINISHED ⓘ J. E. Littlewood NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.