public class Riemann extends Object
| Modifier and Type | Field and Description |
|---|---|
static double |
EPSILON
The predefined accuracy up to which infinite sums are approximated.
|
| Modifier and Type | Method and Description |
|---|---|
static double[] |
chi(double[] s)
Returns the value χ(s) for a complex number
s ∈ ℂ,
such that ζ(s) = χ(s) ζ(1 - s).
|
static double |
theta(double t)
Riemann-Siegel theta function.
|
static double |
Z(double t)
Riemann-Siegel Z-function Z(t).
|
static double[] |
zeta(double[] s)
Riemann zeta function ζ(s)
for s ∈ ℂ.
|
public static final double EPSILON
zeta(double[]),
Constant Field Valuespublic static double[] chi(double[] s)
| χ(s) | = | 2s πs - 1 sin(s π/2) Γ(1 - s) | = | πs - 1/2 |
|
We have χ(s) χ(1 - s) = 1. [Eqs. (2.1.10)-(2.1.12) in E.C. Titchmarsh: The Theory of the Riemann Zeta-function. 2nd Edition, Oxford University Press, Oxford 1986], https://books.google.com/books?id=1CyfApMt8JYC&pg=PA16
Moreover χ is related to the Riemann-Siegel theta function θ
by the equation
χ(½ + it) = e-2iθ(t),
see E.C. Titchmarsh: The Theory of the Riemann Zeta-function. 2nd Edition, Oxford University Press, Oxford 1986, p. 89 https://books.google.com/books?id=1CyfApMt8JYC&pg=PA89.
s - a complex valuezeta(double[]),
theta(double),
Complex.gamma(double[])public static double[] zeta(double[] s)
| ζ(s) | = |
|
|
| if Re s > 0, | |||||||||||||||||||||
| ζ(s) | = |
|
|
|
|
| otherwise. | |||||||||||||||||||
However, in this method the algorithm is used which is documented as “Algorithm 1” in Borwein et al, The Riemann Hypothesis, Springer, Berlin 2008, p 35 (https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA35)
The functions ζ, Z and θ are related by the equality Z(t) = ei θ(t) ζ(½ + it). Cf. H.M. Edwards: Riemann's Zeta Function. Academic Press, New York 1974, §6.5 (https://books.google.de/books?id=ruVmGFPwNhQC&pg=PA119).
s - the argumentZ(double),
theta(double)public static double Z(double t)
| Z(t) | = | 2 |
|
| + O(|t|-1/4). | ||||||||
where
| m = m(t) | = | ⎣ |
| ⎦ |
and θ denotes the Riemann-Siegel theta function.
The functions ζ, Z and θ are related by the equality Z(t) = ei θ(t) ζ(½ + it). Cf. H.M. Edwards: Riemann's Zeta Function. Academic Press, New York 1974, §6.5 (https://books.google.de/books?id=ruVmGFPwNhQC&pg=PA119).
t - value on the critical line s = ½ + it.zeta(double[]),
theta(double)public static double theta(double t)
| θ(t) | = |
| ln |
| - |
| - |
| + |
| + |
| + |
| + |
| + R(t) with |R(t)| < |
|
The functions ζ, Z and θ are related by the equality Z(t) = ei θ(t) ζ(½ + it). Cf. H.M. Edwards: Riemann's Zeta Function. Academic Press, New York 1974, §6.5 (https://books.google.de/books?id=ruVmGFPwNhQC&pg=PA119).
t - value on the critical line s = ½ + it.zeta(double[]),
Z(double)