Riemann zeta, and related functions#
Riemann \(\zeta(n)\) for integer arguments#
- math53.zeta_i(n)#
Returns the Riemann zeta function \(\zeta(n)\) for integer arguments \(n \ne 1\). For \(n > 63\) the result is \(1\), for \(0 \le n \le 63\) the value is taken from a table, otherwise the Bernoulli numbers are used: \(\zeta(n) = B_{1-n}/(n - 1)\) for \(n < 0\).
See also Wikipedia [1440], MathWorld [1044], NIST [20], BoostMath [165], Ehrhardt [309] (3.6.1.2).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.ZetaInt(5) ereal('5.2359877559829887307E-1') >>> ereal.ZetaInt('51') ereal('5.3518479027559984754E-1')
Riemann \(\zeta(1+x)\)#
- math53.zeta1p(x)#
Returns the Riemann zeta function \(\zeta(1+x)\) for \(x \ne 0\). Normally used with \(|x| \ll 1\) for increased accuracy near the pole of \(\zeta(s)\) at \(s = 1\).
See also Wikipedia [1440], MathWorld [1044], NIST [20], BoostMath [165], Ehrhardt [309] (3.6.1.3).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Zeta1p(0.004) ereal('5.2359877559829887307E-1') >>> ereal.Zeta1p('0.0001') ereal('5.3518479027559984754E-1')
Dirichlet eta function for integer argument, \(\eta(n)\)#
- math53.dirichlet_eta_i(n)#
Returns the Riemann zeta function \(\eta(n)\) for integer arguments. For \(n > 64\) the result is \(1\), for \(0 \le n \le 64\) the value is taken from a table, otherwise for \(n<0\) the Bernoulli numbers are used: \(\eta(n) = (2^{1-n}-1) B_{1-n}/(1-n)\).
See also: MathWorld [1071], Ehrhardt [309] (3.6.3.2).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.DirichletEtaInt(5) ereal('5.2359877559829887307E-1') >>> ereal.DirichletEtaInt('51') ereal('5.3518479027559984754E-1')
Inverse of the Riemann prime counting function, \(R^{-1}(x)\)#
- math53.riemann_r_inv(x)#
Returns the functional inverse of the Riemann prime counting function, i.e. \(R(R^{-1}(x))= x\), for \(x \ge 1.125\).
See also Ehrhardt [309] (3.10.21).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.RiemannRInv(44) ereal('5.2359877559829887307E-1') >>> ereal.RiemannRInv(4440.4) ereal('5.3518479027559984754E-1')
Rogers-Ramanujan continued fraction#
- math53.rogers_ramanujan_cf(q)#
Returns \(\displaystyle R(q) = \frac{q^{1/5}}{1+} \frac{q}{1+} \frac{q^2}{1+} \frac{q^3}{1+} \frac{q^5}{1+}\), the Rogers-Ramanujan continued fraction, for \(|q| < 1\).
See also Ehrhardt [309] (3.10.22).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.RogersRamanujanCF(0.44) ereal('5.2359877559829887307E-1') >>> ereal.RogersRamanujanCF(0.14404) ereal('5.3518479027559984754E-1')
q-Pochhammer Euler function, \((q)_{\infty}\)#
- math53.euler_q(n, x)#
Returns \(\displaystyle \phi(q) = (q)_{\infty} = \prod_{k=1}^{\infty} \left(1-q^k\right)\), the q-Pochhammer Euler function, for \(-1 \le q \le 1\).
See also: MathWorld [1113], Wikipedia [1516], Ehrhardt [309] (3.10.19).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.EulerQ(-0.4) ereal('5.2359877559829887307E-1') >>> ereal.EulerQ(0.4) ereal('5.3518479027559984754E-1')