Related to Lerch’s phi

Mpmath: Bernoulli number as fraction

mpm.bernfrac(n)

Returns the Bernoulli number as fraction. See also Mpmath [609].

See also: bernoulli_fmpq_ui in ARB

Returns a tuple of integers \((p, q)\) such that \(p/q = B_n\) exactly, where \(B_n\) denotes the \(n\)-th Bernoulli number. The fraction is always reduced to lowest terms. Note that for \(n > 1\) and \(n\) odd, \(B_n = 0\), and \((0, 1)\) is returned.

Examples

This function works for arbitrarily large \(n\):

>>> p, q = bernfrac(10**4)
>>> print(q)
2338224387510
>>> print(len(str(p)))
27692
>>> mp.dps = 15
>>> print(mpf(p) / q)
-9.04942396360948e+27677
>>> print(bernoulli(10**4))
-9.04942396360948e+27677

Note

bernoulli() computes a floating-point approximation directly, without computing the exact fraction first. This is much faster for large \(n\).

Mpmath: Dirichlet L-Series

ctx.dirichlet_l(s, chi, derivative=0)

where ctx is dec, mpm, fpm, gmp or apm.

Returns the Dirichlet L-function. See also Wikipedia [1467], MathWorld [1072], NIST [12], Mpmath [734], Mpmath [734].

Evaluates the Dirichlet L-function

\[L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}.\]

where \(\chi\) is a periodic sequence of length \(q\) which should be supplied in the form of a list \([\chi(0), \chi(1), \ldots, \chi(q-1)]\). Strictly, \(\chi\) should be a Dirichlet character, but any periodic sequence will work.

For example, dirichlet(s, [1]) gives the ordinary Riemann zeta function and dirichlet(s, [-1,1]) gives the alternating zeta function (Dirichlet eta function).

Also the derivative with respect to \(s\) (currently only a first derivative) can be evaluated.

The ordinary Riemann zeta method:

>>> from xlcalcnet import *
>>> mp.dps = 25; mp.pretty = True
>>> dirichlet(3, [1]); zeta(3)
1.202056903159594285399738
1.202056903159594285399738
>>> dirichlet(1, [1])
+inf

The alternating zeta method:

>>> dirichlet(1, [-1,1]); ln(2)
0.6931471805599453094172321
0.6931471805599453094172321

Mpmath: Number of zeros of the Riemann zeta function

ctxflint.zeta_nzeros(n)

Returns the number of zeros of the Riemann zeta function. See also Wikipedia [1473], MathWorld [1090], NIST [15], Flint [836], Mpmath [740].

This calls acb_dirichlet_zeta_nzeros.

Computes the number of zeros of the Riemann zeta function in \((0,1) \times (0,t]\), usually denoted by \(N(t)\).

An example :

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; t = 10000
>>> \mathrm{d}z = dec.nzeros(t); mz = mpm.nzeros(t); gz = gmp.nzeros(t)
>>> fz = fpm.nzeros(t); az = apm.nzeros(t)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az],  aligned=True)
dec: 1.0142000000000000000E+4
mpm: 1.0142000000000000000e+4
gmp: 1.0142000000000000000E+04
fpm: 1.01420000000000E+04
apm: 1.0142000000000000000e+4 (0.0%)

Mpmath: Secondary zeta function

ctx.secondzeta(s, a=0.015)

where ctx is dec, mpm, fpm, gmp or apm.

Returns the secondary zeta function. See also Mpmath [745], Voros [861], Voros [862], Voros [862].

Evaluates the secondary zeta function \(Z(s)\), defined for \(\mathrm{Re}(s)>1\) by

\[Z(s) = \sum_{n=1}^{\infty} \frac{1}{\tau_n^s}\]

where \(\frac12+i\tau_n\) runs through the zeros of \(\zeta(s)\) with imaginary part positive.

\(Z(s)\) extends to a meromorphic function on \(\mathbb{C}\) with a double pole at \(s=1\) and simple poles at the points \(-2n\) for \(n=0\), 1, 2, …

Examples

>>> from mpfunlab import *
>>> mp.pretty = True; mp.dps = 15
>>> secondzeta(2)
0.023104993115419
>>> xi = lambda s: 0.5*s*(s-1)*pi**(-0.5*s)*gamma(0.5*s)*zeta(s)
>>> Xi = lambda t: xi(0.5+t*j)
>>> chop(-0.5*diff(Xi,0,n=2)/Xi(0))
0.023104993115419

We may ask for an approximate error value:

>>> secondzeta(0.5+100j, error=True)
((-0.216272011276718 - 0.844952708937228j), 2.22044604925031e-16)