Number-Theoretic Functions

class sage.functions.transcendental.DickmanRho

Bases: sage.symbolic.function.BuiltinFunction

Dickman’s function is the continuous function satisfying the differential equation

\[x \rho'(x) + \rho(x-1) = 0\]

with initial conditions \(\rho(x)=1\) for \(0 \le x \le 1\). It is useful in estimating the frequency of smooth numbers as asymptotically

\[\Psi(a, a^{1/s}) \sim a \rho(s)\]

where \(\Psi(a,b)\) is the number of \(b\)-smooth numbers less than \(a\).

ALGORITHM:

Dickmans’s function is analytic on the interval \([n,n+1]\) for each integer \(n\). To evaluate at \(n+t, 0 \le t < 1\), a power series is recursively computed about \(n+1/2\) using the differential equation stated above. As high precision arithmetic may be needed for intermediate results the computed series are cached for later use.

Simple explicit formulas are used for the intervals [0,1] and [1,2].

EXAMPLES:

sage: dickman_rho(2)
0.306852819440055
sage: dickman_rho(10)
2.77017183772596e-11
sage: dickman_rho(10.00000000000000000000000000000000000000)
2.77017183772595898875812120063434232634e-11
sage: plot(log(dickman_rho(x)), (x, 0, 15))
Graphics object consisting of 1 graphics primitive

AUTHORS:

  • Robert Bradshaw (2008-09)

REFERENCES:

  • G. Marsaglia, A. Zaman, J. Marsaglia. “Numerical Solutions to some Classical Differential-Difference Equations.” Mathematics of Computation, Vol. 53, No. 187 (1989).
approximate(x, parent=None)

Approximate using de Bruijn’s formula

\[\rho(x) \sim \frac{exp(-x \xi + Ei(\xi))}{\sqrt{2\pi x}\xi}\]

which is asymptotically equal to Dickman’s function, and is much faster to compute.

REFERENCES:

  • N. De Bruijn, “The Asymptotic behavior of a function occurring in the theory of primes.” J. Indian Math Soc. v 15. (1951)

EXAMPLES:

sage: dickman_rho.approximate(10)
2.41739196365564e-11
sage: dickman_rho(10)
2.77017183772596e-11
sage: dickman_rho.approximate(1000)
4.32938809066403e-3464
power_series(n, abs_prec)

This function returns the power series about \(n+1/2\) used to evaluate Dickman’s function. It is scaled such that the interval \([n,n+1]\) corresponds to x in \([-1,1]\).

INPUT:

  • n - the lower endpoint of the interval for which this power series holds
  • abs_prec - the absolute precision of the resulting power series

EXAMPLES:

sage: f = dickman_rho.power_series(2, 20); f
-9.9376e-8*x^11 + 3.7722e-7*x^10 - 1.4684e-6*x^9 + 5.8783e-6*x^8 - 0.000024259*x^7 + 0.00010341*x^6 - 0.00045583*x^5 + 0.0020773*x^4 - 0.0097336*x^3 + 0.045224*x^2 - 0.11891*x + 0.13032
sage: f(-1), f(0), f(1)
(0.30685, 0.13032, 0.048608)
sage: dickman_rho(2), dickman_rho(2.5), dickman_rho(3)
(0.306852819440055, 0.130319561832251, 0.0486083882911316)
class sage.functions.transcendental.Function_HurwitzZeta

Bases: sage.symbolic.function.BuiltinFunction

class sage.functions.transcendental.Function_stieltjes

Bases: sage.symbolic.function.GinacFunction

Stieltjes constant of index n.

stieltjes(0) is identical to the Euler-Mascheroni constant (sage.symbolic.constants.EulerGamma). The Stieltjes constants are used in the series expansions of \(\zeta(s)\).

INPUT:

  • n - non-negative integer

EXAMPLES:

sage: _ = var('n')
sage: stieltjes(n)
stieltjes(n)
sage: stieltjes(0)
euler_gamma
sage: stieltjes(2)
stieltjes(2)
sage: stieltjes(int(2))
stieltjes(2)
sage: stieltjes(2).n(100)
-0.0096903631928723184845303860352
sage: RR = RealField(200)
sage: stieltjes(RR(2))
-0.0096903631928723184845303860352125293590658061013407498807014

It is possible to use the hold argument to prevent automatic evaluation:

sage: stieltjes(0,hold=True)
stieltjes(0)

sage: latex(stieltjes(n))
\gamma_{n}
sage: a = loads(dumps(stieltjes(n)))
sage: a.operator() == stieltjes
True
sage: stieltjes(x)._sympy_()
stieltjes(x)

sage: stieltjes(x).subs(x==0)
euler_gamma
class sage.functions.transcendental.Function_zeta

Bases: sage.symbolic.function.GinacFunction

Riemann zeta function at s with s a real or complex number.

INPUT:

  • s - real or complex number

If s is a real number the computation is done using the MPFR library. When the input is not real, the computation is done using the PARI C library.

EXAMPLES:

sage: zeta(x)
zeta(x)
sage: zeta(2)
1/6*pi^2
sage: zeta(2.)
1.64493406684823
sage: RR = RealField(200)
sage: zeta(RR(2))
1.6449340668482264364724151666460251892189499012067984377356
sage: zeta(I)
zeta(I)
sage: zeta(I).n()
0.00330022368532410 - 0.418155449141322*I
sage: zeta(sqrt(2))
zeta(sqrt(2))
sage: zeta(sqrt(2)).n()  # rel tol 1e-10
3.02073767948603

It is possible to use the hold argument to prevent automatic evaluation:

sage: zeta(2,hold=True)
zeta(2)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: a = zeta(2,hold=True); a.simplify()
1/6*pi^2

The Laurent expansion of \(\zeta(s)\) at \(s=1\) is implemented by means of the Stieltjes constants:

sage: s = SR('s')
sage: zeta(s).series(s==1, 2)
1*(s - 1)^(-1) + euler_gamma + (-stieltjes(1))*(s - 1) + Order((s - 1)^2)

Generally, the Stieltjes constants occur in the Laurent expansion of \(\zeta\)-type singularities:

sage: zeta(2*s/(s+1)).series(s==1, 2)
2*(s - 1)^(-1) + (euler_gamma + 1) + (-1/2*stieltjes(1))*(s - 1) + Order((s - 1)^2)
class sage.functions.transcendental.Function_zetaderiv

Bases: sage.symbolic.function.GinacFunction

Derivatives of the Riemann zeta function.

EXAMPLES:

sage: zetaderiv(1, x)
zetaderiv(1, x)
sage: zetaderiv(1, x).diff(x)
zetaderiv(2, x)
sage: var('n')
n
sage: zetaderiv(n,x)
zetaderiv(n, x)
sage: zetaderiv(1, 4).n()
-0.0689112658961254
sage: import mpmath; mpmath.diff(lambda x: mpmath.zeta(x), 4)
mpf('-0.068911265896125382')
sage.functions.transcendental.hurwitz_zeta(s, x, **kwargs)

The Hurwitz zeta function \(\zeta(s, x)\), where \(s\) and \(x\) are complex.

The Hurwitz zeta function is one of the many zeta functions. It is defined as

\[\zeta(s, x) = \sum_{k=0}^{\infty} (k + x)^{-s}.\]

When \(x = 1\), this coincides with Riemann’s zeta function. The Dirichlet L-functions may be expressed as linear combinations of Hurwitz zeta functions.

EXAMPLES:

Symbolic evaluations:

sage: hurwitz_zeta(x, 1)
zeta(x)
sage: hurwitz_zeta(4, 3)
1/90*pi^4 - 17/16
sage: hurwitz_zeta(-4, x)
-1/5*x^5 + 1/2*x^4 - 1/3*x^3 + 1/30*x
sage: hurwitz_zeta(7, -1/2)
127*zeta(7) - 128
sage: hurwitz_zeta(-3, 1)
1/120

Numerical evaluations:

sage: hurwitz_zeta(3, 1/2).n()
8.41439832211716
sage: hurwitz_zeta(11/10, 1/2).n()
12.1038134956837
sage: hurwitz_zeta(3, x).series(x, 60).subs(x=0.5).n()
8.41439832211716
sage: hurwitz_zeta(3, 0.5)
8.41439832211716

REFERENCES:

sage.functions.transcendental.zeta_symmetric(s)

Completed function \(\xi(s)\) that satisfies \(\xi(s) = \xi(1-s)\) and has zeros at the same points as the Riemann zeta function.

INPUT:

  • s - real or complex number

If s is a real number the computation is done using the MPFR library. When the input is not real, the computation is done using the PARI C library.

More precisely,

\[xi(s) = \gamma(s/2 + 1) * (s-1) * \pi^{-s/2} * \zeta(s).\]

EXAMPLES:

sage: zeta_symmetric(0.7)
0.497580414651127
sage: zeta_symmetric(1-0.7)
0.497580414651127
sage: RR = RealField(200)
sage: zeta_symmetric(RR(0.7))
0.49758041465112690357779107525638385212657443284080589766062
sage: C.<i> = ComplexField()
sage: zeta_symmetric(0.5 + i*14.0)
0.000201294444235258 + 1.49077798716757e-19*I
sage: zeta_symmetric(0.5 + i*14.1)
0.0000489893483255687 + 4.40457132572236e-20*I
sage: zeta_symmetric(0.5 + i*14.2)
-0.0000868931282620101 + 7.11507675693612e-20*I

REFERENCE: