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Glaisher

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Glaisher

is the Glaisher-Kinkelin constant A, with numerical value ~= 1.28243.

D[Glaisher, x]

is 0. N[Glaisher, prec] evaluates it to any precision.

Notes Glaisher's constant satisfies Log\[A\] == 1/12 - Zeta'\[-1\], where Zeta is the Riemann zeta function. It is a mathematical constant: it has attributes Constant and Protected, NumericQ\[Glaisher\] is True, and

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

In[1]:= N[Glaisher]
Out[1]= 1.28243

In[2]:= N[Glaisher, 40]
Out[2]= 1.2824271291006226368753425688697917277676

In[3]:= NumericQ[Glaisher]
Out[3]= True

In[4]:= D[Glaisher, x]
Out[4]= 0

Implementation notes

Definition. Glaisher is the Glaisher–Kinkelin constant A ≈ 1.2824271291, the constant appearing in the asymptotics of the hyperfactorial and the Barnes G function. It satisfies Log[A] == 1/12 - Zeta'[-1], equivalently Log[A] = (gamma + Log[2 Pi])/12 - Zeta'[2]/(2 Pi^2). It is a constant symbol: row { "Glaisher", 1.2824271291…, fill_mpfr_glaisher } of kConstants[] in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[Glaisher] is {Constant, Protected}. Interned name SYM_Glaisher (src/sym_names.c), docstring in src/info.c. No DownValues; D[Glaisher, x] -> 0 follows from Constant.

Representation & numeric value. A bare EXPR_SYMBOL, kept exact under evaluation. N[Glaisher] returns the tabled machine double. N[Glaisher, k] runs the hand-rolled fill_mpfr_glaisher: MPFR has no ζ′ primitive, so Zeta'[2] = -Sum_{n>=1} Log[n]/n^2 is evaluated by Euler–Maclaurin summation (head sum to N ~ bits/6, the integral tail (ln N + 1)/N, the f(N)/2 term, and a divergent asymptotic correction series whose Bernoulli numbers come from mpfr_zeta_ui, truncated at its smallest term); Log[A] is then assembled and exponentiated at bits + 64 guard precision. Machine value 1.28243; N[Glaisher, 50] = 1.282427129100622636875342568869791727767688927325.

Usage & limits. Glaisher is the closed form behind hyperfactorial / Barnes-G asymptotics; NumericQ[Glaisher] is True (whitelisted in is_numeric_quantity, src/core.c). The only implementation limit is that its filler is a bespoke Euler–Maclaurin routine (not a library constant): the guard bits and smallest-term truncation are sized so the asymptotic floor lies below any requested precision.

  • Attributes Constant, Protected. Attributes[Glaisher] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[Glaisher] is True and D[Glaisher, x] = 0.
  • N[Glaisher] gives the machine value 1.28243; N[Glaisher, prec] gives any precision, e.g. N[Glaisher, 50] = 1.282427129100622636875342568869791727767688927325.

Arbitrary precision is computed from ln A = (γ + ln(2π))/12 − ζ'(2)/(2π²), with ζ'(2) evaluated by Euler–Maclaurin summation of −Σ ln(n)/n² (the even Bernoulli numbers obtained from Zeta[2k]). Verified to 250 digits.

Attributes: Constant, Protected.

References

See also: Zeta

Notes & additional examples

Notes

Glaisher is the Glaisher-Kinkelin constant A, defined by Log[A] == 1/12 - Zeta'[-1] and appearing in the asymptotics of the hyperfactorial and in many Zeta-derivative identities. It is held symbolic (attributes Constant and Protected, so D[Glaisher, x] is 0 and NumericQ is True) until N forces a value; N[Glaisher, 40] returns it to 40 digits via its MPFR series.