Displaying 1-10 of 29 results found.
Decimal expansion of Glaisher-Kinkelin constant A.
+10
447
1, 2, 8, 2, 4, 2, 7, 1, 2, 9, 1, 0, 0, 6, 2, 2, 6, 3, 6, 8, 7, 5, 3, 4, 2, 5, 6, 8, 8, 6, 9, 7, 9, 1, 7, 2, 7, 7, 6, 7, 6, 8, 8, 9, 2, 7, 3, 2, 5, 0, 0, 1, 1, 9, 2, 0, 6, 3, 7, 4, 0, 0, 2, 1, 7, 4, 0, 4, 0, 6, 3, 0, 8, 8, 5, 8, 8, 2, 6, 4, 6, 1, 1, 2, 9, 7, 3, 6, 4, 9, 1, 9, 5, 8, 2, 0, 2, 3, 7, 4, 3, 9, 4, 2, 0, 6, 4, 6, 1, 2, 0
COMMENTS
Arises in expressions such as A002109(n) = 1^1*2^2*3^3*...*n^n which is asymptotic to A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4). See A002109 for more references and links. Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928) and the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 15 2021
REFERENCES
Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, p. 135.
Konrad Knopp, Theory and applications of infinite series, Dover, p. 555.
LINKS
Ovidiu Furdui, proposer, Problem 11494, Amer. Math. Monthly, Vol. 118, No. 9 (2011), 850-852.
FORMULA
A = 2^(1/36)*Pi^(1/6)*exp(1/3*(-Gamma/4 + s(2)/3 - s(3)/4 + ...)) where s(k) denotes Sum_{n>=0} 1/(2n+1)^k.
Closed expressions for A are exp(-zeta'(2)/2/Pi^2 + log(2*Pi)/12 + Gamma/12) or exp(1/12-zeta'(-1)).
Equals (2*Pi)^(1/4) / limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2). - Vaclav Kotesovec, Dec 02 2023 Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(2)/2 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024 Equals e^(-1/4 + Integral_{x=1..2} x*log(sqrt(2*Pi)) - B_2(x) + x^2*Psi(x)/2 dx), where B_2(x) is the second Bernoulli polynomial and Psi(x) is the digamma function. - Andrea Pinos, Apr 16 2024 Equals Product_{k>=1} 2^(10^(-k) + 3/13^k)((2*k)/(2*k + 1))^((k/3 + 1/12))((2*k + 2)/(2*k + 1))^((k/3 + 1/4)). - Antonio Graciá Llorente, May 20 2024 Equals exp(1/12 - 2*Integral_{x=0..oo} x*log(x)/(exp(2*Pi*x) - 1) dx) = exp(1/3 + 7*log(2)/36 - log(Pi)/6 + (2/3)*Integral_{x=0..1/2} log(Gamma(x+1)) dx) (see Finch). - Stefano Spezia, Dec 01 2024
EXAMPLE
1.2824271291006226368753425688697917277676889273250011920637400217404...
MAPLE
evalf(limit(product(k^k, k=1..n)/(n^(n^2/2+n/2+1/12)*exp(-n^2/4)), n=infinity), 120); # Vaclav Kotesovec, Oct 23 2014
PROG
(PARI) x=10^(-100); exp(1/12-(zeta(-1+x)-zeta(-1))/x)
Decimal expansion of the generalized Glaisher-Kinkelin constant A(3).
+10
31
9, 7, 9, 5, 5, 5, 5, 2, 6, 9, 4, 2, 8, 4, 4, 6, 0, 5, 8, 2, 4, 2, 1, 8, 8, 3, 7, 2, 6, 3, 4, 9, 1, 8, 2, 6, 4, 4, 5, 5, 3, 6, 7, 5, 2, 4, 9, 5, 5, 2, 9, 9, 0, 2, 2, 5, 7, 7, 1, 7, 1, 4, 2, 7, 9, 7, 5, 8, 8, 5, 6, 7, 2, 4, 8, 1, 5, 5, 9, 6, 1, 4, 9, 4, 4, 4, 4, 4, 3, 5, 3, 8, 3, 3, 2, 1, 9, 6
COMMENTS
Also known as the third Bendersky constant.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
FORMULA
A(k) = exp(B(k+1)/(k+1)*H(k) - zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(3) = exp(-11/720 - zeta'(-3)).
Equals exp(3*zeta'(4)/(4*Pi^4) - gamma/120) / (2*Pi)^(1/120), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015 Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^4-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(4)/4 = -1/120 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
0.97955552694284460582421883726349...
MATHEMATICA
RealDigits[Exp[-11/720 - Zeta'[-3]], 10, 98] // First
RealDigits[Exp[(BernoulliB[4]/4) * (EulerGamma + Log[2 * Pi] - (Zeta'[4]/Zeta[4]))], 10, 100] // First (* G. C. Greubel, Dec 31 2015 *)
CROSSREFS
Cf. A019727, A074962, A243262, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).
+10
28
9, 9, 2, 0, 4, 7, 9, 7, 4, 5, 2, 5, 0, 4, 0, 2, 6, 0, 0, 1, 3, 4, 3, 6, 9, 7, 7, 6, 2, 5, 4, 4, 3, 3, 5, 6, 7, 3, 6, 9, 0, 4, 8, 5, 1, 2, 7, 6, 1, 8, 8, 0, 8, 9, 3, 5, 2, 0, 9, 4, 6, 1, 4, 9, 1, 5, 5, 4, 1, 4, 5, 3, 8, 5, 3, 8, 9, 4, 5, 9, 7, 6, 1, 8, 0, 5, 7, 7, 3, 6, 1, 7, 2, 9, 5, 6, 4, 3
COMMENTS
Also known as the 4th Bendersky constant.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
FORMULA
A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(4) = exp(-zeta'(-4)) = exp(-3*zeta(5)/(4*Pi^4)).
A(4) = exp((B(4)/4)*(zeta(5)/zeta(4))). - G. C. Greubel, Dec 31 2015
EXAMPLE
0.9920479745250402600134369776254433567369...
MATHEMATICA
RealDigits[Exp[-3*Zeta[5]/(4*Pi^4)], 10, 98] // First
RealDigits[Exp[N[(BernoulliB[4]/4)*(Zeta[5]/Zeta[4]), 100]]] // First (* G. C. Greubel, Dec 31 2015 *)
CROSSREFS
Cf. A019727, A074962, A243262, A243263, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Decimal expansion of the generalized Glaisher-Kinkelin constant A(5).
+10
27
1, 0, 0, 9, 6, 8, 0, 3, 8, 7, 2, 8, 5, 8, 6, 6, 1, 6, 1, 1, 2, 0, 0, 8, 9, 1, 9, 0, 4, 6, 2, 6, 3, 0, 6, 9, 2, 6, 0, 3, 2, 7, 6, 3, 4, 7, 2, 1, 1, 5, 2, 4, 9, 1, 8, 4, 6, 0, 9, 2, 4, 7, 2, 1, 5, 6, 2, 3, 0, 1, 4, 2, 5, 0, 0, 3, 4, 1, 0, 0, 3, 2, 7, 7, 0, 1, 5, 0, 5, 6, 5, 9, 6, 5, 2, 7, 6, 4, 5, 5, 5, 9, 4
COMMENTS
Also known as the 5th Bendersky constant.
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
FORMULA
A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(5) = exp(137/15120-zeta'(-5)).
Equals exp(gamma/252 - 15*Zeta'(6)/(4*Pi^6)) * (2*Pi)^(1/252), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015 Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^6-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(6)/6 = 1/252 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
1.00968038728586616112008919046263...
MATHEMATICA
RealDigits[Exp[137/15120-Zeta'[-5]], 10, 103] // First
RealDigits[Exp[N[(BernoulliB[6]/6)*(EulerGamma + Log[2*Pi] - Zeta'[6]/Zeta[6]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)
CROSSREFS
Cf. A019727, A074962, A243262, A243263, A243264, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Decimal expansion of zeta'(-2) (the derivative of Riemann's zeta function at -2).
+10
23
0, 3, 0, 4, 4, 8, 4, 5, 7, 0, 5, 8, 3, 9, 3, 2, 7, 0, 7, 8, 0, 2, 5, 1, 5, 3, 0, 4, 7, 1, 1, 5, 4, 7, 7, 6, 6, 4, 7, 0, 0, 0, 4, 8, 3, 5, 4, 4, 9, 7, 3, 9, 3, 6, 2, 5, 2, 9, 7, 1, 8, 8, 9, 8, 5, 9, 0, 3, 7, 8, 1, 7, 9, 4, 4, 9, 3, 6, 8, 9, 8, 6, 7, 7, 7, 9, 4, 5, 8, 4, 8, 8, 0, 8, 7, 4, 4, 9, 5, 9, 7, 0, 3, 6
FORMULA
zeta'(-2) = -zeta(3)/(4*Pi^2).
EXAMPLE
-0.030448457058393270780251530471154776647000483544973936252971889859...
MATHEMATICA
Join[{0}, RealDigits[-Zeta[3]/(4*Pi^2), 10, 103] // First]
Decimal expansion of the generalized Glaisher-Kinkelin constant A(13).
+10
20
1, 2, 2, 2, 9, 4, 4, 2, 5, 1, 8, 0, 8, 1, 3, 3, 8, 7, 2, 6, 4, 7, 8, 9, 9, 9, 6, 0, 7, 2, 7, 7, 1, 7, 9, 8, 8, 5, 6, 1, 2, 6, 5, 8, 0, 3, 1, 2, 9, 5, 3, 2, 9, 5, 0, 1, 0, 8, 3, 7, 2, 8, 1, 0, 3, 4, 4, 6, 0, 6, 4, 2, 2, 7, 6, 8, 6, 6, 2, 0, 3, 0, 3, 0, 0, 1, 2, 6, 4, 2, 6, 9, 2, 1, 7, 5, 1, 1, 4, 2, 6, 1, 2, 4, 4, 9, 1, 8, 3, 6, 0, 0, 2, 0, 9
COMMENTS
Also known as the thirteenth Bendersky constant.
FORMULA
A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th Harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(13) = exp((1/14)*HarmonicNumber(13)*Bernoulli(14) - RiemannZeta'(-13)).
A(13) = exp((B(14)/14)*(EulerGamma + Log(2*Pi) - (zeta'(14)/zeta(14)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^14-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(14)/14 = 1/12 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
1.2229442518081338726478999607277179885...
MATHEMATICA
N[Exp[(1/14)*HarmonicNumber[13]*BernoulliB[14] - Zeta'[-13]], 100]
Exp[N[(BernoulliB[14]/14)*(EulerGamma + Log[2*Pi] - Zeta'[14]/Zeta[14]), 200]]
CROSSREFS
Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(7).
+10
20
9, 8, 9, 9, 7, 5, 6, 5, 3, 3, 3, 3, 4, 1, 7, 0, 9, 4, 1, 7, 5, 3, 9, 6, 4, 8, 3, 0, 5, 8, 8, 6, 9, 2, 0, 0, 2, 0, 8, 2, 4, 7, 1, 5, 1, 4, 3, 0, 7, 4, 5, 3, 0, 5, 1, 2, 8, 5, 5, 3, 8, 6, 2, 4, 2, 3, 7, 7, 4, 6, 4, 2, 9, 5, 9, 6, 1, 6, 7, 5, 7, 4, 2, 7, 5, 6, 6, 8, 7, 7, 6, 3, 6
COMMENTS
Also known as the 7th Bendersky constant.
FORMULA
A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(7) = exp(H(7)*B(8)/8 - zeta'(-7)) = exp((B(8)/8)*(EulerGamma + log(2*Pi) - (zeta'(8)/zeta(8)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^8-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(8)/8 = -1/240 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
0.9899756533334170941753964830588692002082471514307453051285538624....
MATHEMATICA
Exp[N[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 200]]
CROSSREFS
Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(9).
+10
20
1, 0, 1, 8, 4, 6, 9, 9, 2, 9, 9, 2, 0, 9, 9, 2, 9, 1, 2, 1, 7, 0, 6, 5, 9, 0, 4, 9, 3, 7, 6, 6, 7, 2, 1, 7, 2, 3, 0, 8, 6, 1, 0, 1, 9, 0, 5, 6, 4, 0, 7, 4, 9, 2, 0, 3, 8, 0, 0, 7, 0, 5, 7, 3, 6, 7, 5, 4, 7, 6, 1, 9, 4, 9, 4
COMMENTS
Also known as the 9th Bendersky constant.
FORMULA
A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(9) = exp(H(9)*B(10)/10 - zeta'(-9)) = exp((B(10)/10)*(EulerGamma + log(2*Pi) - (zeta'(10)/zeta(10)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^10-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(10)/10 = 1/132 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
1.018469929920992912170659049376672172308610190564074920380...
MATHEMATICA
Exp[N[(BernoulliB[10]/10)*(EulerGamma + Log[2*Pi] - Zeta'[10]/Zeta[10]), 200]]
CROSSREFS
Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(11).
+10
20
9, 5, 0, 3, 3, 1, 2, 4, 8, 4, 5, 3, 2, 8, 8, 8, 6, 6, 5, 1, 4, 2, 3, 3, 8, 4, 1, 0, 1, 5, 3, 3, 1, 2, 7, 1, 5, 9, 7, 5, 6, 6, 4, 0, 3, 4, 5, 6, 1, 7, 3, 0, 4, 0, 8, 6, 1, 0, 8, 8, 8, 8, 1, 1, 6, 2, 2, 9, 7, 8, 4, 9, 1, 7, 7, 3, 4, 4, 4, 5, 1
COMMENTS
Also known as the 11th Bendersky constant.
FORMULA
A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(11) = exp(H(11)*B(12)/12 - zeta'(-11)) = exp((B(12)/12)*(EulerGamma + log(2*Pi) - (zeta'(12)/zeta(12)))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^12-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(12)/12 = -691/32760 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
0.950331248453288866514233841015331271597566403456173040861088881...
MATHEMATICA
Exp[N[(BernoulliB[12]/12)*(EulerGamma + Log[2*Pi] - Zeta'[12]/Zeta[12]), 200]]
CROSSREFS
Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
Decimal expansion of the generalized Glaisher-Kinkelin constant A(15).
+10
20
3, 4, 2, 8, 3, 0, 8, 0, 6, 1, 3, 2, 8, 1, 6, 7, 3, 6, 5, 7, 1, 7, 1, 1, 1, 4, 6, 3, 4, 0, 6, 7, 2, 3, 7, 8, 1, 4, 1, 7, 2, 6, 9, 4, 5, 4, 8, 3, 2, 3, 6, 8, 7, 7, 2, 5, 1, 0, 7, 6, 1, 6, 4, 2, 4, 1, 9, 2, 6, 5, 5, 3, 5, 8, 7, 9, 7, 1, 1, 2, 8, 5, 2, 1, 3, 8, 4, 9, 6, 0, 2, 5, 9, 3
COMMENTS
Also known as the 15th Bendersky constant.
FORMULA
A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(15) = exp(H(15)*B(16)/16 - zeta'(-15)) = exp((B(16)/16)*(EulerGamma + log(2*Pi) - (zeta'(16)/zeta(16))).
Equals (2*Pi*exp(gamma) * Product_{p prime} p^(1/(p^16-1)))^c, where gamma is Euler's constant ( A001620), and c = Bernoulli(16)/16 = -3617/8160 (Van Gorder, 2012). - Amiram Eldar, Feb 08 2024
EXAMPLE
0.342830806132816736571711146340672378141726945483236877251076164....
MATHEMATICA
Exp[N[(BernoulliB[16]/16)*(EulerGamma + Log[2*Pi] - Zeta'[16]/Zeta[16]), 200]]
CROSSREFS
Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266560 (A(14)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).
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