A059750 -id:A059750 - cp's OEIS frontend

A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

5, 5, 3, 0, 5, 1, 2, 9, 3, 3, 5, 7, 5, 9, 5, 1, 8, 6, 7, 7, 9, 9, 5, 1, 0, 3, 7, 0, 8, 7, 1, 2, 4, 7, 7, 4, 5, 5, 0, 8, 0, 2, 8, 5, 7, 6, 0, 1, 9, 6, 6, 1, 7, 7, 6, 3, 3, 0, 4, 0, 7, 0, 9, 7, 0, 5, 9, 5, 3, 8, 7, 8, 8, 4, 0, 7, 7, 1, 2, 5, 4, 1, 6, 8, 7, 0, 5, 3, 7, 3, 2, 6, 3, 1, 6, 8, 2, 9, 1, 3

Offset: 0

Jean-François Alcover, May 19 2014

			-0.5530512933575951867799510370871247745508...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 8.8 p. 509.

A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere
Eric Weisstein's World of Mathematics, Thomson Problem
Wikipedia, Thomson problem

Cf. A059750, A080865, A242088.

c = Sqrt[3]*Sqrt[Sqrt[3]/(8*Pi)]*Zeta[1/2]*(Zeta[1/2, 1/3] - Zeta[1/2, 2/3]); RealDigits[c, 10, 100] // First

sqrt(sqrt(27)/8/Pi)*zeta(1/2)*(zetahurwitz(1/2, 1/3) - zetahurwitz(1/2, 2/3)) \\ Charles R Greathouse IV, Jan 31 2018

sqrt(3)*sqrt(sqrt(3)/(8*Pi))*zeta(1/2)*(zeta(1/2, 1/3) - zeta(1/2, 2/3)).

A251734 Decimal expansion of the absolute value of zeta(1/3).

Original entry on oeis.org

9, 7, 3, 3, 6, 0, 2, 4, 8, 3, 5, 0, 7, 8, 2, 7, 1, 5, 4, 6, 8, 8, 8, 6, 8, 6, 2, 4, 4, 7, 8, 9, 6, 5, 7, 0, 7, 7, 2, 8, 2, 9, 6, 3, 1, 7, 4, 3, 0, 5, 3, 3, 3, 9, 9, 4, 5, 3, 5, 8, 1, 4, 4, 6, 2, 1, 0, 8, 5, 1, 8, 2, 8, 1, 3, 3, 9

Offset: 0

R. J. Mathar, Dec 07 2014

			zeta(1/3) = -0.97336024835078271546888....

Wikipedia, Riemann Zeta Function.
Index entries for constants related to Zeta

Cf. A002117, A013661, A059750.

Maple
```
Zeta(1/3) ; evalf(%) ;
```

RealDigits[Zeta[1/3], 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

PARI
```
-zeta(1/3) \\ Jinyuan Wang, Jun 20 2020
```

Lim_{k->infinity} (3/2*k^(2/3) - Sum_{i=1..k} i^(-1/3)). - Jinyuan Wang, Jun 20 2020

A261805 Decimal expansion of M_8, the 8th Madelung constant (negated).

Original entry on oeis.org

2, 0, 5, 2, 4, 6, 6, 8, 2, 7, 2, 6, 9, 2, 7, 1, 2, 2, 8, 1, 7, 6, 3, 3, 7, 7, 9, 9, 1, 7, 3, 3, 8, 3, 9, 9, 1, 7, 0, 8, 3, 7, 7, 5, 2, 9, 9, 6, 5, 5, 8, 2, 1, 9, 3, 2, 3, 7, 3, 2, 4, 5, 7, 7, 5, 3, 4, 9, 9, 4, 1, 3, 2, 8, 7, 5, 2, 7, 0, 6, 1, 4, 6, 9, 8, 5, 1, 9, 8, 8, 3, 9, 4, 1, 3, 1, 7, 5, 1, 0, 8, 8, 1

Offset: 1

Jean-François Alcover, Sep 01 2015

			-2.052466827269271228176337799173383991708377529965582...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.10 Madelung's constant, p. 77.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's MathWorld, Madelung Constants

Cf. A059750, A088537, A085469, A090734, A247040, A261804.

M8 = (15/(4*Pi^3))*(8*Sqrt[2] - 1)*Zeta[1/2]*Zeta[7/2]; RealDigits[M8, 10, 103] // First

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^8-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

M_8 = (15/(4*Pi^3))*(8*sqrt(2) - 1)*zeta(1/2)*zeta(7/2).

A211113 Decimal expansion of -zeta(-1/2).

Original entry on oeis.org

2, 0, 7, 8, 8, 6, 2, 2, 4, 9, 7, 7, 3, 5, 4, 5, 6, 6, 0, 1, 7, 3, 0, 6, 7, 2, 5, 3, 9, 7, 0, 4, 9, 3, 0, 2, 2, 2, 6, 2, 6, 8, 5, 3, 1, 2, 8, 7, 6, 7, 2, 5, 3, 7, 6, 1, 0, 1, 1, 3, 5, 5, 7, 1, 0, 6, 1, 4, 7, 2, 9, 1, 9, 3, 2, 2, 9, 2, 3, 4, 0, 4, 8, 7, 5, 4, 3, 2, 6, 6, 9, 4, 0, 7, 3, 3, 2, 1, 5, 6, 4, 3, 1, 0, 9, 9, 7, 5, 6

Offset: 0

Stanislav Sykora, May 17 2012

			0.207886224977354566017306725397049302226...

Wikipedia, Riemann zeta function

Cf. A059750, A078434.

Maple
```
-Zeta(-1/2) ; evalf(%) ;
```

RealDigits[-Zeta[-1/2], 10, 100][[1]] (* Amiram Eldar, Mar 24 2021 *)

-zeta(-1/2) \\ Charles R Greathouse IV, May 18 2012

Equals -zeta(-1/2) = zeta(3/2)/(4*Pi) = A078434/ (10*A019694).

A161688 Continued fraction for zeta(1/2) (negated).

Original entry on oeis.org

1, 2, 5, 1, 4, 6, 1, 1, 2, 6, 1, 1, 2, 1, 1, 1, 37, 3, 2, 1, 2, 4, 1, 368, 2, 1, 23, 18, 1, 1, 2, 2, 2, 11, 1, 4, 1, 5, 40, 1, 2, 1, 2, 1, 1, 1, 1, 2, 4, 1, 10, 2, 5, 4, 1, 12, 2, 5, 3, 1, 7, 2, 1, 2, 1, 1, 1, 6, 1, 12, 1, 2, 2, 2, 1, 2, 36, 2, 3, 1, 1, 1, 4, 3, 2, 45, 4, 5, 1, 1, 7, 3, 1, 1, 3, 6, 2, 1, 19

Offset: 0

Harry J. Smith, Jun 29 2009

nonn

			1.460354508809586812889499152... = 1 + 1/(2 + 1/(5 + 1/(1 + 1/(4 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..5000

Cf. A059750 Decimal expansion.

{ allocatemem(932245000); default(realprecision, 5400); x=contfrac(-zeta(1/2)); for (n=0, 5000, write("b161688.txt", n, " ", x[n+1])); }

A242616 Decimal expansion of lim_(n->infinity) ((Sum_(k=1..n) 1/sqrt(k)) - (Integral_{x=1..n} 1/sqrt(x))), a generalized Euler constant which evaluates to zeta(1/2) + 2.

Original entry on oeis.org

5, 3, 9, 6, 4, 5, 4, 9, 1, 1, 9, 0, 4, 1, 3, 1, 8, 7, 1, 1, 0, 5, 0, 0, 8, 4, 7, 4, 8, 4, 7, 0, 1, 9, 8, 7, 5, 3, 2, 7, 7, 0, 6, 6, 8, 9, 8, 7, 4, 1, 8, 5, 0, 9, 4, 5, 7, 1, 1, 3, 9, 1, 2, 1, 7, 4, 4, 6, 9, 4, 7, 0, 5, 2, 5, 4, 9, 9, 3, 7, 4, 7, 2, 3, 5, 8, 0, 6, 2, 4, 5, 3, 6, 6, 4, 3, 1, 8, 0, 4

Offset: 0

Jean-François Alcover, May 19 2014

Sometimes called Ioachimescu's constant, after the Romanian mathematician and engineer Andrei Gheorghe Ioachimescu (1868-1943). - Amiram Eldar, Apr 02 2022

			0.53964549119041318711050084748470198753277...

Vasile Berinde and Eugen Păltănea, Gazeta Matematică - A Bridge Over Three Centuries, Romanian Mathematical Society, 2004, pp. 113-114.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.5.3, p. 32.
A. G. Ioachimescu, Problem 16, Gazeta Matematică, Vol. 1, No. 2 (1895), p. 39.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Chao-Ping Chen, Ioachimescu's constant, Research Group in Mathematical Inequalities and Applications, Vol. 13. No. 1 (2010).
Alina Sîntămărian, A Generalisation of Ioachimescu's Constant, The Mathematical Gazette, Vol. 93, No. 528 (2009), pp. 456-467.
Alina Sîntămărian, Regarding a generalisation of Ioachimescu's constant, The Mathematical Gazette, Vol. 94, No. 530 (2010), pp. 270-283.
Alina Sîntămărian, Sequences that converge quickly to a generalized Euler constant, Mathematical and Computer Modelling, Vol. 53, No. 5-6 (2011), pp. 624-630.
Xu You, Di-Rong Chen, and Hong Shi, Some new sequences that converge to the Ioachimescu constant, Journal of Inequalities and Applications, Vol. 2016, No. 1 (2016), Article 148.

Cf. A001620, A059750, A082633.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); 2 + Evaluate(L, 1/2) // G. C. Greubel, Sep 04 2018

RealDigits[Zeta[1/2] + 2, 10, 100] // First

default(realprecision, 100); zeta(1/2)+2 \\ G. C. Greubel, Sep 04 2018

Equals zeta(1/2) + 2.

A252244 Decimal expansion of zeta''(1/2) (negated).

Original entry on oeis.org

1, 6, 0, 0, 8, 3, 5, 7, 0, 1, 3, 9, 2, 8, 6, 6, 1, 4, 2, 2, 6, 9, 1, 3, 0, 6, 5, 0, 5, 9, 4, 4, 9, 6, 2, 7, 8, 5, 1, 8, 5, 5, 9, 3, 6, 1, 9, 6, 3, 6, 3, 5, 4, 5, 3, 5, 3, 0, 9, 2, 9, 5, 7, 5, 3, 6, 6, 7, 8, 0, 9, 2, 4, 6, 0, 1, 4, 4, 9, 8, 0, 1, 3, 3, 8, 0, 6, 8, 0, 6, 2, 7, 6, 3, 5, 6, 3, 8, 8, 5, 5, 4, 8, 4, 9

Offset: 2

Jean-François Alcover, Dec 16 2014

			-16.0083570139286614226913065059449627851855936196363545353...

MathOverflow, Zeta(3) in terms of derivatives of zeta at 1/2 and pi
Eric Weisstein's MathWorld, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A002117, A059750, A114875, A252245.

Maple
```
Zeta(2,1/2) ; evalf(%) ;
```

RealDigits[Zeta''[1/2], 10, 105] // First

zeta(3) = (1/7)*(-Pi^3/4 + (2*zeta'(1/2)^3 - 3*zeta(1/2)*zeta'(1/2)*zeta''(1/2) + zeta(1/2)^2*zeta'''(1/2))/zeta(1/2)^3).

A096616 Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 8, 4, 0, 6, 9, 5, 0, 8, 7, 2, 7, 6, 5, 5, 9, 9, 6, 4, 6, 1, 4, 8, 9, 5, 0, 2, 4, 7, 9, 0, 3, 5, 5, 1, 1, 9, 3, 7, 5, 7, 2, 7, 9, 6, 4, 6, 8, 0, 1, 1, 9, 6, 1, 8, 4, 2, 9, 7, 2, 7, 2, 4, 6, 0, 0, 1, 3, 5, 9, 7, 9, 0, 7, 0, 1, 6, 7, 7, 2, 0, 6, 2, 4, 8, 7, 4, 7, 5, 9, 8, 3, 1, 8, 9, 0, 6, 3, 6, 0, 9, 8

Offset: 0

Eric W. Weisstein, Jun 30 2004

			0.0840695087...

David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke and Victor H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007, pp. 18 and 227.
Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Wellesley, MA: A K Peters, 2004, pp. 15-17.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jonathan M. Borwein and Scott B. Lindstrom, Meetings with Lambert W and other special functions in optimization and analysis, Pure and Applied Functional Analysis, Vol. 1, No. 3 (2016), pp. 361-396, alternative link.
Donald E. Knuth, Problem 10832, The American Mathematical Monthly, Vol. 107, No. 9 (2000), p. 863, A Stirling Series, solution by Cecil C. Rousseau, ibid., Vol. 108, No. 9 (2001), pp. 877-878.
Allen Stenger, Experimental Math for Math Monthly Problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131, alternative link.
Eric Weisstein's World of Mathematics, Knuth's Series.

Cf. A019727, A059750, A231863.

Flatten[{0, RealDigits[2/3 + Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]]}] (* Vaclav Kotesovec, Aug 16 2015 *)

2/3 + zeta(1/2)/sqrt(2*Pi) \\ Michel Marcus, Aug 15 2015

Equals Sum_{k>=1} (1/sqrt(2*Pi*k) - k^k/(k!*exp(k))). - Amiram Eldar, Oct 13 2020

Equals 2/3 - A134469. - R. J. Mathar, Dec 17 2024

A178678 Decimal expansion of the sum of alternating reciprocal square roots, omitting terms where n is a perfect square.

Original entry on oeis.org

0, 8, 8, 2, 4, 8, 5, 3, 7, 1, 3, 8, 3, 1, 4, 9, 3, 9, 1, 6, 9, 9, 6, 6, 2, 0, 7, 2, 2, 2, 2, 2, 1, 0, 6, 8, 3, 1, 5, 7, 3, 7, 5, 8, 9, 2, 3, 0, 0, 0, 7, 8, 7, 3, 7, 4, 2, 1, 3, 3, 3, 6, 1, 4, 1, 1, 2, 0, 6, 3, 6, 8, 4, 7, 4, 6, 3, 4, 3, 5, 8, 2, 7, 8, 4, 5, 9, 3, 7, 0, 0, 7, 8, 0, 6, 9, 1, 3, 3, 1, 5, 8, 9, 6, 7

Offset: 0

Matt Rieckman (mjr162006(AT)yahoo.com), Jun 03 2010

Provides a closed form for the Riemann zeta function of one half: Zeta(1/2) = (1 + sqrt(2))(R - log(2)).

The omitted sum of perfect squares equates to the natural logarithm of 2. Giving the alternating sum of all reciprocal square roots as log(2) - R.

			R=0.0882485371383149391699662072222210683157375892300078737421333614112...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A059750, A002162, A002193.

SetDefaultRealField(RealField(100)); L:=RiemannZeta();  (Sqrt(2)-1)*Evaluate(L, 1/2) +Log(2); // G. C. Greubel, Jan 27 2019

RealDigits[(Sqrt[2] -1)*Zeta[1/2] +Log[2], 10, 100][[1]]

default(realprecision, 100); (sqrt(2)-1)*zeta(1/2)+log(2) \\ G. C. Greubel, Jan 27 2019

numerical_approx((sqrt(2)-1)*zeta(1/2)+log(2), digits=100) # G. C. Greubel, Jan 27 2019

R = Sum_{n>=2} (-1)^n/sqrt(n) for n that are not a perfect square.
R = 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(5) + 1/sqrt(6) - 1/sqrt(7) + 1/sqrt(8) + ...
R = Sum_{n>=2} (-1)^(n+1)*(1-sqrt(n))/n.

Minor correction, simplified description, and additional comments Matt Rieckman (mjr162006(AT)yahoo.com), Jun 28 2010

A252245 Decimal expansion of zeta'''(1/2) (negated).

Original entry on oeis.org

9, 6, 0, 0, 3, 3, 0, 9, 2, 4, 5, 3, 1, 9, 0, 7, 0, 0, 9, 7, 3, 8, 9, 7, 6, 7, 2, 2, 0, 6, 9, 5, 4, 5, 9, 3, 0, 2, 5, 1, 4, 0, 1, 8, 8, 4, 6, 5, 5, 5, 7, 2, 8, 0, 5, 4, 2, 9, 9, 9, 0, 8, 0, 6, 5, 6, 7, 0, 9, 1, 9, 4, 4, 1, 8, 7, 6, 3, 1, 6, 0, 3, 4, 0, 6, 5, 5, 6, 9, 3, 2, 4, 6, 2, 3, 8, 8, 1, 1, 2, 0, 1, 0, 1

Offset: 2

Jean-François Alcover, Dec 16 2014

			-96.003309245319070097389767220695459302514018846555728...

MathOverflow, Zeta(3) in terms of derivatives of zeta at 1/2 and pi
Eric Weisstein's MathWorld, Riemann Zeta Function

Cf. A002117, A059750, A114875, A252244.

RealDigits[Zeta'''[1/2], 10, 104] // First

zeta(3) = (1/7)*(-Pi^3/4 + (2*zeta'(1/2)^3 - 3*zeta(1/2)*zeta'(1/2)*zeta''(1/2) + zeta(1/2)^2*zeta'''(1/2))/zeta(1/2)^3).

cp's OEIS Frontend

A242617 Decimal expansion of Kuijlaars-Saff constant, a constant related to Tammes' constants, Thomson's electron problem and Fekete points.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A251734 Decimal expansion of the absolute value of zeta(1/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A261805 Decimal expansion of M_8, the 8th Madelung constant (negated).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A211113 Decimal expansion of -zeta(-1/2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A161688 Continued fraction for zeta(1/2) (negated).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A242616 Decimal expansion of lim_(n->infinity) ((Sum_(k=1..n) 1/sqrt(k)) - (Integral_{x=1..n} 1/sqrt(x))), a generalized Euler constant which evaluates to zeta(1/2) + 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A252244 Decimal expansion of zeta''(1/2) (negated).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs