A059750 -id:A059750 - cp's OEIS frontend

A078434 Decimal expansion of zeta(3/2).

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

Offset: 1

Robert G. Wilson v, Dec 30 2002

			2.6123753486854883433485675679240716305708006524000634075733...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Alexander Sakhnovich and Lev Sakhnovich, Nonlinear Fokker-Planck equation: stability, distance and the corresponding extremal problem in the spatially inhomogeneous case, in: D. Alpay and B. Kirstein (eds.), Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes, Birkhäuser, Cham, 2015, pp. 379-394; arXiv preprint, arXiv:1307.1126 [math.AP], 2013-2015.
Wikipedia, Bose-Einstein condensate.

Cf. A019694, A019707, A033461, A078470, A059750, A211113.

Mathematica
```
RealDigits[ Zeta[3/2], 10, 110][[1]]
```

zeta(3/2) \\ Charles R Greathouse IV, Feb 06 2015

Equals (2/sqrt(Pi))*Integral_{x>=0} sqrt(x)/(exp(x)-1) dx. - Jean-François Alcover, Nov 12 2013
Equals Gamma(-1/2)*zeta(-1/2)*tau(-1/2) where tau(s) = (2*Pi*i)^(-s) + (-2*Pi*i)^(-s). This follows from the functional equation of the Riemann zeta function. (Cf. A059750, A211113, A019707). - Peter Luschny, May 13 2020
Equals -4*Pi*zeta(-1/2) = 10 * A019694 * A211113. - Amiram Eldar, May 29 2021

A113024 Decimal expansion of Sum_{k>=1} -(-1)^k/sqrt(k).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 11 2005

			1 - 1/sqrt(2) + 1/sqrt(3) - 1/sqrt(4) + 1/sqrt(5) - 1/sqrt(6) + 1/sqrt(7) ... =
0.60489864342163037024726591423595549975976254513024738037854664808...

Stephen Fletcher Hewson, A Mathematical Bridge: An Intuitive Journey In Higher Mathematics, World Scientific, NJ, 2003, p. 83.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Rick Kreminski, Using Simpson's rule to approximate sums of infinite series, Coll. Math. J. 28 (5) (1997), p 368-376, Table 1.
Eric Weisstein's World of Mathematics, Zeta Function.

Cf. A002193, A059750, A263192, A263193, A265162.

Zeta(0,1/2,1/2); evalf(%) ; # R. J. Mathar, Dec 17 2024

RealDigits[(1 - Sqrt[2])Zeta[1/2], 10, 111][[1]]

(1-sqrt(2))*zeta(1/2) \\ G. C. Greubel, Apr 09 2018

Equals (1-sqrt(2))*zeta(1/2) = (-1+A002193) * A059750.
A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 03 2015
Equals -zeta(1/2, 1/2). - Peter Luschny, Nov 03 2020

A088537 Decimal expansion of Madelung's constant M2.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Nov 16 2003

			M2 = -1.61554262671282472386792333275861809...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 76-81.

I. J. Zucker, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A: Math., Gen. vol. 7 (1974) no. 13, pp. 1568-1575.

Cf. A059750.

M2:=evalf(4*(sqrt(2)-1)*Zeta(1/2)*sum('(-1)^n/sqrt(2*n+1)','n'=0..infinity),120); # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009

(2-2*I)*(Sqrt[2]-1)*Zeta[1/2]*(PolyLog[1/2, -I]-Zeta[1/2, 1/4]) // Re // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 15 2013 *)

DirBet=sumalt(n=0, (-1)^n/sqrt(2*n+1)); print(4.0*(sqrt(2)-1)*zeta(0.5)*DirBet) ; \\ R. J. Mathar, Jul 20 2007

M2 = Sum_{ -oo < i < oo, -oo < j < oo, (i,j) != (0,0) } (-1)^(i + j)/sqrt(i^2 + j^2).

M2 = 4*(sqrt(2) - 1)*zeta(1/2)*beta(1/2) (beta=Dirichlet beta function).

More terms from R. J. Mathar, Jul 20 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009

A247041 Decimal expansion of zeta(5/2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 10 2014

Zeta(5/2) appears in the expression of the 6th Madelung constant (A247040).

			1.3414872572509171797567696933486121366230376295059865...

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

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

Cf. A059750, A078434, A247040.

Mathematica
```
RealDigits[Zeta[5/2], 10, 102] // First
```
PARI
```
zeta(5/2) \\ Michel Marcus, Feb 23 2023
```

A090734 Decimal expansion of 4th Madelung constant (negated).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 18 2004

			-1.83939908404504706624730547956723047642278359481773...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 77.

Cf. A059750, A088537, A085469.

RealDigits[8*(5 - 3*Sqrt[2])*Zeta[1/2]*Zeta[-1/2], 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) \\ Charles R Greathouse IV, Jun 07 2016

M_4 = -8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) = -8 *(A157122-6) * A059750 * A211113.

A247040 Decimal expansion of M_6, the 6th Madelung constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 10 2014

			-1.9655570390090782813123135557351853678689767284464645117...

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, Dirichlet Beta Function
Eric Weisstein's MathWorld, Madelung Constants

Cf. A059750, A088537, A085469, A090734.

beta[x_] := (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x; M6 = (3/Pi^2)*(4*(Sqrt[2]-1)*Zeta[1/2]*beta[5/2] - (4*Sqrt[2]-1)*Zeta[5/2]*beta[1/2]); RealDigits[M6, 10, 104][[1]]

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 07 2016

th4(x)=1+2*sumalt(n=1, (-1)^n*x^n^2)
intnum(x=0, [oo, 1], (th4(exp(-x))^6-1)/sqrt(Pi*x)) \\ Charles R Greathouse IV, Jun 06 2016

M6 = (3/Pi^2)*(4*(sqrt(2)-1)*zeta(1/2)*beta(5/2) - (4*sqrt(2)-1)*zeta(5/2)*beta(1/2)), where beta is Dirichlet's "beta" function.

A054040 a(n) terms of series {1/sqrt(j)} are >= n.

Original entry on oeis.org

1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 202, 217, 232, 247, 263, 280, 297, 314, 332, 351, 370, 389, 409, 430, 451, 472, 494, 517, 540, 563, 587, 612, 637, 662, 688, 715, 741, 769, 797, 825

Offset: 1

Asher Auel, Apr 13 2000

nonn

In many cases the first differences have the form {2k, 2k, 2k, 2k+1} (A004524). In such cases the second differences are {0, 0, 1, 1}. See A082915 for the exceptions. In as many as these, the first differences have the form {2k-1, 2k-1, 2k-1, 2k}. - Robert G. Wilson v, Apr 18 2003 [Corrected by Carmine Suriano, Nov 08 2013]
a(100)=2574, a(1000)=250731 & a(10000)=25007302 which differs from Sum{i=4..104}A004524(i)=2625, Sum{i=4..1004}A004524(i)=251250 & Sum{i=4..10004}A004524(i)=25012500. - Robert G. Wilson v, Apr 18 2003
A054040(n) <= A011848(n+2), A054040(10000)=25007302 and A011848(n+2)=25007500. - Robert G. Wilson v, Apr 18 2003

			Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
.k.......1....2.....3.....4.....5.....6.....7
-------------------------------------------------
b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
For A019529 we have:
n=0: smallest k is a(0) = 1 since 1.00 > 0
n=1: smallest k is a(1) = 2 since 1.71 > 1
n=2: smallest k is a(2) = 3 since 2.28 > 2
n=3: smallest k is a(3) = 5 since 3.23 > 3
n=4: smallest k is a(4) = 7 since 4.01 > 4
For this sequence we have:
n=1: smallest k is a(1) = 1 since 1.00 >= 1
n=2: smallest k is a(2) = 3 since 2.28 >= 2
n=3: smallest k is a(3) = 5 since 3.23 >= 3
n=4: smallest k is a(4) = 7 since 4.01 >= 4

Sela Fried, On the partial sums of the Zeta function Sum_{n>=1} 1/n^s for 0 < s < 1, 2024.
Hugo Pfoertner, Plot of a(n)-(1/4)*(n^2-2*zeta(1/2)*n), n=1..1000.

Cf. A002387, A004080, A059750, A082915, A054044, A011848, A082915.

See A019529 for a different version.

f[n_] := Block[{k = 0, s = 0}, While[s < n, k++; s = N[s + 1/Sqrt[k], 50]]; k]; Table[f[n], {n, 1, 60}]

a(n)=if(n<0,0,t=1;z=1;while(zBenoit Cloitre, Sep 23 2012

Let f(n) = (1/4)*(n^2-2*zeta(1/2)*n) then we have a(n) = f(n) + O(1). More precisely we claim that for n >= 2 we have a(n) = floor(f(n)+c) where c > Max{a(n)-f(n) : n>=1} = a(153) - f(153) = 1.032880076066608813953... and we believe we can take c = 1.033. - Benoit Cloitre, Sep 23 2012

Definition and offset modified by N. J. A. Sloane, Sep 01 2009

A261804 Decimal expansion of zeta(7/2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 01 2015

Zeta(7/2) appears in the expression of the 8th Madelung constant (A261805).

			1.126733867317056646427812491854984272221996957403602963842396...

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

Eric Weisstein's MathWorld, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A059750, A078434, A247041, A261805.

Mathematica
```
RealDigits[Zeta[7/2], 10, 104] // First
```

zeta(7/2) \\ Charles R Greathouse IV, Jun 07 2016

A134469 Decimal expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

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

Offset: 0

Hans J. H. Tuenter, Oct 27 2007

This number is the limiting expected overshoot over a boundary for the sum of independent and identically distributed normal variables with unit variance, as their positive mean approaches zero. It has applications in sequential analysis.

			0.58259715793901067020517716418763115472909387019865...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 326.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Joseph T. Chang and Yuval Peres, Ladder heights, Gaussian random walks and the Riemann zeta function, Annals of Probability, 25(2) (1997) 787-802.
Alain Comtet and Satya N. Majumdar, Precise Asymptotics for a Random Walker’s Maximum, J. Stat. Mech. Theor. Exp. 06 (2005) P06013, arXiv:cond-mat/0506195 [cond-mat.stat-mech], 2005.
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.
Robert A. Wijsman, Overshoot in the Case of Normal Variables, Sequential Analysis, 23(2):275-284, 2004.

Cf. A134470 (continued fraction), A134471 (Numerators of continued fraction convergents), A134472 (Denominators of continued fraction convergents).

Digits:=100; evalf(-Zeta(1/2)/sqrt(2*Pi));

RealDigits[-Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]] (* G. C. Greubel, Mar 27 2018 *)

-zeta(1/2)/sqrt(2*Pi) \\ Charles R Greathouse IV, Mar 10 2016

-zeta(1/2)/sqrt(2*Pi)= A059750/A019727.

More decimals from Vaclav Kotesovec, Mar 21 2016

A114875 Decimal expansion of -zeta'(1/2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 03 2006

			3.92264613920915172747153144671459951373032397150650...

B. K. Choudhury, The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond. A 445 (1995) 477, Table 3.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function.

Cf. A001620, A059750.

Zeta(1,1/2) ;evalf(%) ; # R. J. Mathar, May 03 2021

RealDigits[-Zeta'[1/2], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

-zeta'(1/2) \\ Charles R Greathouse IV, Mar 28 2012

-(2*Euler+Pi+2*log(8*Pi))*zeta(1/2)/4 \\ Charles R Greathouse IV, Mar 28 2012

Equals ((2*gamma + Pi + 2*log(8*Pi))*zeta(1/2))/4, where gamma is Euler's constant (A001620).

cp's OEIS Frontend

A078434 Decimal expansion of zeta(3/2).

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A113024 Decimal expansion of Sum_{k>=1} -(-1)^k/sqrt(k).

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A088537 Decimal expansion of Madelung's constant M2.

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A247041 Decimal expansion of zeta(5/2).

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A090734 Decimal expansion of 4th Madelung constant (negated).

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A247040 Decimal expansion of M_6, the 6th Madelung constant.

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A054040 a(n) terms of series {1/sqrt(j)} are >= n.

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