A113024 -id:A113024 - cp's OEIS frontend

A059750 Decimal expansion of zeta(1/2) (negated).

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

Offset: 1

Peter Walker (peterw(AT)aus.ac.ae), Feb 11 2001

zeta(1/2) can be calculated as a limit similar to the limit for the Euler-Mascheroni constant or Euler gamma. - Mats Granvik Nov 14 2012

The WolframAlpha link gives 3 series and 3 integrals for zeta(1/2). - Jonathan Sondow, Jun 20 2013

			-1.4603545088095868128894991525152980124672293310125814905428860878...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Harry J. Smith, Table of n, a(n) for n = 1..5000
B. K. Choudhury, The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond A 445 (1995) 477, Table 3.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
Fredrik Johansson, zeta(1/2) to 1 million digits.
Fredrik Johansson, Rapid computation of special values of Dirichlet L-functions, arxiv:2110.10583 [math.NA], 2021.
Hisashi Kobayashi, Some results on the xi(s) and Xi(t) functions associated with Riemann's zeta(s) function, arXiv preprint arXiv:1603.02954 [math.NT], 2016.
Lutz Mattner and Irina Shevtsova, An optimal Berry-Esseen type theorem for integrals of smooth functions, arXiv:1710.08503 [math.PR], 2017.
J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function
WolframAlpha, zeta(1/2)

Cf. A161688 (continued fraction), A078434, A014176, A113024.

Maple
```
Digits := 120; evalf(Zeta(1/2));
```

RealDigits[ Zeta[1/2], 10, 111][[1]] (* Robert G. Wilson v, Oct 11 2005 *)
RealDigits[N[Limit[Sum[1/Sqrt[n], {n, 1, k}] - 2*Sqrt[k], k -> Infinity], 90]][[1]] (* Mats Granvik Nov 14 2012 *)

default(realprecision, 5080); x=-zeta(1/2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b059750.txt", n, " ", d)); \\ Harry J. Smith, Jun 29 2009

zeta(1/2) = lim_{k->oo} ( Sum_{n=1..k} 1/n^(1/2) - 2*k^(1/2) ) (according to Mathematica 8). - Mats Granvik Nov 14 2012
From Magri Zino, Jan 05 2014 - personal communication: (Start)
The previous result is the case q=2 of the following generalization:
zeta(1/q) = lim_{k->oo} (Sum_{n=1..k} 1/n^(1/q) - (q/(q-1))*k^((q-1)/q)), with q>1. Example: for q=3/2, zeta(2/3) = lim_{k->oo} (Sum_{n=1..k} 1/n^(2/3) - 3*k^(1/3)) = -2.447580736233658231... (End)
Equals -A014176*A113024. - Peter Luschny, Oct 25 2021

Sign of the constant reversed by R. J. Mathar, Feb 05 2009

A263192 Decimal expansion of Sum_{n >= 1} cos(n)/sqrt(n), negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Oct 11 2015

A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).

			-0.1941089350921820497391492449281947266353205526340478...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT].

Cf. A113024, A121225, A263193.

evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) + Zeta(0, 1/2, 1-1/(2*Pi))), 120);

N[(Zeta[1/2, 1/(2*Pi)] + Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
RealDigits[Re[(1/2)*(PolyLog[1/2, E^(-I)] + PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)

zetahurwitz(1/2, 1/Pi/2)/2 + zetahurwitz(1/2, 1-1/Pi/2)/2 \\ Charles R Greathouse IV, Jan 30 2018

(Zeta(1/2, 1/(2*Pi)) + Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.

A263193 Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Oct 11 2015

A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).

			1.043982102849161527453294872586750459790907144722612...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT].

Cf. A096444, A113024, A263192.

evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) - Zeta(0, 1/2, 1-1/(2*Pi))), 120);

N[(Zeta[1/2, 1/(2*Pi)] - Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
RealDigits[Re[(1/2)*I*(PolyLog[1/2, E^(-I)] - PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)

(Zeta(1/2, 1/(2*Pi)) - Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.

A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

Original entry on oeis.org

1, -1, 1, 0, 1, -1, 1, -2, 2, -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -1, 1, -2, 2, -1, 2, 0, 1, -1, 1, -3, 1, -1, 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 2, -1, 1, -1, 2, -2, 1, 0, 1, -2, 1, -2, 1, -1, 1, 0, 1, -1, 2, -2, 1, -1, 1, 0, 1, -1, 1, -4, 1, -1, 2, 0, 1, -1, 1, -1, 3, -1, 1, 0, 1, -1, 1, -2, 1, -2, 1, 0, 1, -1, 1

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..11025
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000290, A010052, A046951, A048272, A113024, A300853.

seq(coeff(series(add(x^(k^2)/(1+x^(k^2)),k=1..n), x,n+1),x,n),n=1..100); # Muniru A Asiru, Jul 30 2018

nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}]
f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

A317529(n) = sumdiv(n,d,((-1)^(1+(n/d)))*issquare(d)); \\ Antti Karttunen, Nov 07 2018

G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)).
L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d).
If n is odd, a(n) = A046951(n).
Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = -(sqrt(2)-1) * zeta(1/2) = 0.604898... (A113024). (End)

A367309 Decimal expansion of area under the curve (1-2^(1-x))*zeta(x) from 0 to 1.

Original entry on oeis.org

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

Offset: 0

Alejandro Malla, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to g(x) = (1 - 2^(1-x))*zeta(x) on the open interval (0, 1). This series can be described as an alternating version of the 'p-series' when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x. Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined, but has the limit value log(2). Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			0.60211234931037155497112632...

Cf. A113024, A367310, A367311.

y = NIntegrate[(1 - 2^(1-x)) Zeta[x], {x, 0, 1}, WorkingPrecision -> 200]
RealDigits[y][[1]]

intnum(x=0, 1, (1-2^(1-x))*zeta(x)) \\ Michel Marcus, Nov 14 2023

A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Maximum curvature = 0.0641392820642571684220887165127181687393..., which occurs at x = 0.6827548440370203586269... .

Cf. A113024, A367309, A367312.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
c[x_] := Abs[f''[x]]/(1 + f'[x]^2)^(3/2)
y = FindMaximum[{c[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

One initial 0 inserted by Artur Jasinski, Aug 04 2025

A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 03 2015

Differentiation of Sum_{k>=1} (-1)^k/k^s = -(2^s-2)*zeta(s)/2^s with respect to s gives -Sum_{k>=1} (-1)^k*log(k)/k^s = -2^(1-s)*log(2)*zeta(s) - (1-2^(1-s))*zeta'(s), where zeta(.) and zeta'(.) are the Riemann zeta function and its derivative. - R. J. Mathar, Apr 17 2019, typo in the first formula corrected by Vaclav Kotesovec, Jan 11 2024

			0.1932888316392827389646154593552381142952702225292219922936048103344...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Cf. A091812, A113024.

evalf(sum((-1)^k*log(k)/sqrt(k), k=1..infinity), 120);

RealDigits[((3-Sqrt[2])*Log[2]/2 - (Sqrt[2]-1)*(2*EulerGamma + Pi + 2*Log[Pi])/4) * Zeta[1/2], 10, 106][[1]]
RealDigits[DirichletEta'[1/2], 10, 110][[1]] (* Eric W. Weisstein, Jan 08 2024 *)

((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*Euler + Pi + 2*log(Pi))/4)* zeta(1/2) \\ G. C. Greubel, Apr 15 2018

Equals ((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*gamma + Pi + 2*log(Pi))/4) * zeta(1/2), where gamma is the Euler-Mascheroni constant A001620.

A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2.

A367312 Minimum value of 2nd derivative of (1 - 2^(1-x)) zeta(x), for 0 < x < 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling and Peter J. C. Moses, Nov 13 2023

The series Sum_{n >= 1} (-1)^(n+1)/n^x converges nonuniformly to (1 - 2^(1-x)) zeta(x) (0,1). This series can be described as an alternating version of the "p-series" when 0 < p < 1. Let f(x) = Sum_{n >= 1} (-1)^(n+1)/n^x and g(x) = (1 - 2^(1-x)) zeta(x). Then f(0+) = g(0) = 1/2 and f(1) = log(2), whereas g(1) is undefined. Also, f(1/2) = g(1/2) = A113024 = 0.604898643421... .

			Minimum value of f"(x), where f(x) = (1 - 2^(1-x)) zeta(x), for 0 < x < 1:
0.0641392820642571684220887165127181687393656828446464013955957700...,
which occurs for x = 0.59737100658235275929541785444598... .

Cf. A113024, A367309, A367311.

f[x_] := (1 - 2^(1 - x)) Zeta[x];
y = FindMinimum[{f''[x], 0 < x < 1}, {x, 1/2}, WorkingPrecision -> 1000]
RealDigits[y][[1]][[1]]

A251735 Decimal expansion of Sum_{n>=1} (-1)^(n+1)/n^(1/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 07 2014

Cubic root analog of A113024.

			0.57175283382527766493...

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

Cf. A005480, A113024, A251734.

Maple
```
Zeta(1/3)*(1-root[3](4)) ; evalf(%) ;
```

RealDigits[-Zeta[1/3]*(4^(1/3) - 1), 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)

-zeta(1/3)*(4^(1/3)-1) \\ Charles R Greathouse IV, Apr 20 2016

Equals 1 - 1/A002580 + 1/A002581 - 1/A005480 + ... = A251734 *(1 - A005480).

A331239 Decimal expansion of Sum_{k>=0} (-1)^k/AGM(1, 1+k).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Jan 13 2020

AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.

This series is closely related to A188859 (Sum_{k>=0} (-1)^k/((1+(1+k))/2)) and A113024 (Sum_{k>=0} (-1)^k/sqrt(1+k)). The denominators of these alternating series differ by being arithmetic, geometric, or arithmetic-geometric means of 1 and k.

			0.6092151504524492287304733713491660511183939228565999735...

Cf. A016627, A188859, A113024.

PARI
```
sumalt(k=0, (-1)^k/agm(1,1+k))
```

cp's OEIS Frontend

A059750 Decimal expansion of zeta(1/2) (negated).

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A263192 Decimal expansion of Sum_{n >= 1} cos(n)/sqrt(n), negated.

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A263193 Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).

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A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

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A367309 Decimal expansion of area under the curve (1-2^(1-x))*zeta(x) from 0 to 1.

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A367311 Maximum curvature of the curve (1 - 2^(1-x)) zeta(x) from 0 to 1.

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A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).

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