A020777 -id:A020777 - cp's OEIS frontend

A256778 Decimal expansion of the generalized Euler constant gamma(1,4).

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

Offset: 0

Jean-François Alcover, Apr 10 2015

			0.71028979306409369731376647579508261030406104249693294...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.3, p. 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Cf. A001620 (EulerGamma), A016627, A020777, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256779-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

R:=RealField(100); (2*EulerGamma(R) + Pi(R) + 2*Log(2))/8; // G. C. Greubel, Aug 27 2018

RealDigits[EulerGamma/4 + Pi/8 + Log[2]/4, 10, 103] // First

default(realprecision, 100); (2*Euler + Pi + 2*log(2))/8 \\ G. C. Greubel, Aug 27 2018

Equals (2*EulerGamma + Pi + 2*log(2))/8.
Equals Sum_{n>=0} (1/(4n+1) - 1/2*arctanh(2/(4n+3))).
Equals -(psi(1/4) + log(4))/4 = (A020777 - A016627)/4. - Amiram Eldar, Jan 07 2024

A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Feb 18 2020

			0.0231049931154189707889338104303390140033817603974220901231825...

J. P. Gram, "Note sur le calcul de la fonction zeta(s) de Riemann", Det Kgl. Danske Vid. Selsk. Overs., 1895, pp. 303-308. p.307 (16 decimal digits).
Charles Jean De La Vallée Poussin, Sur La Fonction de Riemann Et Le Nombre Des Nombres Premiers Inférieurs à Une Limite Donnee, 1899.

Jesús Guillera, Some sums over the non-trivial zeros of the Riemann zeta function, arXiv:1307.5723v7 [math.NT], 2013-2014; see p.9 eq.(21).
Kano Kono, Vieta's Formulas on Completed Riemann Zeta, Alien's Mathematics p. 13 eq. 2.3(2).
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen I, 1909, pp. 310-342.
J. J. Y. Liang and John Todd, The Stieltjes Constants, Journal of Research of the National Bureau of Standards, 1972, pp. 175-176.
André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
André Voros, Zeta functions for the Riemann zeros, 2001(2008) p.20 Table 1.
André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

Cf. A006752, A074760, A332614.

evalf((-32 - log(Pi)^2 + Psi(0, 1/4)^2 + Psi(1, 1/4) + 4*(Psi(0, 1/4) * Zeta(1, 1/2) + Zeta(2, 1/2)) / Zeta(1/2)) / 8, 120); # Vaclav Kotesovec, Feb 19 2020

Join[{0}, RealDigits[N[-4 + Catalan + Pi^2/8 + (Zeta''[1/2]/Zeta[1/2] - (Zeta'[1/2] / Zeta[1/2])^2)/2, 105]][[1]]]
N[SeriesCoefficient[Log[s*(s-1)*Pi^(-s/2)*Gamma[s/2]*Zeta[s]/2], {s, 1/2, 2}], 105] (* Vaclav Kotesovec, Feb 19 2020 *)

Equals -4 + G + Pi^2/8 + (1/2)(zeta''(1/2)/zeta(1/2) - (zeta'(1/2)/zeta(1/2))^2) where G is the Catalan constant A006752.
Equals G - 4 + (Pi^2 - (gamma + Pi/2 + log(8*Pi))^2) / 8 + zeta''(1/2) / (2*zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620 and G is the Catalan constant A006752. - Vaclav Kotesovec, Feb 19 2020
Also equals (-32 - log(Pi)^2 + psi(0, 1/4)^2 + psi(1, 1/4) + 4*(psi(0, 1/4) * zeta'(1/2) + zeta''(1/2)) / zeta(1/2)) / 8, where psi(0, 1/4) = -A020777 and psi(1, 1/4) = A282823. - Vaclav Kotesovec, Feb 19 2020

A200134 Decimal expansion of the negated value of the digamma function at 3/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/4) = -1.085860879786472169626886762817...

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A001620, A020759, A020777, A047787, A301816.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) + Pi(R)/2 - 3*Log(2); // G. C. Greubel, Aug 29 2018

Maple
```
evalf(-gamma+Pi/2-3*log(2)) ;
```

RealDigits[ -PolyGamma[3/4], 10, 97] // First (* Jean-François Alcover, Feb 20 2013 *)
N[StieltjesGamma[0, 3/4], 99] (* Peter Luschny, May 16 2018 *)

-psi(3/4) \\ Charles R Greathouse IV, Nov 22 2011

Psi(3/4) = -gamma + Pi/2 - 3*log(2) = A000796 - A020777 = 3.14159... - 4.22745...

Pi = gamma(0,1/4) - gamma(0,3/4) = A020777 - A200134, where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A200064 Decimal expansion of the negated value of the digamma function at 2/3.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			psi(2/3) = -1.3182344157865884724023408166...

Cf. A020759, A047787, A020777

RealDigits[ -PolyGamma[2/3], 10, 98] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(2/3) \\ Charles R Greathouse IV, Nov 22 2011

psi(2/3) = -gamma+Pi*sqrt(3)/6-3*log(3)/2 = A093602 - A047787 = 1.813799 -3.132033...

A200138 Decimal expansion of the negated value of the digamma function at 4/5.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 13 2011

			Psi(4/5) = -0.965008566706138459391297633...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200137.

-gamma+Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[ -PolyGamma[4/5], 10, 87] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(4/5) \\ Charles R Greathouse IV, Nov 22 2011

Psi(4/5) = -gamma + Pi*sqrt(1+2/sqrt 5)/2 -5*log(5)*log((3+sqrt 5)/2)/4.

A250129 Decimal expansion of the negated value of the digamma function at 1/8.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 15 2015

			Psi(1/8) = -8.388492663295854867802742923086343000051446...

Eric Weisstein's MathWorld, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A222457, A222458.

RealDigits[PolyGamma[1/8], 10, 105] // First

-psi(1/8) \\ Charles R Greathouse IV, Jan 15 2015

Psi(1/8) = -gamma - (1/2)*(1+sqrt(2))*Pi - sqrt(2)*arccoth(sqrt(2)) - 4*log(2).

A306716 Decimal expansion of the negated value of the digamma function at 1/10.

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Aug 22 2019

			Equals 10.4237549404110767951682162190100254042916425624441889203263920841...

Eric Weisstein's World of Mathematics, Digamma Function
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200135, A222457, A250129.

Maple
```
evalf(-Psi(1/10), 102);
```

RealDigits[-PolyGamma[1/10], 10, 105][[1]]

PARI
```
-psi(1/10)
```

Psi(1/10) = -gamma - Pi*5^(1/4)*(sqrt(2 + sqrt(5))/2) - 2*log(2) - 5*log(5)/4 - 3*sqrt(5)*log((1 + sqrt(5))/2)/2, where gamma is the Euler-Mascheroni constant A001620.

Equals gamma - H(-9/10), H(z) the harmonic number. - Peter Luschny, Aug 22 2019

A200136 Decimal expansion of the negated value of the digamma function at 2/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(2/5) = -2.5613845445851161457306754820475...

Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A200135-A200138.

-gamma-Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)*log((3+sqrt(5))/2)/4 ; evalf(%) ;

RealDigits[ PolyGamma[2/5], 10, 84] // First (* Jean-François Alcover, Feb 21 2013 *)

-psi(2/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(2/5) = -gamma -Pi*sqrt(1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

A200137 Decimal expansion of the negated digamma function at 3/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/5) = -1.540619213893190414760663948806231941510...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200138

-gamma+Pi*sqrt(1-2/sqrt(5))/2-5*log(5)/4+sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[-PolyGamma[3/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(3/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(3/5) = -gamma +Pi*sqrt( 1-2/sqrt 5)/2 -5*log(5)/4 +sqrt(5)*log((3+sqrt 5)/2)/4.

More terms from Jean-François Alcover, Feb 11 2013

cp's OEIS Frontend

A256778 Decimal expansion of the generalized Euler constant gamma(1,4).

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A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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A200134 Decimal expansion of the negated value of the digamma function at 3/4.

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A200064 Decimal expansion of the negated value of the digamma function at 2/3.

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A200138 Decimal expansion of the negated value of the digamma function at 4/5.

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A250129 Decimal expansion of the negated value of the digamma function at 1/8.

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