A068466 -id:A068466 - cp's OEIS frontend

A249189 Decimal expansion of Hayman's constant in Landau's Theorem.

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

Offset: 1

Jean-François Alcover, Oct 23 2014

Named after the British mathematician Walter Kurt Hayman (1926-2020). - Amiram Eldar, Apr 15 2021

			4.37687923045295327767353988140892908651874544565...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 421.

Steven Finch, Goldberg’s Zero-One Constants, May 21, 2014. [Cached copy, with permission of the author]
W. K. Hayman, Some remarks on Schottky's theorem, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 43, No. 4 (1947), pp. 442-454.
Wan Tzei Lai, The exact value of Hayman's constant in Landau's Theorem, Scientia Sinica, Vol. 22, No. 2 (1979), pp. 129-134.
Wikipedia, Théorème de Landau, [in French].

Cf. A068466.

K = (1/(4*Pi^2))*Gamma[1/4]^4; RealDigits[K, 10, 102] // First

(1/(4*Pi^2))*gamma(1/4)^4 \\ Michel Marcus, Oct 23 2014

K = (1/(4*Pi^2))*Gamma(1/4)^4.

A256591 Decimal expansion of Xi''(1/2) = 0.02297..., the second derivative of the Riemann Xi function at 1/2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 03 2015

As mentioned in the paper by Borwein et al., the Riemann hypothesis is equivalent to a positivity condition on every even-order derivative of the Xi function at the point s = 1/2.

			0.022971944315145437535249876497632170264593013837589...
Are also listed in the Borwein paper the Xi derivatives of order 4 and 6:
Xi^(4)(1/2) = 0.002962848433687632165368...
Xi^(6)(1/2) = 0.000599295946597579491843...

H. M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0) pp. 16-18

Jonathan M. Borwein, David M. Bradley, Richard E. Crandall, Computational strategies for the Riemann zeta function, Journal of Computational and Applied Mathematics 121 (2000) p. 289.
Eric Weisstein's MathWorld, Xi-Function
Wikipedia, Riemann Xi function

Cf. A020777 (PolyGamma(1/4)), A059750 (zeta(1/2)), A068466 (Gamma(1/4)), A114720 (Xi(1/2)), A114875 (zeta'(1/2)), A252244 (zeta''(1/2)).

d2 = (-(32*Pi^(1/4))^(-1))*Gamma[1/4]*((-32 + (Log[Pi] - PolyGamma[1/4])^2 + PolyGamma[1, 1/4])*Zeta[1/2] + 4*((-Log[Pi] + PolyGamma[1/4])*Zeta'[1/2] + Zeta''[1/2])); Join[{0}, First[RealDigits[d2, 10, 102]]]

Xi(s) = 1/2*s*(s-1)*Pi^(-s/2)*Gamma(s/2)*zeta(s).

Xi''(1/2) = (-(32*Pi^(1/4))^(-1))*Gamma(1/4)*((-32 + (log(Pi) - PolyGamma(1/4))^2 + PolyGamma(1, 1/4))*zeta(1/2) + 4*((-log(Pi) + PolyGamma(1/4))*zeta'(1/2) + zeta''(1/2))).

A377542 Decimal expansion of Gamma(1/4)^4/(16*Pi^2).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 31 2024

			1.09421980761323831941838497035223227162968636141...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 134.
Index to sequences related to the Gamma function.
Index entries for transcendental numbers.

Cf. A002388, A068465, A068466, A068467, A254794.

RealDigits[Gamma[1/4]^4/(16Pi^2),10,100][[1]]

Equals Product_{n>=1} (1 - 1/(2*n + 1)^2)^(-1)^n (see Finch).
Equals Product_{n>=1} (4*n - 1)^2*((4*n + 1)^2 - 1)/(((4*n - 1)^2 - 1)*(4*n + 1)^2) (see Shamos).
Equals 2*(Gamma(5/4)/Gamma(3/4))^2.
Equals A254794/2. - Hugo Pfoertner, Oct 31 2024

A382242 Decimal expansion of Gamma(1/4)^2/(8sqrt(2Pi)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 19 2025

			0.6555143885730299526162098974727798534...

Leyda Almodovar, Victor H. Moll, Hadrian Quan, Fernando Roman, Eric Rowland, and Michole Washington, Infinite products arising in paperfolding, JIS 19 (2016), Article 16.5.1, eq. (72).

Cf. A034937, A068466, A231863, A005843, A005408.

Digits := 120 ; GAMMA(1/4)^2/8/sqrt(2*Pi) ; evalf(%) ;

RealDigits[Gamma[1/4]^2/(8*Sqrt[2*Pi]), 10, 120][[1]] (* Amiram Eldar, Mar 20 2025 *)

Equals A068466^2 *A231863 /8.

Equals Product_{n>=1} (A005843(n)/A005408(n))^A034947(n).

A384563 Decimal expansion of Beta(1/4,1/4).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jun 03 2025

			7.416298709205487673735401388781040184870395294...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 406.

Similar constants Beta(1/k,1/k): A000796 (k=2), A197374 (k=3).

Cf. A002161, A068466, A175576, A377731.

Mathematica
```
RealDigits[Beta[1/4,1/4],10,100][[1]]
```

Equals Gamma(1/4)^2/sqrt(Pi) = A068466^2/A002161.

Equals 2*A175576 = 3*A377731. - Hugo Pfoertner, Jun 03 2025

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A249189 Decimal expansion of Hayman's constant in Landau's Theorem.

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A256591 Decimal expansion of Xi''(1/2) = 0.02297..., the second derivative of the Riemann Xi function at 1/2.

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A377542 Decimal expansion of Gamma(1/4)^4/(16*Pi^2).

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A382242 Decimal expansion of Gamma(1/4)^2/(8*sqrt(2*Pi)).

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A384563 Decimal expansion of Beta(1/4,1/4).

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A382242 Decimal expansion of Gamma(1/4)^2/(8sqrt(2Pi)).