A126689 -id:A126689 - cp's OEIS frontend

A143301 Decimal expansion of the Hall-Montgomery constant.

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

Offset: 0

Eric W. Weisstein, Aug 05 2008

Named after Richard Roxby Hall and Hugh Lowell Montgomery. - Amiram Eldar, Aug 25 2020

			0.17150049314153606586...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 205-207.

Thomas Bloom, Problem 121, Erdős Problems.
Andrew Granville and Kannan Soundararajan, The spectrum of multiplicative functions, Annals of Mathematics, Vol. 153, No. 2 (2001), pp. 407-470.
R. R. Hall, Proof of a conjecture of Heath-Brown concerning quadratic residues, Proceedings of the Edinburgh Mathematical Society, Vol. 39, No. 3 (1996), pp. 581-588.
Terence Tao, On product representations of squares, arXiv:2405.11610 [math.NT], May 2024. See page 2, (1.2).
Terence Tao, Erdős problem database, see no. 121.
Eric Weisstein's World of Mathematics, Hall-Montgomery Constant.

Cf. A126689, A246849.

RealDigits[1+Pi^2/6+2PolyLog[2,-Sqrt[E]],10,120][[1]] (* Harvey P. Dale, Jul 18 2013 *)

from mpmath import mp, pi, e, polylog, sqrt
mp.dps=106
print([int(z) for z in list(str(1 + pi**2/6 + 2* polylog(2, - sqrt(e)))[2:-1])]) # Indranil Ghosh, Jul 04 2017

Equals 1 + Pi^2/6 + 2*PolyLog(2, -sqrt(e)).
Equals (1 - A126689)/2. - Amiram Eldar, Aug 25 2020
Equals 1 - A246849. - Hugo Pfoertner, May 25 2024

A165361 Decimal expansion of log 2 times the negative of Granville-Soundararajan constant.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Sep 16 2009

Given as formula and decimal expansion in p. 4 of Tao, citing Granville.

			-0.455397....

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
A. Granville and K. Soundararajan, Negative values of truncations to L(1,chi), Analytic number theory: a tribute to Gauss and Dirichlet, Clay Math. Proc. Volume 7, 2007.
Terence Tao, A remark on partial sums involving the Mobius function, Bull. Aust. Math. Soc. 81 (2010), no. 2, 343-349.

RealDigits[ -Log[2]*(4*PolyLog[2, -Sqrt[E]] + Pi^2/3 + 1) , 10, 105] // First (* Jean-François Alcover, Feb 15 2013, after R. J. Mathar *)

-log(2)*(Pi^2/3+4*polylog(2, -exp(1/2))+1) \\ Charles R Greathouse IV, Jul 18 2014

from mpmath import mp, pi, sqrt, polylog, log
mp.dps=106
C = -log(2)*(4*polylog(2, -sqrt(e)) + pi**2/3 + 1)
print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 03 2017

Equals log(2)*A126689 = A002162*A126689.

More digits from R. J. Mathar, Sep 21 2009

cp's OEIS Frontend

A143301 Decimal expansion of the Hall-Montgomery constant.

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