A365281 Decimal expansion of the least real solution x > 0 of Gamma(1/4 + x/2)/(Pi^x*Gamma(1/4 - x/2)) = 1.
1, 8, 5, 6, 7, 7, 5, 0, 8, 4, 7, 0, 6, 9, 6, 6, 2, 0, 7, 2, 7, 9, 1, 4, 5, 8, 3, 6, 5, 6, 2, 3, 4, 4, 7, 3, 0, 3, 3, 8, 4, 2, 0, 1, 7, 3, 2, 6, 5, 8, 5, 3, 9, 8, 3, 3, 4, 7, 4, 6, 1, 7, 7, 8, 5, 4, 3, 6, 0, 0, 6, 4, 1, 7, 3, 5, 7, 9, 7, 2, 7, 1, 1, 7, 3, 1, 5, 9, 1, 4, 0, 1, 2, 1, 0, 6, 5
Offset: 2
Examples
18.56775084706966207279145836562344730...
Programs
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Mathematica
FindRoot[-1 + Gamma[1/4 - x/2]/(Pi^(-x) Gamma[1/4 + x/2]) == 0, {x, 18.5569, 18.5739}, WorkingPrecision -> 100]
Formula
Let x be this constant:
Gamma(1/4 - x/2)/(Pi^x*Gamma(1/4 + x/2)) = 1.
zeta((1/2) + x) = zeta((1/2) - x), where zeta is the Riemann zeta function.
(2*Pi)^(-1/2 - x)*(2*x - 1)*cos(Pi/4 + (Pi*x)/2)*Gamma(x - 1/2) = 1.
2^(1/2 - x)*Pi^(-1/2 - x)*sin(Pi/2 - (Pi*x)/2)*Gamma(1/2 + x) = 1.
Z(i*x) = -zeta((1/2) + x) = -zeta((1/2) - x), where Z is the Riemann-Siegel Z function and i is the imaginary unit. From this follows that theta(i*-x) = theta(i*x) is an odd multiple of Pi where theta is the Riemann-Siegel theta function. This can also be seen if we consider Hardy's definition of the Z function: Z(s) = Pi^(-i*s/2)*zeta((1/2) + i*s)*Gamma((1/4)+(i*s/2))^(1/2)/Gamma((1/4) - (i*s/2))^(1/2).