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%I #14 Sep 13 2023 22:59:58
%S 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,
%T 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,
%U 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
%N Decimal expansion of the least real solution x > 0 of Gamma(1/4 + x/2)/(Pi^x*Gamma(1/4 - x/2)) = 1.
%F Let x be this constant:
%F Gamma(1/4 - x/2)/(Pi^x*Gamma(1/4 + x/2)) = 1.
%F zeta((1/2) + x) = zeta((1/2) - x), where zeta is the Riemann zeta function.
%F (2*Pi)^(-1/2 - x)*(2*x - 1)*cos(Pi/4 + (Pi*x)/2)*Gamma(x - 1/2) = 1.
%F 2^(1/2 - x)*Pi^(-1/2 - x)*sin(Pi/2 - (Pi*x)/2)*Gamma(1/2 + x) = 1.
%F 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).
%e 18.56775084706966207279145836562344730...
%t FindRoot[-1 + Gamma[1/4 - x/2]/(Pi^(-x) Gamma[1/4 + x/2]) == 0, {x, 18.5569, 18.5739}, WorkingPrecision -> 100]
%Y Cf. A059750, A114720, A257870.
%K nonn,cons
%O 2,2
%A _Thomas Scheuerle_, Aug 31 2023
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