%I #30 Aug 30 2023 07:27:56

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

%T 5,4,5,7,0,5,0,7,7,6,0,9,7,8,7,0,4,7,2,1,6,0,9,8,0,8,9,8,9,0,7,7,7,4,

%U 6,8,4,0,5,6,7,8,7,4,9,2,5,7,0,2,8,9,6,3,9,2,7,9,3,3,6,0,8,8,0,2

%N Decimal expansion of q = 0.193072033..., which is the value of q which gives the maximum of the Dedekind eta function eta(q) := q^(1/12) * Product_{n>=1} (1 - q^(2n)) for q between 0 and 1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.

%F Equals sqrt(A211342). - _Vaclav Kotesovec_, Jul 02 2017

%e 0.19307203395741097892294168542126225457050776097870...

%t RealDigits[FindRoot[D[q^(1/12)*Product[(1-q^(2 n)), {n, 100}], q] == 0, {q, 0.2}, WorkingPrecision -> 200][[1,2]]][[1]]

%t q /. Last @ FindMaximum[ DedekindEta[ -I*Log[q]/Pi], {q, 1/5}, WorkingPrecision -> 200] // RealDigits[#][[1]][[1 ;; 100]]& (* _Jean-François Alcover_, Feb 05 2013 *)

%t q0 = q /. FindMaximum[q^(1/12)*QPochhammer[q^2], {q, 1/5}, WorkingPrecision -> 200][[2]]; RealDigits[q0, 10, 100][[1]] (* _Jean-François Alcover_, Nov 25 2015 *)

%Y Cf. A211342.

%K cons,nonn,easy,nice

%O 0,2

%A Peter L. Walker (peterw(AT)aus.ac.ae), Nov 24 2000

%E More terms from _Vladeta Jovovic_, Jun 19 2004