cp's OEIS Frontend

A057823 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.

%I A057823 #31 Feb 16 2025 08:32:43
%S A057823 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 A057823 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 A057823 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 A057823 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 A057823 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>.
%F A057823 Equals sqrt(A211342). - _Vaclav Kotesovec_, Jul 02 2017
%e A057823 0.19307203395741097892294168542126225457050776097870...
%t A057823 RealDigits[FindRoot[D[q^(1/12)*Product[(1-q^(2 n)), {n, 100}], q] == 0, {q, 0.2}, WorkingPrecision -> 200][[1,2]]][[1]]
%t A057823 q /. Last @ FindMaximum[ DedekindEta[ -I*Log[q]/Pi], {q, 1/5}, WorkingPrecision -> 200] // RealDigits[#][[1]][[1 ;; 100]]&  (* _Jean-François Alcover_, Feb 05 2013 *)
%t A057823 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 A057823 Cf. A211342.
%K A057823 cons,nonn,easy,nice
%O A057823 0,2
%A A057823 Peter L. Walker (peterw(AT)aus.ac.ae), Nov 24 2000
%E A057823 More terms from _Vladeta Jovovic_, Jun 19 2004