|
%I #16 Sep 25 2022 04:57:28
%S 1,8,8,7,7,9,0,9,2,6,7,0,8,1,1,8,9,2,7,1,9,6,3,2,1,5,4,2,0,3,5,1,1,6,
%T 6,6,8,2,2,3,4,7,0,1,2,6,0,2,8,0,1,6,4,7,9,8,0,9,1,5,4,3,8,0,9,5,5,4,
%U 6,7,3,4,7,1,4,4,1,5,3,3,8,1,8,8,8,1,0,8,4,2,6,6,7
%N Decimal expansion of the unique real number in [1, 2] satisfying ((2^x)/(2^x-1))*((3^x+1)/(3^x-1)) = zeta(x).
%C This constant is named the Defantstant by Zubrilina (see link).
%C See the Defant link for more explanations on this constant.
%C The set of numbers {sigma_{-r}(n) | n>=1}, where sigma_{-r}(n) = Sum_{d|n} d^(-r), is dense in [1, zeta(r)) if and only if r <= this constant (Defant, 2015). - _Amiram Eldar_, Sep 25 2022
%H Colin Defant, <a href="https://www.nntdm.net/papers/nntdm-21/NNTDM-21-3-80-87.pdf">On the Density of Ranges of Generalized Divisor Functions</a>, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 3 (2015), pp. 80-87; <a href="https://arxiv.org/abs/1506.05432">arXiv preprint</a>, arXiv:1506.05432 [math.NT], 2015.
%H Nina Zubrilina, <a href="https://arxiv.org/abs/1711.02871">On the Number of Connected Components of Ranges of Divisor Functions</a>, arXiv:1711.02871 [math.NT], 2017.
%e 1.8877909267081189271963215420351166682234701260280164798091543809554673...
%t RealDigits[x /. FindRoot[2^x*(3^x + 1)/((2^x - 1)*(3^x - 1)) == Zeta[x], {x, 3/2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Sep 25 2022 *)
%o (PARI) solve(x=1.5, 2, 2^x*(3^x+1)/((2^x-1)*(3^x-1)) - zeta(x))
%K nonn,cons
%O 1,2
%A _Michel Marcus_, Nov 09 2017
|
