Is 0.1010010001000010000010000001…0.1010010001000010000010000001 \ldots transcendental?

Does anyone know if this number is algebraic or transcendental, and why?



The number 0.1010010001000010000010000001 is transcendental.

Consider following three Jacobi theta series defined by
and for any mZ+, k{2,3,4}, use
Dmθk(q) as a shorthand for

Based on Corollary 52 of a survey article Elliptic functions and Transcendence by M. Waldschmidt in 2006,

Let i,j and k{2,3,4} with ij. Let qC
satisfy 0<|q|<1. Then each of the two fields
Q(q,θi(q),θj(q),Dθk(q)) and Q(q,θk(q),Dθk(q),D2θk(q))
has transcendence degree 3 over Q

We know for any non-zero algebraic q with |q|<1, the three θk(q), in particular θ2(q) is transcendental. Since


and using the fact 110 and 8102 are both algebraic, we find the number at hand is transcendental.

Source : Link , Question Author : Raffaele , Answer Author : Grigorios Kostakos

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