My new article proves by using a Dirichlet theorem that Theta-3 Functions give only irrational outputs when the inputs are rationals. The proof is all calculus and Analysis without any Algebra. However, in my previous article I proved that if all the outputs of Theta-3 Functions are irrational then this will contradict the continuity of these functions.

Can you exploit these results otherwise?

I wait for your collaboration. Here is the link:

https://www.researchgate.net/publication/377658268_The_proof_that_the_rational_numbers_as_input_values_in_Theta-3_Functions_give_only_irrational_numbers_as_output_in_these_functions