Algebraic results for certain values of the Jacobi theta-constant θ_3(τ) - Elsner, Carsten; Tachiya, Yohei (Mathematica Scandinavica, 2018)
In its most elaborate form, the Jacobi theta function is defined for two complex variables z and τ by θ (z|τ ) =_ u =-∞ ^∞ e^π i u ^2τ + 2π i u z, which converges for all complex number z, and τ in the upper half-plane. The special case \[ θ _3(τ ):=θ (0|τ )= 1+2_ u =1^∞ e^π i u ^2 τ \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function θ (z|τ ). In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant θ _3(τ ). For example, the three values θ _3(τ ), θ _3(nτ ), and Dθ _3(τ ) are algebraically independent over Q for any τ such that q=e^π iτ is an algebraic number, where n≥ 2 is an integer and D:=(π i)^-1d/dτ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values θ _3(τ ) and θ _3(2^mτ ) for m≥ 1. As an application of our main theorem, the algebraic dependence over Q of the three values θ _3(τ ), θ _3(mτ ), and θ _3(nτ ) for integers ,m,n≥ 1 is also presented.