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- W2891423093 abstract "In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $theta (z|tau ) =sum _{nu =-infty }^{infty } e^{pi inu ^2tau + 2pi inu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case [ theta _3(tau ):=theta (0|tau )= 1+2sum _{nu =1}^{infty } e^{pi inu ^2 tau } ] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $theta (z|tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $theta _3(tau )$. For example, the three values $theta _3(tau )$, $theta _3(ntau )$, and $Dtheta _3(tau )$ are algebraically independent over $mathbb{Q} $ for any τ such that $q=e^{pi itau }$ is an algebraic number, where $ngeq 2$ is an integer and $D:=(pi i)^{-1}{d}/{dtau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $theta _3(tau )$ and $theta _3(2^mtau )$ for $mgeq 1$. As an application of our main theorem, the algebraic dependence over $mathbb{Q} $ of the three values $theta _3(ell tau )$, $theta _3(mtau )$, and $theta _3(ntau )$ for integers $ell ,m,ngeq 1$ is also presented." @default.
- W2891423093 created "2018-09-27" @default.
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- W2891423093 date "2018-09-05" @default.
- W2891423093 modified "2023-09-27" @default.
- W2891423093 title "Algebraic results for certain values of the Jacobi theta-constant $theta_3(tau)$" @default.
- W2891423093 doi "https://doi.org/10.7146/math.scand.a-105465" @default.
- W2891423093 hasPublicationYear "2018" @default.
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