Linear independence of special values of Jacobi-theta constants

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Let m≥2 be any integer and let β>11$$\\end{document}]]> be a real algebraic integer such that all its other Galois conjugates have absolute value less than or equal to 1. Let a1,a2,…,am be distinct positive integers. In this article we prove that the following infinite sums 1,∑n=1∞1βa1n2,∑n=1∞1βa2n2,…,∑n=1∞1βamn2are Q(β)-linearly independent. As a consequence, we prove the linear independence of special values of Jacobi-theta constants.