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arXiv · 1603.01306

Zeros of certain combinations of Eisenstein series

Sarah Reitzes·Polina Vulakh·Matthew P. Young
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Abstract

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_k(z) E_{\ell}(z) - E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. We moreover find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $x = \pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

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Sarah Reitzes, Polina Vulakh, Matthew P. Young. 2016-03-03. Zeros of certain combinations of Eisenstein series. https://doi.org/10.1112/s0025579317000110

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