Epstein zeta-functions, subconvexity, and the purity conjecture

Valentin Blomer

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arXiv · 1602.02326

Epstein zeta-functions, subconvexity, and the purity conjecture

Valentin Blomer

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Abstract

Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.

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Valentin Blomer. 2016-02-07. Epstein zeta-functions, subconvexity, and the purity conjecture. https://arxiv.org/abs/1602.02326

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