Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

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EPSTEIN ZETA 函数、次凸性和纯度猜想

对于一般 Epstein zeta 函数，证明了临界线上的次凸边界$k$-ary 二次形式。这与酉爱森斯坦级数的超范数界有关$\文本{GL}(k)$与类型的最大抛物线相关联$(k-1,1)$，并且精确的上范数指数被确定为$(k-2)/8$为了$k\geqslant 4$. 特别是，如果$k$很奇怪，这个指数不在$\frac{1}{4}\mathbb{Z}$，这与 Sarnak 的纯度猜想有关，并表明它通常不能直接推广到 Eisenstein 级数。