1 results
EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 2 / March 2020
- Published online by Cambridge University Press:
- 02 April 2018, pp. 581-596
- Print publication:
- March 2020
-
- Article
- Export citation
-
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.
