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9 votes
1 answer
204 views

Which unimodular lattices $L\subset \mathbb R^2$ minimize $f_t(L):=\sum_{ v\in L} e^{-t \|v\|_2}$? (for parameters $t>0$)

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Consider the lattices in $\SL(2,\mathbb R)(\mathbb Z^2)$ up to rotation. The space of such lattices can be identified with the modular surface $\...
2 votes
1 answer
229 views

Upper bound for a higher dimensional Ramanujan sum

Fix an integer vector $\mathbf m\in \mathbb Z^k$. Let $q$ be a positive integer. Is there a "good" upper bound in terms of $q,\bf m$ for the exponential sum: \[\sum_{\mathbf n} e\left(\frac{\langle m,...
16 votes
1 answer
1k views

On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...