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Tagged with exponential-sums lattices
3 questions
16
votes
1
answer
1k
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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 ...
- 5,169
9
votes
1
answer
204
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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 $\...
- 241
2
votes
1
answer
229
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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,...
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