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taylor
taylor
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For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define $$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$

It is known that $\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|\gamma \|^4}$ is finite (can be proved by estimating the number of lattice points in a ball of $\text{SL}(2,\mathbb Z)$ and use Euler-Maclaurin/Abel sum method).

I wonder how $f(g)$ depends on $g\in \text{SL}(2,\mathbb R)$. Is it true/how to show that What $f(g)$ is not a constant function in $g$? If it is not a constant, then what are $\sup_{g\in \text{SL}(2,\mathbb R)}f(g)$, $\inf_{g\in \text{SL}(2,\mathbb R)}f(g)$?

For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define $$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$

It is known that $\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|\gamma \|^4}$ is finite (can be proved by estimating the number of lattice points in $\text{SL}(2,\mathbb Z)$ and use Euler-Maclaurin/Abel sum method).

I wonder how $f(g)$ depends on $g\in \text{SL}(2,\mathbb R)$. Is it true/how to show that What $f(g)$ is not a constant function in $g$? If it is not a constant, then what are $\sup_{g\in \text{SL}(2,\mathbb R)}f(g)$, $\inf_{g\in \text{SL}(2,\mathbb R)}f(g)$?

For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define $$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$

It is known that $\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|\gamma \|^4}$ is finite (can be proved by estimating the number of lattice points in a ball of $\text{SL}(2,\mathbb Z)$ and use Euler-Maclaurin/Abel sum method).

I wonder how $f(g)$ depends on $g\in \text{SL}(2,\mathbb R)$. Is it true/how to show that $f(g)$ is not a constant function in $g$? If it is not a constant, then what are $\sup_{g\in \text{SL}(2,\mathbb R)}f(g)$, $\inf_{g\in \text{SL}(2,\mathbb R)}f(g)$?

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taylor
taylor
  • 457
  • 2
  • 6

The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$

For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define $$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$

It is known that $\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|\gamma \|^4}$ is finite (can be proved by estimating the number of lattice points in $\text{SL}(2,\mathbb Z)$ and use Euler-Maclaurin/Abel sum method).

I wonder how $f(g)$ depends on $g\in \text{SL}(2,\mathbb R)$. Is it true/how to show that What $f(g)$ is not a constant function in $g$? If it is not a constant, then what are $\sup_{g\in \text{SL}(2,\mathbb R)}f(g)$, $\inf_{g\in \text{SL}(2,\mathbb R)}f(g)$?