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Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $\Re(s)>1+\varepsilon$? or even for $\Re(s)>1+\delta$ for a quite small $\delta$?

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $\Re(s)>1$? or even for $\Re(s)>1+\delta$ for a quite small $\delta$?

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $s>1+\varepsilon$? or even $s>1+\delta$ for a quite small $\delta$?

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $\Re(s)>1+\varepsilon$? or even for $\Re(s)>1+\delta$ for a quite small $\delta$?

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $s>1$? This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if

$$\sum_{n>0} \frac{|a_n|}{n^s}$$

and

$$\sum_{n>0} \frac{\left( \sum_{k \mid n} |a_k|\right)^2}{n^s}$$

converge for $s>1+\varepsilon$? or even $s>1+\delta$ for a quite small $\delta$? 

This essentially amounts to say that the coefficients are constant on average. It seems to be known for Gelbart-Jacquet lifts by Rankin-Selberg properties, but is it known or at least expected for non-Gelbart-Jacquet lifts?