Return to Question

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$. Although it remains unproved in general, there are lots of partial results of the form $A(n) \ll n^\theta$ towards the Ramanujan-Petersson conjecture. The best general result I'm aware of is due to Kim and Sarnak, showing that $\theta < \frac{7}{64} + \epsilon$.

I'm currently investigating some bounds involving Maass forms $\mu_j$ of level $N$ of the the form $$ \mu_j(z) = \sum_{n \neq 0} \rho_j(n)y^{\frac{1}{2}}K_{it_j}(2\pi \lvert n \rvert y)e^{2\pi i n x} $$ and I wonder: what is the best known Ramanujan-Petersson-style bound for the coefficients $\rho_j(n)$ for such a Maass form, and how does it depend on the level? Is it known that $\rho_j(n) \ll n^{\frac{7}{64} + \epsilon}$, with no dependence on the level $N$?

It seems more likely to me that we would currently have a bound of the form $\rho_j(n) \ll N^? n^{\frac{7}{64} + \epsilon}$, but I cannot seem to find what it may be.

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$.

For more general cusp forms, there are lots of partial results of the form $A(n) \ll n^\theta$ towards the Ramanujan-Petersson conjecture. The best general result I'm aware of is due to Kim and Sarnak, showing that $\theta < \frac{7}{64} + \epsilon$.

I'm currently investigating some bounds involving Maass forms $\mu_j$ of level $N$ of the the form $$ \mu_j(z) = \sum_{n \neq 0} \rho_j(n)y^{\frac{1}{2}}K_{it_j}(2\pi \lvert n \rvert y)e^{2\pi i n x} $$ and I wonder: what is the best known Ramanujan-Petersson-style bound for the coefficients $\rho_j(n)$ for such a Maass form, and how does it depend on the level? Is it known that $\rho_j(n) \ll n^{\frac{7}{64} + \epsilon}$, with no dependence on the level $N$?

It seems more likely to me that we would currently have a bound of the form $\rho_j(n) \ll N^? n^{\frac{7}{64} + \epsilon}$, but I cannot seem to find what it may be.

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ automorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$. Although it remains unproved in general, there are lots of partial results of the form $A(n) \ll n^\theta$ towards the Ramanujan-Petersson conjecture. The best general result I'm aware of is due to Kim and Sarnak, showing that $\theta < \frac{7}{64} + \epsilon$.

I'm currently investigating some bounds involving Maass forms of level $N$, and I wonder: what is the best known Ramanujan-Petersson-style bound for the coefficients $\rho_j(n)$ for such a Maass form, and how does it depend on the level? Is it known that $\rho_j(n) \ll n^{\frac{7}{64} + \epsilon}$, with no dependence on the level $N$?

It seems more likely to me that we would currently have a bound of the form $\rho_j(n) \ll N^? n^{\frac{7}{64} + \epsilon}$, but I cannot seem to find what it may be.

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture says roughly that $A(n) \ll n^{\epsilon}$ for any $\epsilon > 0$. Although it remains unproved in general, there are lots of partial results of the form $A(n) \ll n^\theta$ towards the Ramanujan-Petersson conjecture. The best general result I'm aware of is due to Kim and Sarnak, showing that $\theta < \frac{7}{64} + \epsilon$.

I'm currently investigating some bounds involving Maass forms $\mu_j$ of level $N$ of the the form $$ \mu_j(z) = \sum_{n \neq 0} \rho_j(n)y^{\frac{1}{2}}K_{it_j}(2\pi \lvert n \rvert y)e^{2\pi i n x} $$ and I wonder: what is the best known Ramanujan-Petersson-style bound for the coefficients $\rho_j(n)$ for such a Maass form, and how does it depend on the level? Is it known that $\rho_j(n) \ll n^{\frac{7}{64} + \epsilon}$, with no dependence on the level $N$?

It seems more likely to me that we would currently have a bound of the form $\rho_j(n) \ll N^? n^{\frac{7}{64} + \epsilon}$, but I cannot seem to find what it may be.