8
votes

Accepted

### Abscissa of convergence of the $\tau$ Dirichlet series

Consider the sum
$$S(x):=\sum_{n\leq x}\frac{\tau(n)}{n^{11/2}}.$$
By Theorem 1.3 in Montgomery-Vaughan: Multiplicative number theory I, the abscissa of convergence equals
$$\sigma_c=\frac{11}{2}+\...

5
votes

Accepted

### Uniqueness of the $J$ invariant

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...

3
votes

Accepted

### Automorphic representation of GL(1)

Using your definition of automorphic representation, the answer to Question 1 is yes, tautologically. If $V_\omega$ is the space of automorphic forms "with character $\omega$" then by ...

2
votes

### Properties of the ring of all holomorphic modular forms

It may be easier to consider this question for genus zero groups, where we can construct modular forms vanishing at given points in terms of Hauptmoduln. I don't know how to find the divisors in ...

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