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}+\...
- 94.9k
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 ...
- 35.7k
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 ...
- 546
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 ...
- 21
Only top scored, non community-wiki answers of a minimum length are eligible