Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \varepsilon}$$ for any $\varepsilon > 0$. Unlike the integer-weight case, this is unresolved as far as I am aware.

I am interested to know what is currently the best bound on the growth of $a(n)$. The bound I am most familiar with is due to Iwaniec (Theorem 1 of this article) giving $a(n) \ll n^{k/2 - 2/7 +\varepsilon}$. But it seems that Conrey and Iwaniec proved a better bound of the form $a(n) \ll n^{k/2 - 1/3 + \varepsilon}$ here as Corollary 1.3 (it is unclear to me whether this holds generally)

Both of the articles I linked to are fairly old. I would be grateful if an expert could inform me about what has happened since then.


The Conrey-Iwaniec bound is still the best known one. Originally it was restricted to $N=1$, but Petrow and Young extended the result to all square-free $N$'s (this is highly nontrivial). Also, in these results (including Iwaniec's) $n$ itself is assumed to be square-free. Obtaining a bound for all $n$'s is somewhat tricky: see Corollary 2 and the references in Blomer and Harcos. Note that this paper features a weaker bound but in a more general situation (for example, there is no restriction on $N$).

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