# A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights

Denote $\operatorname{dim}(M_k(\Gamma_0(N)))$ by $m(k,N)$ and $\operatorname{dim}(S_k(\Gamma_0(N)))$ by $s(k,N)$. Let $N$ any positive multiple of $4$ and $j \ge 1$. $$a(N) := \frac1j \left(m \left(\frac{4j+1}{2},N\right) +s\left(\frac{4j+3}{2}, N\right) \right)$$ For $N = 4,8,12, \ldots$ this gives the first table in H. Cohen and J. Oesterlé (see below) and was registered by Steven Finch in OEIS A159634.

Recently E. P. Herrero conjectured the identity $$a(N) = \frac{1}{3}\Psi\left(\frac{N}{2}\right). \qquad ({\Psi \text{ denotes the Dedekind Psi function.)}}$$

The question is: Are -- apart from numerical evidence -- reasons known which support this conjecture?

Meanwhile W. Meeussen made a related conjecture with regard to $$b(N) = m\left(\frac{k}{2}, N\right) - s\left(\frac{k}{2}, N\right),$$

where $N$ is any positive multiple of 4 and $k \ge 5$ is odd.

Let [] denote the Iverson brackets, $\varphi$ Euler's totient function and $$\chi(n) = \sum_{d|n} \varphi \left(\operatorname{gcd} \left(d, \frac {n}{d} \right) \right).$$

Then Meeussen conjectured $$b(N) = 2 \chi\left(\frac{N}{2}\right) - \frac 12 \chi\left(\frac{N}{2}\right)\, \left[ \frac{N}{2}+2 \bmod 4 = 0\right] .$$

This is OEIS sequence A159633.

Ref.:

H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.

S. R. Finch, Primitive Cusp Forms, April 27, 2009.

http://oeis.org/A159634

http://oeis.org/A159633

This is easy to determine by the known closed formulæ for $\dim \mathcal{S}_k\left(\Gamma_0(N)\right)$ and $\dim \mathcal{M}_k\left(\Gamma_0(N)\right)$ of Cohen and Oesterlé, namely for $k \in 1/2 + \mathbb{Z}$ with $k > 3/2$ and $N \in \mathbb{N}$, \begin{align} \dim \mathcal{S}_k\left(\Gamma_0(4N)\right) & = \frac{k - 1}{12} 4N \prod_{p|4N}\left(1 + \frac{1}{p}\right) - \frac{\zeta(k,4N)}{2} \prod_{\substack{p|4N \\ p > 2}} \lambda(r_p,p),\\ \dim \mathcal{M}_k\left(\Gamma_0(4N)\right) & = \frac{k - 1}{12} 4N \prod_{p|4N}\left(1 + \frac{1}{p}\right) + \frac{\zeta(2 - k,4N)}{2} \prod_{\substack{p|4N \\ p > 2}} \lambda(r_p,p), \end{align} where if $r_p$ is the power of $p$ dividing $4N$, then \begin{align} \lambda(r_p,p) &= \begin{cases} p^{r_p/2}\left(1 + \frac{1}{p}\right) & \text{if $r_p$ is even,}\\ 2 p^{(r_p - 1)/2} & \text{if $r_p$ is odd,} \end{cases}\\ \zeta(k,4N) & = \begin{cases} 2^{(r_2 + 1)/2} & \text{if $r_2 \geq 5$ is odd,}\\ 2^{r_2/2 + 1} & \text{if $r_2 \geq 4$ is even,}\\ 3 & \text{if $r_2 = 3$,}\\ 2 & \text{if $r_2 = 2$ and $r_p$ is odd for some prime $p|4N$ with $p \equiv 3\pmod{4}$,}\\ 3/2 & \text{if $r_2 = 2$, $r_p$ is even for all $p \equiv 3\pmod{4}$, and $k \in 1/2 + 2\mathbb{Z}$,}\\ 5/2 & \text{if $r_2 = 2$, $r_p$ is even for all $p \equiv 3\pmod{4}$, and $k \in 3/2 + 2\mathbb{Z}$.} \end{cases} \end{align} This is Théorème 2 of the paper of Cohen and Oesterlé, which reappears as Theorem 1.56 of The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and Q-Series by Ken Ono (we take $\chi = \chi_0$ in this case, so that $s_p = 0$ for all $p$). As an aside, it must be pointed out that the proof of this theorem has actually never appeared in print!
From here it is clear that for $k \in \mathbb{N}$, $$\dim \mathcal{S}_{2k+1/2}\left(\Gamma_0(4N)\right) + \dim \mathcal{M}_{2k + 3/2}\left(\Gamma_0(4N)\right) = \frac{4kN}{3} \prod_{p|4N}\left(1 + \frac{1}{p}\right)$$ and similarly $$\dim \mathcal{S}_{2k+3/2}\left(\Gamma_0(4N)\right) + \dim \mathcal{M}_{2k + 1/2}\left(\Gamma_0(4N)\right) = \frac{4kN}{3} \prod_{p|4N}\left(1 + \frac{1}{p}\right),$$ as required.
For your second question, it is a simple exercise to show that $$\sum_{d|N}\varphi\left(\left(d,\frac{N}{d}\right)\right) = \prod_{p^r||N}\sum_{m = 0}^r \varphi\left(p^{\min\{m,r-m\}}\right) = \prod_{p|N}\lambda(r_p,p),$$ and so again the conjecture follows easily by the fact that $$\dim \mathcal{M}_k\left(\Gamma_0(4N)\right) - \dim \mathcal{S}_k\left(\Gamma_0(4N)\right) = \frac{\zeta(2 - k,4N) - \zeta(k,4N)}{2} \prod_{\substack{p|4N \\ p > 2}} \lambda(r_p,p)$$ and analysing each case of $r_2$ as necessary.