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?


*Addition:*


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>



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


