The Wikipedia article Dedekind psi function indicates the Dedekind psi function defined in formula (1) below was introduced by Richard Dedekind in connection with modular functions.
(1) $\quad\psi(n)=n \prod\limits_{p|n}\left(1+\frac{1}{p}\right)$
I've searched the Internet a bit and the only connection I've found between $\psi(n)$ and modular functions is documented in the MathOverflow question A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights but the MathOverflow question implies this is a recent conjecture.
Question (1): What connection did Richard Dedekind establish between modular functions and the Dedekind psi function ψ(n)? Was it the same as the conjecture in the MathOverflow question linked above?
I've found the Dedekind psi summatory function defined in formula (2) below to be interesting as I've derived a number of conjectured relationships with other number theoretic functions such as those illustrated in formulas (3) to (9) below. The Dedekind psi function is related to the Riemann zeta function $\zeta(s)$ as illustrated in formula (2) below and the Riemann hypothesis. I've read the Riemann hypothesis is satisfied if-and-only-if $f(n)=\psi(n)/n-e^\gamma\log\log n < 0$ for all integers $n > n_0 = 30$ (see Riemann hypothesis from the Dedekind psi function). The derivation of this result is based on the relationship $\psi(n)=\sigma_1(n)$ when $n$ is a square-free number which also perhaps provides insight into the conjectured relationships illustrated in formulas (8) and (9) below.
(2) $\quad\Psi(x)=\sum\limits_{n\le x}\psi(n)\qquad\qquad\qquad\sum\limits_{n=1}^\infty\frac{\psi(n)}{n^s}=\frac{\zeta(s-1)\,\zeta (s)}{\zeta (2\,s)},\quad\Re(s)>2$
(3) $\quad U(x)=\sum\limits_{n\le x}\delta_{1,n}\overset{?}{=}\sum\limits_{n\le x}\frac{rad(n)\,\lambda(n)\,\psi(n)}{n}\Psi\left(\frac{x}{n}\right)$
(4) $\quad Q(x)=\sum\limits_{n\le x}\left|\mu(n)\right|\overset{?}{=}\sum\limits_{n\le x}\mu(n)\,n\,\Psi\left(\frac{x}{n}\right)$
(5) $\quad\Psi(x)\overset{?}{=}\sum\limits_{n\le x} n\,Q\left(\frac{x}{n}\right)$
(6) $\quad\Phi(x)=\sum\limits_{n\le x}\phi(n)\overset{?}{=}\sum\limits_{n\le x}\lambda(n)\,2^{\,\nu(n)}\,\Psi\left(\frac{x}{n}\right)$
(7) $\quad\Psi(x)\overset{?}{=}\sum\limits_{n\le x} 2^{\nu(n)}\,\Phi\left(\frac{x}{n}\right)$
(8) $\quad \Sigma_1(x)=\sum\limits_{n\le x}\sigma_1(n)\overset{?}{=}\sum\limits_{n\le x}\left( \begin{array}{cc} \{ & \begin{array}{cc} 1 & \sqrt{n}\in \mathbb{Z} \\ 0 & \text{True} \\ \end{array} \\ \end{array} \right)\Psi\left(\frac{x}{n}\right)$
(9) $\quad\Psi(x)\overset{?}{=}\sum\limits_{n\le x}\left( \begin{array}{cc} \{ & \begin{array}{cc} \mu \left(\sqrt{n}\right) & \sqrt{n}\in \mathbb{Z} \\ 0 & \text{True} \\ \end{array} \\ \end{array} \right) \Sigma_1\left(\frac{x}{n}\right)$
In the formulas above:
- $\delta_{1,n}$ is the Kronecker delta function
- $rad(n)$ is the radical of $n$ also referred to as the square-free kernel of $n$
- $\lambda(n)$ is the Liouville function
- $\mu(n)$ is the Möbius function
- $\nu(n)$ is the number of distinct primes in the factorization of $n$
- $\sigma_1(n)$ is the divisor function
Question (2): Are any of the conjectured relationships illustrated in formulas (3) to (9) above known proven relationships?