Auxiliary solution-count conjectures for the discrete logarithm equation

Let $p$ be prime. For the auxiliary equation studied in the paper, let $C_{h\,A,a\,B}(p)$ denote the number of solutions with the indicated restrictions, and let $\phi$ be Euler's totient function. If $p-1$ is squarefree, write its prime divisors as $q$; in general write $q^\alpha\Vert p-1$ when $q^\alpha$ is the exact power of $q$ dividing $p-1$.

Auxiliary solution-count conjectures.

$$C_{h\,\mathsf{ANY},a\,\mathsf{ANY}}(p)\approx (p-1)+\sum_{m\mid p-1}\frac{\phi(m)}{m^2}\left(\sum_{d\mid (p-1)/m}\frac{\phi(dm)}{d}\right)^2.$$

If $p-1$ is squarefree, then

$$C_{h\,\mathsf{ANY},a\,\mathsf{ANY}}(p)\approx (p-1)+\prod_{q\mid p-1}\left(q+1-\frac1q\right).$$

In general,

$$C_{h\,\mathsf{ANY},a\,\mathsf{ANY}}(p)\approx (p-1)+\prod_{q^\alpha\Vert p-1}\left(\left[\left(1-\frac1q\right)\alpha+1\right]^2+\left(1-\frac1q\right)^3\left[(\alpha+1)^2\frac{q^{\alpha+1}-q}{q-1}-2(\alpha+1)\frac{\alpha q^{\alpha+2}-(\alpha+1)q^{\alpha+1}+q}{(q-1)^2}+\frac{\alpha^2q^{\alpha+3}-(2\alpha^2+2\alpha-1)q^{\alpha+2}+(\alpha^2+2\alpha+1)q^{\alpha+1}-q^2-q}{(q-1)^3}\right]\right).$$

The product is over primes $q$ dividing $p-1$. Additionally,

$$C_{h\,\mathsf{PR},a\,\mathsf{ANY}}(p)\approx 2\phi(p-1),$$ $$C_{h\,\mathsf{ANY},a\,\mathsf{PR}}(p)\approx 2\phi(p-1),$$ $$C_{h\,\mathsf{PR},a\,\mathsf{PR}}(p)\approx \phi(p-1)+\frac{\phi(p-1)^2}{p-1}.$$

These predictions are derived from the paper's random-map heuristic for $x\mapsto x^x\bmod p$ and remain conjectural in the supplied text.