The affine-linearity conjecture for Dedekind-sum averages over Mersenne subgroups

Let $d_0\thickspace\geq\thickspace 1$ be odd and square-free. Let $N$ be the order of $2$ in the multiplicative group $({\mathbb Z}/d_0{\mathbb Z})^*$. Set $f=2^d-1$ with $d>1$ odd, and let

$$H=\{2^k;\ 0\leq k\leq d-1\},$$

a subgroup of order $d$ of $({\mathbb Z}/f{\mathbb Z})^*$. Assume $\gcd(f,d_0)=1$.

Affine-linearity conjecture. Then

$$N_{d_0}'(f,H)=A_1(d)d+A_0(d),$$

where $A_1(d)$ and $A_0(d)$ are rational numbers depending only on $d$ modulo $N$, equivalently only on $f$ modulo $d_0$. Hence, for a prime $p\geq 3$, one expects

$$M_{d_0}(p,H)=\frac{\pi^2}{2}\left\{\prod_{q\mid d_0}\left(1-\frac{1}{q^2}\right)\right\}\left(1+\frac{A_1(d)d}{p}+\frac{A_0(d)}{p}\right).$$

The assertion is verified in the paper for $d_0\in\{1,3,5,15,105\}$, and is presented as evidence that the restriction on $d$ in the stated asymptotic theorem should be sharp.