Let $f$ be a transcendental holomorphic curve from the complex plane into the two dimensional complex projective space of which defect relation over a set $X$ in $N$-subgeneral position is extremal. Then, there are $N-1$ vectors in $X$ whose deficiency with respect to $f$ is equal to 1.

We construct the absolute Frobenius operator as a matrix of infinite size for each prime. We calculate the characteristic polynomial and the trace. We investigate the composition of two absolute Frobenius operators also.

In this paper, we will prove that for every prime $p \equiv 1 \pmod{4}$, $E_{(p-1)/2} \not\equiv 0 \pmod{p}$.

Let $a$ be a positive integer which is not a perfect $b$-th power with $b \geq 2$, and $Q_a(x; k,l)$ be the set of primes $p \leq x$ such that the residual order of $a \pmod{p}$ in $\mathbf{Z}/p\mathbf{Z}^*$ is congruent to $l \pmod{k}$. In this paper, which is a sequel of our previous paper [1], under the assumption of the Generalized Riemann Hypothesis, we prove that for any residue class $l \pmod{k}$ the set $Q_a(x; k,l)$ has the natural density $\Delta_a(k,l)$, and the values of $\Delta_a(k,l)$ are effectively computable. We also consider some number theoretical properties of $\Delta_a(k,l)$ as a number theoretical function of $k$ and $l$.