Distribution of Leading Digits of Imaginary Parts of Riemann Zeta Zeros Ii

Let \( \gamma_n, n = 1,2,\ldots \), be the sequence of all positive imaginary
parts of non-trivial zeros of the Riemann zeta function \( \zeta(s) \) in ascending order.
In this paper we give a quantitative result on the distribution of the leading
block of digits of \( \gamma_n \).

For
\[
N_i = \#\{n \ge 1 : \gamma_n < 10^i\} \ge 2 \] and an \(r\)-digit number \[ D = d_1 d_2 \cdots d_r, \] we show that the frequency

\[
\frac{\#\{n \le N_i : \text{leading block of } r \text{ digits of } \gamma_n = D\}}
{N_i}
\]

is approximated by

\[
\frac{1}{9\cdot10^{r-1}}
+
\frac{\log 10}{18\pi\,10^{r-1}}
\left(
(D+1)\log_{10}(D+1)
-
D\log_{10}D
-
r
-
\frac{1}{9}
\right)
\frac{10^i-10}{N_i}
\]

with the explicit error bound

\[
0.516\,\frac{i^2}{N_i}
+
1.112\,\frac{i\log i}{N_i}
+
10.452\,\frac{i}{N_i}
+
1.668\,\frac{\log i}{N_i}
-
1.474\,\frac{1}{N_i}.
\]