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

Let γ_n, n = 1,2, …, be the sequence of all positive imaginary parts of non-trivial zeros of the Riemann zeta function 𝜁(s) in ascending order. n this paper we give a quantitative result on the distribution of leading block of digits of γ_n, n = 1,2, …. For N_i =# {n ≥ 1: γ_n < 10ⁱ}(≥ 2) and an r-digits number D = d₁d₂ ··· d_r, we show that the frequency.

#{ n≤Ni:(leading block of r digits of γn )=D }Ni {{\# \left\{ {n \le {N_i}:\left( {\text{leading block of} \,r\,\text{digits of}\,{\gamma _n}\,} \right) = D} \right\}} \over {{N_i}}} is approximated by 19⋅10r-1+log1018π10r-1((D+1)log10(D+1)-Dlog10D-r-19)10i-10Ni {1 \over {9 \cdot {{10}^{r - 1}}}} + {{\log 10} \over {18\pi {{10}^{r - 1}}}}\left( {\left( {D + 1} \right){{\log }_{10}}\left( {D + 1} \right) - D{{\log }_{10}}D - r - {1 \over 9}} \right){{{{10}^i} - 10} \over {{N_i}}} with the explicit error bound 0.516i2Ni+1.112ilogiNi+10.452iNi+1.668logiNi-1.4741Ni. 0.516{{{i^2}} \over {{N_i}}} + 1.112{{i\log i} \over {{N_i}}} + 10.452{i \over {{N_i}}} + 1.668{{\log i} \over {{N_i}}} - 1.474{1 \over {{N_i}}}.