Details of the first region of integers $x$ with $\pi _{3,2}(x)<\pi _{3,1}(x)$

Abstract:

Since the time of Chebyshev [4] there has been interest in the magnitude of the smallest integer x with ${\pi _{3,2}}(x) < {\pi _{3,1}}(x)$, where ${\pi _{b,c}}(x)$ denotes the number of positive primes $\leqslant x$ and $\equiv c\;\pmod b$. The authors have recently reached this threshold with the discovery that ${\pi _{3,2}}(608981813029) - {\pi _{3,1}}(608981813029) = - 1$. This paper includes a detailed numerical and graphical description of values of ${\pi _{3,2}}(x) - {\pi _{3,1}}(x)$ in the vicinity of this long sought number.