Logarithmic bias and sign-change mechanics in arithmetic progressions π(x; q, a) - π(x; q, b)
Chebyshev's bias describes the tendency for prime numbers to be slightly more abundant in quadratic non-residue classes than residue classes. For example, primes of the form $4k + 3$ systematically lead primes of the form $4k + 1$ across large domain spans, despite Dirichlet's Theorem guaranteeing equal asymptotic distribution.
The prime race difference $\Delta(x) = \pi(x; q, a) - \pi(x; q, b)$ is governed by the non-trivial zeros of the associated Dirichlet $L$-functions $L(s, \chi)$. The Knapowski-Turán theory proves that $\Delta(x)$ changes sign infinitely many times, though the first crossover for modulus 4 occurs at extreme logarithmic thresholds ($x \approx 1.84 \times 10^{361}$).
This visualization tracks the running difference between prime counts across selected moduli. The persistent positive displacement illustrates the underlying non-residue bias, while high-frequency oscillations reflect zero-sum contributions from non-principal characters.