Silent Primes and the Variance of the Collision Count
Silent primes
Fix a rotation angle (lag) and ask: which primes have zero digit matches at that angle? Only finitely many. The "silent" primes at each lag are the prime factors of specific integers: N(l, u) = b^{l+1} + u(1 - b^l) for u = 1,...,b - 1. These form an arithmetic sequence with common difference -(b^l - 1).
At lag 1 in base 10: the integers are 91, 82, 73, 64, 55, 46, 37, 28, 19. Their prime factors above 10 are {11, 13, 19, 23, 37, 41, 73}. For every other prime, lag 1 produces matches.
The sawtooth form
The collision count C(g) has an exact expression as a signed sawtooth sum:
C(g) = Q + sum of sgn(((mc/p)) - ((mb/p))) for m = 1,...,Q
where ((x)) = {x} - 1/2 is the sawtooth function and c = b(1-g)^{-1} mod p. This places the collision count in the family of Dedekind-type sums.
The variance theorem
The collision count fluctuates across non-deranging multipliers. The variance scales as Q:
var(C) = lambda_b * Q + O(1)
giving standard deviation proportional to sqrt(Q) = sqrt(p/b). In base 10, lambda is approximately 0.89. The square-root scaling is the characteristic fluctuation of equidistributed sequences modulo p, the same pattern that governs prime counting under the Riemann Hypothesis.
Toward the fluctuation sum
The Dirichlet series L^G(s, l) decomposes into a mean part (smooth, converging for Re(s) > 2) and a fluctuation part F(s, l) whose individual terms are O(sqrt(p/b)). The fluctuation sum converges for Re(s) > 3/2, deeper into the critical strip. Whether F admits continuation to Re(s) > 1 is the bridge to the zeros of zeta.
The paper
Silent Primes and the Variance of the Collision Count (PDF)
.:.