The Centered Collision Sum

The prime number theorem gives an asymptotic formula for the count of primes: π(x) = Li(x) + error, with the error controlled. This paper proves a different kind of result: the centered collision deviations, summed over primes with weight $1/p$, converge. The divergent part cancels exactly.

The class mean

Each spectral class $R = (p-1) \bmod b$ has a mean collision deviation. I found this mean has a closed form for prime bases:

$$\bar{S}(R) = \frac{R+1}{b} - 1$$

The grand mean across all classes is $-1/2$, consistent with the reflection identity from Paper A. The class means are rational, evenly spaced, and determined by elementary arithmetic.

Centering

Subtract the class mean from each prime's deviation:

$$S^\circ(p) = S(p) - \bar{S}(R(p))$$

The centered deviation removes the predictable part. What remains is the fluctuation around the class-level structure.

The convergence

The centered sum converges:

$$F^\circ(1) = \sum_{p > b} \frac{S^\circ(p)}{p} < \infty$$

Computation through ten million primes confirms this in bases 3, 7, 10, and 12. The sum stabilizes. The centered deviations cancel across primes.

The raw sum diverges. The centered sum converges. The difference is one formula: the class mean.

Connection to Paper 1

The class mean at $R = 0$ in base 3 is $\bar{S}(0) = 1/3 - 1 = -2/3$. The alignment limit $2/3$ from Paper 1 reappears here, inverted in sign, as the collision bias of the class that contains $p \equiv 1 \pmod{3}$. The same constant that the golden ratio selected in Paper 1 governs the centering in Paper 16.

The program returns to where it began.

Try it yourself

./nfield 7               # class R = (7-1) mod 10 = 6
./nfield 97              # class R = (97-1) mod 10 = 6

Code: github.com/alexspetty/nfield

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Alexander S. Petty
April 2025
.:.