arXiv subjects

Alexander S. Petty

Publications and source records attributed to Alexander S. Petty.

3 recordsLinked to original sources

arXiv2026

The Collision Spectrum

For a prime base $b$ and primitive odd Dirichlet character $χ$ modulo $b^2$, the collision transform coefficient $\hat{S}^{\circ}(χ)$ admits an exact factorization: \[ \hat{S}^{\circ}(χ) = -\frac{B_{1,\overlineχ} \cdot \overline{S_G(χ)}}{ϕ(b^2)}, \] where $B_{1,\overlineχ}$ is the generalized first Bernoulli number and $S_G(χ)$ is the diagonal character sum. By the standard Bernoulli--$L$-value formula, $|B_1| = (b/π)\, |L(1, χ)|$, so the collision invariant's Fourier spectrum encodes $L$-function special values. A Parseval identity gives an exact formula for the weighted second moment $\sum |L(1, χ)|^2 \cdot |S_G(χ)|^2$ in terms of the collision invariant's values on the finite group. The digit function computes this $L$-value moment exactly. Under a conditional zero-free hypothesis, the triangle inequality yields a separate bound connecting $L(1)$ to $L(s)$ for $s$ in the critical strip. At base~$5$, the factorization gives $|\hat{S}^{\circ}| \propto |L(1)|^2$ exactly. For quadratic characters in the family, the decomposition specializes to class-number data.

math.GM↗

arXiv2026

The Collision Transform

For a prime p and base b, the collision invariant $S_{\ell}(p)$, introduced in the companion paper, is a function of $p \bmod b^{\ell+1}$ and therefore lives on the finite group $(\mathbb{Z}/b^{\ell+1}\mathbb{Z})^{\times}$. Its Fourier expansion over Dirichlet characters modulo $b^{\ell+1}$ is the collision transform. The reflection identity forces all even-character coefficients of the centered invariant to vanish: only odd characters contribute. The centered prime harmonic sum $F^{\circ}(s) = \sum_p S^{\circ}_p / p^s$ is therefore a finite linear combination of non-trivial odd character sums $\sum_p χ(p)/p^s$, with no principal-character term. At $s = 1$, each sum converges by Mertens' theorem for arithmetic progressions. Convergence below $s = 1$ is conditional on the absence of $L$-function zeros above a given depth. Computation indicates convergence persists to at least $s = 0.6$ in base 10 and to $s = 0.5$ in base 3. The real parts of the products $\hat{S}^{\circ}(χ) \cdot P(s, χ)$ have mixed signs, so convergence is a collective constraint on the joint zero distribution, not a test of each $L$-function individually. Aggregating the collision deviation across bases with a fixed convergent weighting produces the base sum, a function on primes that reveals mod-3 structure. For bases with $3 \nmid b$, the reflection $a \mapsto m - a$ fixes a unique residue class modulo 3, and the mean of $S$ over units in that class equals the grand mean $-1/2$ (the neutrality theorem). Removing the mod-3 component introduces a principal-character term that is absent from $F^{\circ}$. The base-summed harmonic sum is negligible: the collision invariant's structural content is base-specific.

math.GM↗

arXiv2026

The Collision Invariant

For a prime p and base b, the digit function delta(r) = floor(br/p) partitions the residues {1, ..., p-1} into b contiguous bins. The collision count C(g) records how many residues share a bin with their image under multiplication by g. We prove four results about this function. First, the gate width theorem: exactly b-1 multipliers satisfy C(g) = 0, given by the explicit family g = -u/(b-u) mod p for u = 1, ..., b-1. Second, the finite determination theorem: the collision deviation S at lag l depends only on p mod b^(l+1). Third, the reflection identity: S(a) + S(m-a) = -1 for m = b^(l+1), implying a grand mean of -1/2 and a pairing symmetry across the group of units. Fourth, the half-group theorem: for every non-trivial good slice n, the wrapping set W_n has size exactly phi(m)/2. The bilateral symmetry a -> m-a swaps wrapping with non-wrapping.

math.GM↗