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The Collision Spectrum

Alexander S. Petty

Abstract

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.

The Collision Spectrum

Abstract

For a prime base and primitive odd Dirichlet character modulo , the collision transform coefficient admits an exact factorization: where is the generalized first Bernoulli number and is the diagonal character sum. By the standard Bernoulli---value formula, , so the collision invariant's Fourier spectrum encodes -function special values. A Parseval identity gives an exact formula for the weighted second moment in terms of the collision invariant's values on the finite group. The digit function computes this -value moment exactly. Under a conditional zero-free hypothesis, the triangle inequality yields a separate bound connecting to for in the critical strip. At base~, the factorization gives exactly. For quadratic characters in the family, the decomposition specializes to class-number data.

Paper Structure

This paper contains 6 sections, 5 theorems, 11 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Decomposition Theorem
  3. The $L$-Encoding
  4. Moment Bounds
  5. The Correlation Decay
  6. Remarks

Key Result

Theorem 1

Let $b$ be prime, $m = b^2$, and $\chi$ a primitive odd character modulo $m$. Then where $B_{1,\overline{\chi}} = (1/m) \sum_a a\, \overline{\chi}(a)$ and $S_G(\chi) = \sum_{n \in G} [\overline{\chi}(n{+}1) - \overline{\chi}(n)]$, and $G = \{n \in \{0, \ldots, m{-}1\} : \lfloor n/b \rfloor = n \bmod b\}$ is the diagonal set paperA (elements whose base-$b$ digits coincide), with $|

Theorems & Definitions (14)

  • Theorem 1: Decomposition
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Remark 4: Connection to short partial sums
  • Remark 5
  • Theorem 6: Moment identity
  • proof
  • ...and 4 more