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Fix a base, a positive lag, and a finite prime cutoff. The collision deviation at each prime gives a real signal, and the primes carrying that signal split into the unit residue classes of the base. Dirichlet characters give the standard finite Fourier coordinates of those residue streams. Fourier inversion identifies the total fluctuation with the principal component, pairs conjugate channels, and gives a Parseval identity. No information is lost. A finite base-ten calculation illustrates the decomposition. The transform itself is classical. The arithmetic input is the centered collision coefficient supplied by the digit partition. The result exposes every finite residue channel. The cutoff dependence of each component remains an open question.
In base ten, every prime greater than five ends in 1, 3, 7, or 9. A sum over those primes is therefore four sums before it is one. The four streams may carry the same signal, different signals, or signals that cancel only after they are added.
Dirichlet characters are made for this split. They are multiplicative weights on the unit residue classes, and they form a complete orthogonal basis for functions on those classes. In base ten there are four unit classes and four characters. Passing from the residue streams to character components loses no information. It changes the coordinates in which the finite signal is read.
Character orthogonality, Fourier inversion, and Parseval’s identity on a finite abelian group are classical [2, 3, 4]. Their use below does not define a new transform. It assigns exact coordinates to a collision signal whose coefficients come from the finite digit partition.
Bin Derangements and the Gate Width Theorem [1] gives the exact mean over the positive nonidentity collision counts. That mean supplies the reference level for the collision deviation. The character identities are exact at every fixed cutoff.
Fix an integer base b\geq2. Let p>b+1 be prime, and let [u]_p denote the least positive representative of a nonzero residue modulo p. For 1\leq r<p, define \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor . The values of \delta_{p,b} lie in \{0,\ldots,b-1\} and partition the nonzero residues into b digit bins.
For g\in\mathbb F_p^\times, define the collision count C_{p,b}(g) = \#\left\{ r\in\{1,\ldots,p-1\}\ \middle|\ \delta_{p,b}(r)=\delta_{p,b}([gr]_p) \right\}. A nonidentity multiplier is constructive when its collision count is positive.
The reference level for the fluctuation is the mean collision count over the constructive multipliers. Its exact value is derived here because it centers the collision signal before the character transform.
Lemma 1 (Constructive mean). Write p-1=bQ+R, \qquad 0\leq R<b. There are p-b-1=b(Q-1)+R constructive multipliers, and their mean collision count is \overline C_{p,b} = \frac{Q\bigl(b(Q-1)+2R\bigr)} {b(Q-1)+R}.
Proof. Multiplication by b turns the digit bins into residue classes modulo b. Indeed, if x=[br]_p, then br=p\delta_{p,b}(r)+x, and reduction modulo b shows that two residues have the same digit exactly when their corresponding values of x agree modulo b. Consequently C_{p,b}(g) = \#\{x\in\{1,\ldots,p-1\}\mid x\equiv[gx]_p\pmod b\}.
For g\neq1, put c(g)=[b(1-g)^{-1}]_p. This gives a bijection from \mathbb F_p^\times\setminus\{1\} onto \mathbb F_p^\times\setminus\{b\}. We claim that C_{p,b}(g)=0 exactly when 1\leq c(g)<b.
Suppose a collision exists. Write y=[gx]_p=x+mb. The relation c(g)(1-g)\equiv b\pmod p gives x+mc(g)\equiv0\pmod p. If 1\leq c(g)<b and m\geq0, then 0<x+mc(g)=y-m\bigl(b-c(g)\bigr)<p. If m=-n<0, then 0<x-nc(g)=y+n\bigl(b-c(g)\bigr)<p. Either case produces a positive multiple of p smaller than p, which is impossible. Conversely, if b<c(g)<p, take x=p-c(g) and y=x+b. Both lie between 1 and p-1, x\equiv y\pmod b, and y=[gx]_p. Thus a collision exists. There are exactly b-1 nonidentity multipliers with zero collision count. The remaining (p-2)-(b-1)=p-b-1 nonidentity multipliers are constructive.
Let n_d be the size of the dth digit bin. The interval description in (1) shows that every n_d is Q or Q+1. Since the bin sizes sum to p-1, exactly R bins have size Q+1, and the other b-R have size Q. Counting ordered pairs in a common bin, first by the pair and then by their quotient, gives \sum_{g\in\mathbb F_p^\times}C_{p,b}(g) = \sum_{d=0}^{b-1}n_d^2 = bQ^2+R(2Q+1). The identity multiplier contributes p-1, and the zero-collision multipliers contribute zero. The total constructive collision mass is therefore bQ^2+R(2Q+1)-(p-1) = Q\bigl(b(Q-1)+2R\bigr). Division by the number of constructive multipliers proves (3). ◻
Fix a positive integer lag \ell. The collision deviation at p is \Delta_{p,b}(\ell) = C_{p,b}([b^\ell]_p)-\overline C_{p,b}. This remains defined when b^\ell\equiv1\pmod p. In that case the multiplier is the identity and its collision count is p-1. No primitive-root hypothesis is needed because the collision count is defined on the full multiplier group. The prime p=b+1, when it is prime, is excluded because there are no constructive multipliers and the reference mean is undefined.
Let U_b=(\mathbb Z/b\mathbb Z)^\times and let \widehat U_b be its character group. Its elements are the Dirichlet characters modulo b, restricted to the unit classes [2, 3]. There are \varphi(b) such characters.
For a real cutoff x, a complex parameter s, and r\in U_b, define the residue stream F_{r,x}(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}\\p\equiv r\pmod b}} \frac{\Delta_{p,b}(\ell)}{p^s}. For \chi\in\widehat U_b, define the character component \widehat\Phi_{\chi,x}(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}}} \frac{\chi(p)\Delta_{p,b}(\ell)}{p^s}. The total finite fluctuation is \Phi_x(s,\ell) = \sum_{\substack{b+1<p\leq x\\p\ {\rm prime}}} \frac{\Delta_{p,b}(\ell)}{p^s}. Here p^{-s} is defined using the real logarithm. These sums are finite, so no convergence hypothesis on s is needed.
The following result is the standard finite Fourier transform on U_b, applied to the collision deviation with the normalization fixed above. The proof records the conventions used in the numerical table.
Theorem 2 (Finite character decomposition). For every cutoff x, every s\in\mathbb C, and every positive lag \ell, the residue streams and character components satisfy \begin{aligned} \widehat\Phi_{\chi,x}(s,\ell) &= \sum_{r\in U_b}\chi(r)F_{r,x}(s,\ell), \\ F_{r,x}(s,\ell) &= \frac1{\varphi(b)} \sum_{\chi\in\widehat U_b} \overline{\chi(r)}\,\widehat\Phi_{\chi,x}(s,\ell). \end{aligned} If \chi_0 is the principal character, then \Phi_x(s,\ell) = \sum_{r\in U_b}F_{r,x}(s,\ell) = \widehat\Phi_{\chi_0,x}(s,\ell). The transform also satisfies \sum_{\chi\in\widehat U_b} \left|\widehat\Phi_{\chi,x}(s,\ell)\right|^2 = \varphi(b)\sum_{r\in U_b}|F_{r,x}(s,\ell)|^2.
Proof. Partitioning the prime sum by residue class gives \widehat\Phi_{\chi,x} = \sum_{r\in U_b}\chi(r)F_{r,x}, which proves the forward transform. Character orthogonality gives \frac1{\varphi(b)} \sum_{\chi\in\widehat U_b} \chi(t)\overline{\chi(r)} = \begin{cases} 1,&t=r,\\ 0,&t\neq r \end{cases} \qquad (r,t\in U_b). Substituting the forward transform and applying this identity leaves only F_{r,x}, which proves the inverse transform.
Every prime in the sums is coprime to b, so the principal character equals one on every term. This proves (15). Finally, expand the left side of (16), use the forward transform, and apply \sum_{\chi\in\widehat U_b} \chi(r)\overline{\chi(t)} = \varphi(b)\,\mathbf1_{\{r=t\}}. Only the diagonal terms remain, which proves Parseval’s identity. ◻
Corollary 3 (Conjugate pairing). For every s\in\mathbb C, \widehat\Phi_{\overline\chi,x}(\overline s,\ell) = \overline{\widehat\Phi_{\chi,x}(s,\ell)}. In particular, when s is real, conjugate characters have equal component magnitudes.
Proof. The deviations are real. Conjugating the finite sum in (9) replaces \chi by \overline\chi and s by \overline s. ◻
The group U_{10}=\{1,3,7,9\} is cyclic with generator 3. Label its characters by \chi_j(3)=i^j, \qquad 0\leq j\leq3. Thus \chi_0 is principal, \chi_3=\overline{\chi_1}, and \chi_2 is real.
Table 1 uses lag \ell=1 and exponent s=1. Every prime in 11<p\leq x is included. The calculation forms its p-1 digit values, evaluates multiplication by 10, subtracts (3), and adds the four character weights. The nfield [5] carries out this finite calculation and exposes the residue streams. The prime count is for this eligible range.
| x | primes | \widehat\Phi_{\chi_0,x} | \operatorname{Re}\widehat\Phi_{\chi_1,x} | \operatorname{Im}\widehat\Phi_{\chi_1,x} | \widehat\Phi_{\chi_2,x} |
|---|---|---|---|---|---|
| 100 | 20 | -0.851414 | -0.012410 | -0.168966 | 0.300620 |
| 1000 | 163 | -1.149104 | -0.208657 | -0.217315 | 0.329630 |
| 5000 | 664 | -1.318994 | -0.288890 | -0.234378 | 0.354014 |
| 10000 | 1224 | -1.392595 | -0.316180 | -0.241931 | 0.350235 |
The same nfield calculation reconstructs every residue stream from the four components. In long-double arithmetic, the largest inverse-reconstruction and Parseval residuals in these rows are both below 10^{-17}, after normalization by one plus the magnitude of the corresponding reference value. These finite values verify the calculation and show the four distinct character coordinates, including the conjugate pair; they do not determine cutoff asymptotics.
Each component is a finite prime sum whose collision coefficient is exact at every prime. Its scale across primes is unknown. Fourier inversion names the channel. It does not estimate it.
Question 4. For fixed b, \ell, and \chi, what is the behavior of \widehat\Phi_{\chi,x}(1,\ell) as x tends to infinity? Does it remain bounded, oscillate without a limit, or admit an unbounded leading scale?
Every finite residue stream is recoverable from the character components. The principal component alone misses the nonprincipal channels. The finite transform closes. The cutoff problem remains.
[1]A. S. Petty, Bin Derangements and the Gate Width Theorem, July 2022. \href{https://doi.org/10.5281/zenodo.21850917} {doi:10.5281/zenodo.21850917}.
[2]H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, 2000.
[3]H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, 2004.
[4]A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge University Press, 1999. \href{https://doi.org/10.1017/CBO9780511626265} {doi:10.1017/CBO9780511626265}.
[5]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
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