The Collision Transform
Abstract
At fixed base and lag, the collision deviation is determined by one finite function on the units modulo a power of the base. Centering each reduction fiber removes the coarse base-level signal. Reflection makes the remaining signal odd. Its finite Fourier transform is therefore supported only on odd character channels that do not descend to the base. For a prime base at lag one, only primitive odd characters can survive.
The associated prime Dirichlet series is a finite collision-weighted sum of prime character series. It converges at exponent one. A common zero-free half-plane for the active Dirichlet L-functions gives ordinary convergence in the same open half-plane. The converse leaves an exact obstruction. If the collision series converges at a real point at or to the right of one half, then every active zero farther to the right has zero total multiplicity after weighting by the collision transform. A zero carried by only one effective primitive channel cannot hide in the finite sum.
This gives a direct connection between finite digit-collision geometry and the location of Dirichlet L-function zeros. It does not prove positivity or place any zero on the critical line. Deterministic calculations with nfield record declared finite prime windows only. No finite window is used as evidence of convergence on the critical line.
Introduction
Long division produces a finite partition before any analytic averaging begins. Fix a base and divide the nonzero residues by their leading digit. Multiplication by a power of the base then asks which residues remain inside the same digit bin. After subtracting the natural bin scale, the answer is a bounded arithmetic signal.
The collision invariant makes that signal finite [4]. At lag \ell, every value beyond b^{\ell+1} is read from one table on the units modulo b^{\ell+1}. The transform studied here is the finite Fourier transform of that table after its base-level means have been removed. Finite Fourier analysis is classical. The collision table and the exact route from its symmetries to a constraint on L-function zeros are the new objects.
Two cancellations form the spectral gate. Reflection removes every even character. Centering over reduction fibers removes every character that already lives modulo the base. The prime series therefore sees only the odd fine-scale part of the table.
That gate reaches the zero question in both directions. A common bound on the real parts of the active zeros forces ordinary convergence to the right of that bound. Conversely, ordinary convergence forces every zero in the resulting half-plane to disappear from an exact weighted zero ledger. The second statement is not a numerical pattern. It follows from residues of logarithmic derivatives.
The transform weights contain further arithmetic, but none of it is needed to locate the gate. The surviving channels and their zero obstruction are already exact.
The finite collision table
Fix a base b\geq2 and a lag \ell\geq1. Let N>b be an integer coprime to b. For 1\leq r<N, put \delta_{N,b}(r)=\left\lfloor\frac{br}{N}\right\rfloor. If [x]_N denotes the representative of x\bmod N in \{1,\ldots,N-1\}, define C_{b,\ell}(N)= \#\left\{1\leq r<N\ \middle|\ \delta_{N,b}(r)=\delta_{N,b}([b^\ell r]_N)\right\}. The collision deviation is S_{b,\ell}(N)=C_{b,\ell}(N) -\left\lfloor\frac{N-1}{b}\right\rfloor.
Put q=b^{\ell+1} and let G_{b,\ell}= \left\{0\leq n<q\ \middle|\ \left\lfloor\frac{n}{b^\ell}\right\rfloor=n\bmod b\right\}. The first and last base-b digits of every index in G_{b,\ell} agree. Its middle digits are free, so |G_{b,\ell}|=b^\ell.
Theorem 1 (Finite determination). Let N>q and (N,b)=1. Write N=qt+a with 1\leq a<q. Then S_{b,\ell}(N)=T_{b,\ell}(a), where T_{b,\ell}(a)= -1-\left\lfloor\frac ab\right\rfloor +\sum_{n\in G_{b,\ell}} \left( \left\lfloor\frac{(n+1)a}{q}\right\rfloor -\left\lfloor\frac{na}{q}\right\rfloor \right). Thus the collision deviation depends only on N\bmod q.
Proof. For 1\leq r<N, set n(r)=\lfloor qr/N\rfloor. The identity \left\lfloor\frac{\lfloor x\rfloor}{k}\right\rfloor =\left\lfloor\frac{x}{k}\right\rfloor for positive integral k gives \delta_{N,b}(r)=\left\lfloor\frac{n(r)}{b^\ell}\right\rfloor. Writing b^\ell r as a quotient and remainder modulo N gives \delta_{N,b}([b^\ell r]_N)=n(r)\bmod b. A collision occurs exactly on the slices indexed by G_{b,\ell}.
No interior slice boundary is integral because (N,q)=1. The floor difference \left\lfloor\frac{(n+1)N}{q}\right\rfloor -\left\lfloor\frac{nN}{q}\right\rfloor counts the positive residues in the nth slice. The terminal slice also counts the excluded endpoint N, and q-1 belongs to G_{b,\ell}. Consequently C_{b,\ell}(N)= -1+\sum_{n\in G_{b,\ell}} \left( \left\lfloor\frac{(n+1)N}{q}\right\rfloor -\left\lfloor\frac{nN}{q}\right\rfloor \right). Substitution of N=qt+a contributes t from each of the b^\ell selected slices. Since a is a unit modulo b, \left\lfloor\frac{N-1}{b}\right\rfloor =b^\ell t+\left\lfloor\frac ab\right\rfloor. Subtracting proves (7). ◻
The finite table is not an approximation. Once b and \ell are fixed, it contains every collision deviation beyond q. The formula and the finite-determination theorem also appear in [4].
Reflection and fiber centering
Lemma 2 (Reflection). For every unit a\bmod q, T_{b,\ell}(a)+T_{b,\ell}(q-a)=-1.
Proof. Write D_n(a)= \left\lfloor\frac{(n+1)a}{q}\right\rfloor -\left\lfloor\frac{na}{q}\right\rfloor. For 1\leq n\leq q-2, complementary floors give D_n(a)+D_n(q-a)=1. The paired totals at n=0 and n=q-1 are 0 and 2. Both endpoints belong to G_{b,\ell}, so \sum_{n\in G_{b,\ell}} \bigl(D_n(a)+D_n(q-a)\bigr)=b^\ell. Since a is a unit modulo b, \left\lfloor\frac ab\right\rfloor +\left\lfloor\frac{q-a}{b}\right\rfloor=b^\ell-1. The result follows from (7). ◻
Let U_q=(\mathbb Z/q\mathbb Z)^\times and let \rho\colon U_q\longrightarrow U_b be reduction modulo b. For u\in U_b, write A_u=\{a\in U_q\mid \rho(a)=u\}. Every lift u+kb remains coprime to b, so each fiber contains exactly b^\ell elements. Its mean is \mu_{b,\ell}(u)= \frac1{b^\ell}\sum_{a\in A_u}T_{b,\ell}(a).
Definition 3. The fiber-centered collision table is f_{b,\ell}(a)= T_{b,\ell}(a)-\mu_{b,\ell}(\rho(a)).
This centering removes one exact mean from each base-level digit family. It does not fit a mean to sampled primes.
Proposition 4 (Centered symmetry). For every u\in U_b and a\in U_q, \begin{aligned} \mu_{b,\ell}(u)+\mu_{b,\ell}(-u)&=-1, \\ f_{b,\ell}(-a)&=-f_{b,\ell}(a), \\ \sum_{a\in A_u}f_{b,\ell}(a)&=0. \end{aligned}
Proof. Negation maps A_u bijectively onto A_{-u}. Averaging (8) over A_u proves (11). Subtracting that identity from the pointwise reflection law proves (12). The last identity follows from the definition of the fiber mean. ◻
The spectral gate
For a Dirichlet character \chi\bmod q, define c_\chi=\widehat f_{b,\ell}(\chi) =\frac1{\varphi(q)} \sum_{a\in U_q}f_{b,\ell}(a)\overline{\chi(a)}. Finite Fourier inversion gives f_{b,\ell}(a)=\sum_{\chi\bmod q}c_\chi\chi(a). Call \chi active when c_\chi\ne0, and write \mathcal A_{b,\ell} for the active set.
Theorem 5 (Spectral gate). No even character is active. No character that factors through reduction modulo b is active. Every active character is therefore nonprincipal.
Proof. If \chi(-1)=1, pair a with -a in (14). The two terms cancel by (12).
Suppose instead that \chi=\psi\circ\rho for a character \psi on U_b. Grouping the transform by reduction fibers gives \sum_{a\in U_q}f_{b,\ell}(a)\overline{\chi(a)} =\sum_{u\in U_b}\overline{\psi(u)} \sum_{a\in A_u}f_{b,\ell}(a)=0 by (13). The principal character is even and is already absent. ◻
The gate is exact. It does not say that every remaining character is active. It says that the collision signal has no component outside the odd fine-scale channels.
Corollary 6 (Prime-base gate). If b is prime and \ell=1, every active character modulo b^2 is primitive and odd.
Proof. Every imprimitive character modulo b^2 is induced from a character whose conductor divides b. It therefore factors through reduction modulo b and vanishes by Theorem 5. ◻
Let \chi^* be the unique primitive character inducing an active character \chi. At the fixed modulus q, two characters cannot have the same primitive ancestor. Let \mathcal P_{b,\ell}=\{\chi^*: \chi\in\mathcal A_{b,\ell}\} and assign C_{\chi^*}=c_\chi. Every weight in this primitive set is nonzero.
For every prime p>q, an imprimitive character and its primitive ancestor have the same value at p. Fourier inversion therefore becomes f_{b,\ell}(p\bmod q)= \sum_{\psi\in\mathcal P_{b,\ell}}C_\psi\psi(p).
The collision prime series
Define the ordinary prime Dirichlet series F_{b,\ell}(s)= \sum_{p>q}\frac{f_{b,\ell}(p\bmod q)}{p^s}, where the sum runs over primes. It converges absolutely for \operatorname{Re}(s)>1.
Proposition 7 (The reciprocal-prime boundary). The series F_{b,\ell}(1) converges for every fixed base and lag.
Proof. For each a\in U_q, Mertens’ theorem in arithmetic progressions gives a constant M(q,a) with \sum_{\substack{p\leq x\\p\equiv a\, (\mathrm{mod}\ q)}}\frac1p =\frac1{\varphi(q)}\log\log x+M(q,a)+o(1) [1]. The coefficient of the common \log\log x term is \sum_{a\in U_q}f_{b,\ell}(a), which vanishes by (13). The remaining finite sum has a limit. Removing the primes no larger than q changes only finitely many terms. ◻
For a primitive character \psi, put \begin{aligned} P(s,\psi)&=\sum_p\frac{\psi(p)}{p^s}, \\ H(s,\psi)&=\sum_p\sum_{k\geq2} \frac{\psi(p)^k}{k p^{ks}}. \end{aligned} The series H(s,\psi) converges absolutely and locally uniformly when \operatorname{Re}(s)>1/2. Let E_{b,\ell}(s)= \sum_{\psi\in\mathcal P_{b,\ell}}C_\psi \sum_{\substack{p\leq q}}\frac{\psi(p)}{p^s}. This is an entire function.
Proposition 8 (Euler-product decomposition). For \operatorname{Re}(s)>1, F_{b,\ell}(s)= \sum_{\psi\in\mathcal P_{b,\ell}}C_\psi\log L(s,\psi) -\sum_{\psi\in\mathcal P_{b,\ell}}C_\psi H(s,\psi) -E_{b,\ell}(s). Each logarithm is the Euler-product branch that tends to zero as \operatorname{Re}(s) tends to infinity.
Proof. Equation (16) gives F_{b,\ell}(s)+E_{b,\ell}(s) =\sum_{\psi\in\mathcal P_{b,\ell}}C_\psi P(s,\psi). The Euler product gives P(s,\psi)=\log L(s,\psi)-H(s,\psi) in the half-plane of absolute convergence. Substitution proves (21). ◻
The signs in (21) are structural. The prime term is the first term in the logarithm of the Euler product. All higher prime powers are removed.
Ordinary convergence below one
Analytic continuation of the right side of (21) does not by itself prove convergence of the prime series. Ordinary convergence needs cancellation among the primes. A zero-free half-plane supplies that cancellation.
Theorem 9 (Zero-free penetration). Let 1/2\leq\theta<1. Suppose every nontrivial zero of L(s,\psi) has real part at most \theta for every \psi\in\mathcal P_{b,\ell}. Then the ordinary series (17) converges locally uniformly in \operatorname{Re}(s)>\theta. It therefore defines a holomorphic function there.
Proof. Fix \psi\in\mathcal P_{b,\ell}. The explicit formula under the stated zero bound gives \sum_{n\leq x}\Lambda(n)\psi(n) \ll x^\theta\log^2(qx) for fixed q [1, 3]. Removing prime powers costs at most the square-root scale. Since \theta\geq1/2, \vartheta(x,\psi)= \sum_{p\leq x}\psi(p)\log p \ll x^\theta\log^2(qx). Partial summation gives \sum_{p\leq X}\frac{\psi(p)}{p^s} =\frac{\vartheta(X,\psi)}{X^s\log X} +\int_2^X\vartheta(t,\psi) \frac{s\log t+1}{t^{s+1}(\log t)^2}\,dt. The boundary term tends to zero and the integral converges whenever \operatorname{Re}(s)>\theta. The convergence is uniform on compact subsets of that half-plane. The primitive set is finite, so (16) completes the proof. ◻
Corollary 10 (Conditional critical-strip convergence). Assume every nontrivial zero of every primitive L-function selected by \mathcal P_{b,\ell} lies on \operatorname{Re}(s)=1/2. Then F_{b,\ell}(s) converges locally uniformly for \operatorname{Re}(s)>1/2.
No assertion is made on the line \operatorname{Re}(s)=1/2.
Proposition 11 (Analytic continuation). Let \Omega be a simply connected open subset of \operatorname{Re}(s)>1/2 that meets \operatorname{Re}(s)>1. Assume that this intersection is connected. If every L(s,\psi) with \psi\in\mathcal P_{b,\ell} is nonzero on \Omega, then the right side of (21) has a holomorphic branch on \Omega that continues F_{b,\ell} from \operatorname{Re}(s)>1.
Proof. Each nonvanishing L(s,\psi) has a holomorphic logarithm on the simply connected region \Omega. Choose the branch that agrees with the Euler-product logarithm where \Omega meets \operatorname{Re}(s)>1. The higher-power terms are holomorphic for \operatorname{Re}(s)>1/2, and the finite correction is entire. ◻
A continued function and a convergent prime series are different conclusions.
The collision zero ledger
For any complex number \rho, define its collision-weighted zero multiplicity by \mathcal M_{b,\ell}(\rho)= \sum_{\psi\in\mathcal P_{b,\ell}} C_\psi\operatorname{ord}_{\rho}L(s,\psi). The order is zero when L(\rho,\psi)\ne0.
Theorem 12 (Zero-cancellation obstruction). Suppose the ordinary series F_{b,\ell}(\sigma) converges at one real point \sigma\geq1/2. Then \mathcal M_{b,\ell}(\rho)=0 for every \rho with \operatorname{Re}(\rho)>\sigma.
Proof. Regard (17) as an ordinary Dirichlet series by assigning coefficient zero to every nonprime index. Convergence at the real point \sigma implies local uniform convergence, and hence holomorphy, throughout \operatorname{Re}(s)>\sigma [2].
Differentiate (21) where \operatorname{Re}(s)>1. It gives \sum_{\psi\in\mathcal P_{b,\ell}} C_\psi\frac{L'}{L}(s,\psi) =F'_{b,\ell}(s) +\sum_{\psi\in\mathcal P_{b,\ell}}C_\psi H'(s,\psi) +E'_{b,\ell}(s). The right side is holomorphic for \operatorname{Re}(s)>\sigma. The differentiated higher-power sum is holomorphic there because \sigma\geq1/2, and the finite correction is entire. The difference of the two sides is meromorphic, vanishes on the open half-plane \operatorname{Re}(s)>1, and therefore vanishes throughout the connected half-plane \operatorname{Re}(s)>\sigma.
At a zero \rho, the residue of L'/L(s,\psi) is \operatorname{ord}_{\rho}L(s,\psi). The right side of (27) has no pole. Its residue is zero, so the weighted sum of residues on the left is zero. This is (26). ◻
Corollary 13 (An isolated effective zero cannot hide). Let \psi_0\in\mathcal P_{b,\ell}. Suppose \operatorname{Re}(\rho)>1/2, that L(\rho,\psi_0)=0, and that no other primitive L-function selected by \mathcal P_{b,\ell} vanishes at \rho. Then F_{b,\ell}(\sigma) cannot converge at any real \sigma\in[1/2,\operatorname{Re}(\rho)).
Proof. The zero contributes the nonzero term C_{\psi_0}\operatorname{ord}_{\rho}L(s,\psi_0) to (25). No other effective primitive channel is present to cancel it. Theorem 12 rules out convergence. ◻
The obstruction is collective. Distinct effective L-functions could in principle share a zero, and their collision weights could cancel its multiplicity. No assertion that such sharing occurs is needed. The theorem identifies the exact cancellation that convergence would require.
Finite prime windows
The numerical question is not whether one truncated sum looks flat. The useful finite object is the contribution made by a declared prime window. Table 1 records the final decimal window in a deterministic calculation with nfield [5] at lag one.
| s | base 3 | base 5 | base 7 | base 10 | base 12 |
|---|---|---|---|---|---|
| 1.0 | 0.000042 | 0.000014 | -0.000136 | 0.000065 | 0.000098 |
| 0.9 | 0.000183 | 0.000016 | -0.000580 | 0.000047 | 0.000409 |
| 0.8 | 0.000807 | -0.000122 | -0.002488 | -0.000868 | 0.001714 |
| 0.7 | 0.003629 | -0.001345 | -0.010721 | -0.008671 | 0.007215 |
| 0.6 | 0.016621 | -0.009300 | -0.046418 | -0.060258 | 0.030483 |
| 0.5 | 0.077496 | -0.055134 | -0.201877 | -0.366499 | 0.129262 |
nfield checked every reduction fiber and every unit class for the five displayed bases. This covers 156 unit classes and 468 direct comparisons between the finite table and the defining collision count. It then evaluated every prime in the declared window at all six displayed exponents. The finite calculations are independent checks and are not used as premises in the proofs.
The final window is small at s=1, where convergence is proved. The windows generally grow as the exponent falls, and their signs vary with the base. They do not decide convergence at any point below one.
The boundary at one half
Let \sigma_{b,\ell} be the abscissa of ordinary convergence of (17). Proposition 7 gives \sigma_{b,\ell}\leq1. Theorem 9 moves that boundary left when the effective L-functions have a common zero-free half-plane. Theorem 12 moves in the opposite direction. Every zero to the right of one half with a nonzero collision ledger is a barrier that the ordinary prime series cannot cross.
Conjecture 14 (Collision boundary). If f_{b,\ell} is not identically zero, then \sigma_{b,\ell}=\frac12.
If every effective primitive L-function has its nontrivial zeros on the critical line, Theorem 9 gives the upper bound \sigma_{b,\ell}\leq1/2. That is only one half of the conjecture. The reverse inequality requires a collision-visible obstruction on the line. Neither finite calculation nor zero-free penetration supplies it.
The finite geometry reaches the zero question through an exact ledger. The collision transform determines the channels and their weights. If the prime series converges, every zero to its right must cancel in (25). An isolated effective zero cannot disappear. What remains is the shared-zero case, where several channel weights could cancel at the same point.
References
[1]H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000.
[2]G. H. Hardy and M. Riesz, The General Theory of Dirichlet's Series, Cambridge University Press, 1915.
[3]H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, 2004.
[4]A. S. Petty, The collision invariant, arXiv:2604.00045, 2026.
[5]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield