The Collision Transform and the Critical Strip
Abstract
At fixed base and lag, a digit-collision deviation is determined by one finite function on the units modulo a power of the base. Centering each reduction fiber removes the coarse digit families. Reflection then makes the remaining signal odd, and its finite Fourier transform is supported only on odd character channels that do not descend to the base.
The associated prime Dirichlet series is a collision-weighted sum of prime character series. A zero-free half-plane for the active Dirichlet L-functions gives ordinary convergence in the same open half-plane by the explicit formula and partial summation. 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 L-function zero farther to the right must have zero total multiplicity after weighting by the collision transform. In particular, a zero carried by only one active channel cannot be hidden by the finite sum. The obstruction is unconditional. Shared zeros can cancel only when their total collision weight is zero.
nfield [6] reproduces the exact centered tables and the declared terminal prime window through ten million.
Below the reciprocal-prime edge
At s=1, reciprocal-prime weighting is enough for the fiber-centered collision signal to settle. Replace 1/p by 1/p^s with s<1, and the distant primes become louder. The finite table does not change. The question is how far its prime sampling survives as an ordinary Dirichlet series.
The finite table, its reflection law, and its centered character channels are established in The Centered Collision Sum and displayed cell by cell at lag one in The Collision Periodic Table [4, 5]. Those foundations remain here so that the argument is self-contained. The new object is the prime Dirichlet series formed from the centered table.
Ordinary convergence and analytic continuation are different phenomena. A zero-free half-plane supplies cancellation among the sampled primes. Conversely, ordinary convergence forces an exact balance among the zero multiplicities carried by the active character channels. This places the finite collision geometry directly against the zeros of Dirichlet L-functions.
The finite collision table
Fix a base b\geq 2 and a lag \ell\geq 1. Let p be a positive integer coprime to b. For 1\leq r<p, put \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor. If [x]_p denotes the representative of x\bmod p in \{1,\ldots,p-1\}, define C_{b,\ell}(p)= \#\left\{1\leq r<p\ \middle|\ \delta_{p,b}(r)=\delta_{p,b}([b^\ell r]_p)\right\}. The bounded collision deviation is S_{b,\ell}(p)=C_{b,\ell}(p) -\left\lfloor\frac{p-1}{b}\right\rfloor.
Set 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. The middle digits are free, so |G_{b,\ell}|=b^\ell.
Theorem 1 (Finite determination). Let p>q and \gcd(p,b)=1. Write p=qt+a with 1\leq a<q. Then S_{b,\ell}(p)=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 is determined by p\bmod q.
Proof. For 1\leq r<p, set n(r)=\lfloor qr/p\rfloor. The elementary identity \left\lfloor\frac{\lfloor x\rfloor}{k}\right\rfloor =\left\lfloor\frac{x}{k}\right\rfloor for positive integral k gives \delta_{p,b}(r)=\left\lfloor\frac{n(r)}{b^\ell}\right\rfloor. Writing b^\ell r as a quotient and remainder modulo p gives \delta_{p,b}([b^\ell r]_p)=n(r)\bmod b. A collision therefore occurs exactly on the slices indexed by G_{b,\ell}.
No interior slice boundary is integral because p is coprime to q. The floor difference \left\lfloor\frac{(n+1)p}{q}\right\rfloor -\left\lfloor\frac{np}{q}\right\rfloor counts the positive residues in the nth slice. The terminal slice also counts the excluded endpoint p, and q-1 belongs to G_{b,\ell}. Consequently C_{b,\ell}(p)= -1+\sum_{n\in G_{b,\ell}} \left( \left\lfloor\frac{(n+1)p}{q}\right\rfloor -\left\lfloor\frac{np}{q}\right\rfloor \right). Substitution of p=qt+a contributes t from each of the b^\ell selected slices. Since a is a unit modulo b, \left\lfloor\frac{p-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.
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 corresponding sums 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 now 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 exact 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 mean from each base-level digit family. The mean belongs to the complete fiber and is fixed before any prime is sampled.
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 is the definition of the fiber mean. ◻
The active character channels
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. In particular, every active character is 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, so it is already absent. ◻
The active set is therefore contained in the odd fine-scale channels. Some of those eligible channels may still have zero coefficient.
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. Every such prime is a unit modulo q. The series converges absolutely for \operatorname{Re}(s)>1.
Proposition 6 (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) for each a\in U_q [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 character \chi\bmod q, put \begin{aligned} P(s,\chi)&=\sum_p\frac{\chi(p)}{p^s}, \\ H(s,\chi)&=\sum_p\sum_{k\geq2} \frac{\chi(p)^k}{k p^{ks}}. \end{aligned} The series H(s,\chi) converges absolutely and locally uniformly when \operatorname{Re}(s)>1/2. Let E_{b,\ell}(s)= \sum_{\substack{p\leq q\\p\nmid q}} \frac{f_{b,\ell}(p\bmod q)}{p^s}. This is an entire function.
Proposition 7 (Euler-product decomposition). For \operatorname{Re}(s)>1, F_{b,\ell}(s)= \sum_{\chi\in\mathcal A_{b,\ell}}c_\chi\log L(s,\chi) -\sum_{\chi\in\mathcal A_{b,\ell}}c_\chi H(s,\chi) -E_{b,\ell}(s). Each logarithm is the Euler-product branch that tends to zero as \operatorname{Re}(s) tends to infinity.
Proof. Fourier inversion gives F_{b,\ell}(s)+E_{b,\ell}(s) =\sum_{\chi\in\mathcal A_{b,\ell}}c_\chi P(s,\chi). The Euler product gives P(s,\chi)=\log L(s,\chi)-H(s,\chi) 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 exactly that cancellation.
Theorem 8 (Zero-free penetration). Let 1/2\leq\theta<1. For every \chi\in\mathcal A_{b,\ell}, let \chi^* be the primitive character that induces \chi. Suppose every nontrivial zero of L(s,\chi^*) has real part at most \theta. Then the ordinary series (16) converges locally uniformly in \operatorname{Re}(s)>\theta. It therefore defines a holomorphic function there.
Proof. Begin with a primitive active character. The explicit formula under the stated zero bound gives \psi(x,\chi)= \sum_{n\leq x}\Lambda(n)\chi(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,\chi)= \sum_{p\leq x}\chi(p)\log p \ll x^\theta\log^2(qx). Partial summation now gives \sum_{p\leq X}\frac{\chi(p)}{p^s} =\frac{\vartheta(X,\chi)}{X^s\log X} +\int_2^X\vartheta(t,\chi) \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.
If \chi is imprimitive, it agrees at every prime larger than q with its primitive inducing character. Thus its series in (16) has the same convergence. The active set is finite, so Fourier inversion completes the proof. ◻
Corollary 9 (Conditional critical-strip convergence). Assume the Generalized Riemann Hypothesis for the primitive L-functions selected by \mathcal A_{b,\ell}. Then F_{b,\ell}(s) converges locally uniformly for \operatorname{Re}(s)>1/2.
The corollary covers the open half-plane only. The boundary \operatorname{Re}(s)=1/2 remains open.
Proposition 10 (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 active L(s,\chi) 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,\chi) 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. ◻
Proposition 10 is separate from Theorem 8. 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_{\chi\in\mathcal A_{b,\ell}} c_\chi\operatorname{ord}_{\rho}L(s,\chi). Here \operatorname{ord}_{\rho}L(s,\chi) is the multiplicity of the zero at \rho, and it is zero when L(\rho,\chi)\ne0.
Theorem 11 (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 (16) 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_{\chi\in\mathcal A_{b,\ell}} c_\chi\frac{L'}{L}(s,\chi) =F'_{b,\ell}(s) +\sum_{\chi\in\mathcal A_{b,\ell}}c_\chi H'(s,\chi) +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. By the identity theorem, (27) continues as a meromorphic identity throughout that half-plane.
At a zero \rho, the residue of L'/L(s,\chi) is \operatorname{ord}_{\rho}L(s,\chi). The right side of (27) has no pole. Its residue is zero, so the weighted sum of the residues on the left is zero. This is (26). ◻
Corollary 12 (An isolated active zero cannot hide). Let \chi_0\in\mathcal A_{b,\ell} and let \chi_0^* be its primitive inducing character. Suppose \operatorname{Re}(\rho)>1/2, that L(\rho,\chi_0^*)=0, and that no other primitive L-function underlying an active character vanishes at \rho. Then F_{b,\ell}(\sigma) cannot converge at any real \sigma\in[1/2,\operatorname{Re}(\rho)).
Proof. In the half-plane \operatorname{Re}(s)>1/2, the imprimitive L(s,\chi_0) differs from L(s,\chi_0^*) only by Euler factors whose zeros lie on \operatorname{Re}(s)=0. Their zero multiplicities at \rho are therefore equal. The zero contributes the nonzero term c_{\chi_0}\operatorname{ord}_{\rho}L(s,\chi_0) to (25). No other active channel is present to cancel it, so Theorem 11 rules out convergence. ◻
The obstruction is collective. A shared zero can pass through the ledger only when the collision-weighted sum of its multiplicities is zero. This is the exact cancellation that ordinary convergence requires.
Finite prime windows
The numerical question is not whether a single truncated sum looks flat. The useful finite object is the contribution made by a declared prime window. Table 1 records the last decimal window in a deterministic finite computation performed with nfield [6] 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 |
The same base-ten window has a visible separation by the prime’s true residue modulo three. The 292{,}963 primes congruent to one contribute 0.0002534203 at s=1, while the 293{,}118 primes congruent to two contribute -0.0001883766. The first contribution remains positive and the second remains negative at all six tested exponents. These are genuine prime classes modulo 300, using exactly the fiber-centered coefficients defined above. The opposed signs are a finite observation.
Do the two true residue-three channels carry independent principal terms, or is their opposition carried entirely by a nonprincipal twist?
nfield [6] constructs the finite table from (7) and uses the exact means from (9). It verifies the zero sum of every reduction fiber and the odd reflection of every centered class. It checks 468 direct collision counts against the finite table and then evaluates every prime beyond the relevant square modulus through 10^7. Every mean comes from a complete reduction fiber.
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. nfield supplies the complete checkpoint and window ledger for all six exponents.
The boundary at one half
Let \sigma_{b,\ell} be the abscissa of ordinary convergence of (16). Proposition 6 gives \sigma_{b,\ell}\leq1. Theorem 8 moves that boundary left when the active L-functions have a common zero-free half-plane. Theorem 11 moves in the opposite direction. Every zero with a nonzero collision ledger is a barrier that the ordinary prime series cannot cross.
Conjecture 13 (Collision boundary). If f_{b,\ell} is not identically zero, then \sigma_{b,\ell}=\frac12.
GRH for the active primitive characters gives only the upper bound in this conjecture. The reverse bound requires a collision-visible critical-line obstruction. It is not supplied by finite computation, and it is not implied by the zero-free penetration theorem.
The finite geometry now reaches the zero question at one exact point. The collision transform determines which character channels are present and how strongly each is weighted. Ordinary convergence then forces the zeros carried by those channels to balance in the ledger (25). An isolated active zero cannot disappear into the sum. Any deeper conclusion must explain why the remaining weighted cancellations can or cannot occur.
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 Centered Collision Sum, research note, October 2023, revised August 2026. doi:10.5281/zenodo.21852556.
[5]A. S. Petty, The Collision Periodic Table, research note, December 2023, revised August 2026. doi:10.5281/zenodo.21853017.
[6]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield