Fix a base b\geq2, a lag \ell\geq1, and put q=b^{\ell+1}. The collision deviation on the complete nonzero residue ensemble is one finite function on U_q. In the full-reptend prime case, its underlying count is cyclic-shift Hamming agreement for the digits of 1/p. Centering every reduction fiber removes the base-level signal, while reflection removes the even character channels. At prime base and lag one, only primitive odd channels can survive.
Sampling the centered table at primes gives a finite collision-weighted sum of prime character series. It converges at exponent one. A common bound on the real parts of the active Dirichlet L-function zeros gives ordinary convergence in the corresponding open half-plane. Conversely, convergence at a real point \sigma\geq1/2 forces every zero farther right to have zero total multiplicity after weighting by the collision transform. A zero in only one active primitive channel cannot cancel. The finite table selects the channels and weights. The analytic implication is an exact collective zero-cancellation obstruction, not a zero-location theorem.
What can a finite digit table say about a zero of a Dirichlet L-function? It cannot locate the zero. It can select the character channels in which the zero would have to appear and assign an exact weight to each one. Once the table is sampled at primes, ordinary convergence cannot pass such a zero unless the weighted multiplicities cancel.
The underlying observable already has a classical sequence interpretation. Coordinate agreement under cyclic shift belongs to Hamming correlation theory [1]. Kak and Chatterjee studied prime-reciprocal digit sequences as communication codes and obtained bounds for Hamming distance from cyclic shifts and for autocorrelation [2]. When b generates \mathbb F_p^\times, the lag collision count below is exactly that cyclic shift agreement count. The complete-residue construction retains the same observable without a primitive-root hypothesis and, for its finite table, without primality.
Digit values along reciprocal orbits lead to a different but neighboring theory. Girstmair expressed full-period digit variance through a Dedekind sum under a primitive-root hypothesis [3]. Murty and Thangadurai studied digit averages for bases of prescribed order using generalized Bernoulli numbers and Dirichlet L-functions [4]. The signal here is instead an equality indicator over every nonzero residue, centered within the fibers of reduction modulo the base.
Finite Fourier inversion and character orthogonality are standard [5]. Mertens’ theorem in arithmetic progressions, Euler-product logarithms, explicit-formula estimates, partial summation, and the residue of L'/L at a zero are also classical [6, 7]. The contribution specific to the collision table is its exact assembly. The all-lag finite formula and reflection law feed a literal fiber centering. Reflection eliminates every even channel. Fiber centering eliminates every channel inherited from the base. The surviving finite coefficients then weight both directions of the convergence problem.
One direction is sufficient. A common bound on the real parts of the active zeros gives ordinary convergence to its right. The other is necessary. Ordinary convergence makes every zero farther right disappear from the collision-weighted multiplicity ledger. The theorem identifies this exact collective obstruction while leaving zero location itself untouched.
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. For an integer x, let [x]_N be its least nonnegative residue modulo N. Since (N,b)=1, the residue [b^\ell r]_N below is nonzero. 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). ◻
Once b and \ell are fixed, the finite table contains every collision deviation beyond q.
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 takes the exact mean of each complete base-level digit family.
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. ◻
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 confines the transform support to the odd fine-scale channels. Within that sector, activity is decided by the coefficient c_\chi.
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).
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) [6]. The coefficient of the common term \varphi(q)^{-1}\log\log x 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.
The right side of (21) supplies an analytic continuation where compatible logarithm branches exist. Ordinary convergence of the prime series requires 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 [6, 7]. 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.
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.
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 [8].
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). The isolated hypothesis makes this the complete ledger contribution at \rho. Theorem 12 rules out convergence. ◻
The obstruction is collective. Distinct effective L-functions may share a zero, and their collision weights may cancel its multiplicity. The theorem identifies the exact cancellation that ordinary convergence requires.
A declared prime window is the useful finite object. Table 1 records the final decimal window in a deterministic calculation with nfield [9] 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 proofs are symbolic and independent of these calculations.
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 are bounded arithmetic records of the prime series.
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. The reverse inequality requires a collision-visible obstruction on the line. The results here control open half-planes, while the boundary line remains open.
The finite table and the analytic machinery have different responsibilities. The table determines which channels survive and the coefficient attached to each one. Standard prime-series and L-function theory transports those coefficients. It gives unconditional convergence at one, conditional convergence in a common zero-free half-plane, and a necessary cancellation law in the reverse direction.
The last law is the point at which the finite arithmetic becomes visible at a zero. If the prime series converges, every zero to its right must cancel in (25). An isolated effective zero cannot hide. Shared zeros remain the exact unresolved case because several collision weights may meet there with total multiplicity zero.
Conjecture 14 therefore does not follow from the finite transform. Equality at one half requires both an upper and a lower bound on the abscissa. A common zero-free theorem for the active channels would give the upper bound. A collision-visible obstruction on the boundary would give the lower bound. A finite digit table selects the channels, and ordinary convergence is possible past their zeros only through the stated weighted cancellation.
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[4]M. R. Murty and R. Thangadurai, The class number of \mathbb{Q}(\sqrt{-p}) and digits of 1/p, Proc. Amer. Math. Soc. 139 (2011), no. 4, 1277–1289. https://doi.org/10.1090/S0002-9939-2010-10560-9
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[9]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
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