Still preparing the reader…Long documents can take a little longer to load and typeset.
Still preparing the reader…Long documents can take a little longer to load and typeset.
Still preparing the reader…Long documents can take a little longer to load and typeset.
At fixed base and lag, the collision deviation is determined by a finite table on the units modulo a power of the base. Centering each reduction fiber makes the table odd under reflection and gives every fiber sum zero. There are then two different ways to speak of a residue modulo three. One belongs to the canonical representative inside the finite table. The other belongs to the integer, or prime, represented by that class. The quotient layer makes the distinction exact.
Reflection fixes one internal residue-three slice. When the slice is nonempty, its uncentered mean is exactly negative one half, and its centered sum is zero. On the joint modulus, each true nonzero residue class modulo three contains an exact copy of the centered collision table when three does not divide the base. When three divides the base, each true residue class is a union of zero-sum reduction fibers. Thus both true prime channels are independently neutral in every base and at every lag.
Each residue-three reciprocal-prime series therefore converges at one by Mertens’ theorem in arithmetic progressions. At every finite cutoff, the two channels are the half-sum and half-difference of the original collision sum and its nontrivial character twist modulo three. Opposed finite signs measure a surviving twist rather than two principal biases waiting to cancel. Below one, the remaining question is the prime-weighted nonprincipal fluctuation.
The Chinese remainder theorem, finite character orthogonality, and Mertens’ theorem in arithmetic progressions are classical. The collision-specific result is the exact quotient-layer separation and the independent vanishing of the principal coefficient in each true residue-three channel.
In the base-ten prime window 10^6<p\leq10^7, the centered collision contributions at s=1 are +0.0002534 for primes congruent to one modulo three and -0.0001884 for primes congruent to two. The channels point in opposite directions, yet neither carries a principal bias. Each is neutral before the channels meet.
The question is how a visible finite imbalance can sit inside exact independent neutrality. The finite table sees a\bmod m. The true residue of an integer p=mt+a modulo three also depends on the quotient t. On the joint modulus, fixing that true residue either leaves a complete copy of the finite collision table or selects a union of complete zero-sum fibers. Both routes force the principal coefficient to vanish.
The finite table and its centering are developed in The Centered Collision Sum [1]. The character expansion in The Collision Transform and the Critical Strip [2] identifies the zero principal coefficient of the unsplit sum. The question here is whether it still vanishes after selecting either true residue class modulo three.
When the base generates the units modulo a prime, the collision count is periodic Hamming agreement on a reciprocal digit word. Lempel and Greenberger give the general Hamming-correlation setting [3], while Kak and Chatterjee study Hamming distance between prime-reciprocal decimal sequences and their cyclic shifts [4]. Neighboring number-theoretic work measures digit values along multiplicative orbits. Girstmair relates full-period digit variance to Dedekind sums [5]. Murty and Thangadurai treat digit means on possibly proper orbits through generalized Bernoulli numbers attached to odd characters [6].
The statistic here is an equality indicator on the complete nonzero residue system. Neither primality nor a primitive-root hypothesis is required for its finite table. The Chinese remainder theorem and finite character orthogonality used below are classical [7, 9]. The new finite statement is narrower. It separates the representative’s internal residue from the integer’s true residue and proves that both true residue-three channels are independently centered-neutral. Mertens’ theorem then transports that finite cancellation to reciprocal primes.
Fix a base b\geq2 and a lag \ell\geq1. Let p\geq2 be an integer coprime to b. For 1\leq r<p, put \delta_{p,b}(r)=\left\lfloor\frac{br}{p}\right\rfloor. For x\not\equiv0\pmod p, let [x]_p denote its representative in \{1,\ldots,p-1\}, and define the collision count C_{b,\ell}(p)= \#\left\{1\leq r<p\ \middle| \delta_{p,b}(r)=\delta_{p,b}([b^\ell r]_p)\right\} and its bounded deviation S_{b,\ell}(p)=C_{b,\ell}(p) -\left\lfloor\frac{p-1}{b}\right\rfloor.
Set m=b^{\ell+1} and let G_{b,\ell}= \left\{0\leq n<m\ \middle| \left\lfloor\frac{n}{b^\ell}\right\rfloor=n\bmod b\right\}. The first and last base-b digits agree on this set, while the middle digits are free. Hence |G_{b,\ell}|=b^\ell.
Theorem 1 (Finite determination). Let p>m and \gcd(p,b)=1. Write p=mt+a with 1\leq a<m. 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}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor \right). Thus the collision deviation is determined by p\bmod m.
Proof. For 1\leq r<p, set n(r)=\lfloor mr/p\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_{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 occurs exactly on the slices indexed by G_{b,\ell}.
No interior slice boundary is integral because p is coprime to m. The floor difference \left\lfloor\frac{(n+1)p}{m}\right\rfloor -\left\lfloor\frac{np}{m}\right\rfloor counts the positive residues in the nth slice. The terminal slice also counts the excluded endpoint p, and m-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}{m}\right\rfloor -\left\lfloor\frac{np}{m}\right\rfloor \right). Substitution of p=mt+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 exact. Once b and \ell are fixed, every collision deviation beyond m is already present in its unit classes.
Let U_m=(\mathbb Z/m\mathbb Z)^\times. Every element will be written as its canonical representative in \{1,\ldots,m-1\}. Reflection gives the first exact constraint on the table.
Lemma 2 (Reflection). For every a\in U_m, T_{b,\ell}(a)+T_{b,\ell}(m-a)=-1. Consequently, \frac1{\varphi(m)}\sum_{a\in U_m}T_{b,\ell}(a)=-\frac12.
Proof. Write D_n(a)= \left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor. For 1\leq n\leq m-2, complementary floors give D_n(a)+D_n(m-a)=1. The corresponding sums at n=0 and n=m-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(m-a)\bigr)=b^\ell. Since a is a unit modulo b, \left\lfloor\frac ab\right\rfloor +\left\lfloor\frac{m-a}{b}\right\rfloor=b^\ell-1. Equation (8) follows from (7). Negation has no fixed point on U_m. Averaging the reflection identity proves (9). ◻
Let \rho\colon U_m\longrightarrow U_b be reduction modulo b. For u\in U_b, write A_u=\{a\in U_m\mid \rho(a)=u\}. Every lift u+kb remains coprime to b, so each fiber contains exactly b^\ell elements. Define its mean by \mu_{b,\ell}(u)= \frac1{b^\ell}\sum_{a\in A_u}T_{b,\ell}(a) and define the fiber-centered table by f_{b,\ell}(a)=T_{b,\ell}(a)-\mu_{b,\ell}(\rho(a)). This centering is performed on the complete finite table before any integer or prime is sampled.
Proposition 3 (Centered symmetry). For every u\in U_b and every a\in U_m, \begin{aligned} \mu_{b,\ell}(u)+\mu_{b,\ell}(-u)&=-1, \\ f_{b,\ell}(m-a)&=-f_{b,\ell}(a), \\ \sum_{a\in A_u}f_{b,\ell}(a)&=0. \end{aligned} In particular, \sum_{a\in U_m}f_{b,\ell}(a)=0.
Proof. Negation maps A_u bijectively onto A_{-u}. Averaging (8) over A_u proves (12). Subtracting that identity from the pointwise reflection law proves (13). Equation (14) is the definition of the fiber mean. Summing the fiber identities proves (15). ◻
Suppose first that 3\nmid m. The canonical representative a of a class modulo m has an internal label a\bmod3. An integer in that class has the form p=mt+a, \qquad p\equiv a+mt\pmod3. Its true residue modulo three therefore depends on the quotient layer t as well as the representative a. Since m is invertible modulo three, the three possible quotient layers move a fixed a through all three residues modulo three.
The distinction is visible without asymptotics. In base ten at lag one, the modulus is 100. The prime 101 has canonical representative 1. The representative is congruent to one modulo three, while the prime is congruent to two. The representative and the prime therefore occupy different residue-three coordinates in (16).
The collision table retains the internal coordinate. The joint modulus carries the true one.
Assume 3\nmid m. For j\in\mathbb Z/3\mathbb Z, define the internal canonical slice I_j= \left\{a\in\{1,\ldots,m-1\}\ \middle| \gcd(a,m)=1,\ a\equiv j\pmod3\right\}.
Theorem 4 (Internal reflection neutrality). Let j_* be the unique class satisfying 2j_*\equiv m\pmod3. Reflection preserves I_{j_*} and exchanges the other two internal slices. On the fixed slice, \begin{aligned} 2\sum_{a\in I_{j_*}}T_{b,\ell}(a)&=-|I_{j_*}|, \\ \sum_{a\in I_{j_*}}f_{b,\ell}(a)&=0. \end{aligned} When I_{j_*} is nonempty, (19) says that its uncentered mean is -1/2. If j\ne j_* and j'\equiv m-j\pmod3, then \begin{aligned} |I_j|&=|I_{j'}|, \\ \sum_{a\in I_j}T_{b,\ell}(a) +\sum_{a\in I_{j'}}T_{b,\ell}(a)&=-|I_j|, \\ \sum_{a\in I_j}f_{b,\ell}(a) +\sum_{a\in I_{j'}}f_{b,\ell}(a)&=0. \end{aligned}
Proof. Reflection sends the internal class j to m-j modulo three. Its fixed class is exactly the solution of (18). Summing (8) over the fixed slice gives 2\sum_{a\in I_{j_*}}T_{b,\ell}(a)=-|I_{j_*}|. Summing (13) over the same slice gives 2\sum_{a\in I_{j_*}}f_{b,\ell}(a)=0. These are (19) and (20). On either exchanged slice, reflection is a bijection onto the other. Summing the raw and centered reflection identities gives (21), (22), and (23). ◻
Whenever the fixed slice is present, its neutrality is real and exact. It belongs to the chosen representatives inside the collision table. It is distinct from a true prime congruence class, which is encoded on the joint modulus.
Put M=\operatorname{lcm}(m,3) and lift the centered table to U_M by \widetilde f(A)=f_{b,\ell}(A\bmod m). For r\in\{1,2\}, let B_r=\{A\in U_M\mid A\equiv r\pmod3\}. These are the true nonzero residue classes seen by primes greater than three.
Theorem 5 (Independent neutrality). For every base b\geq2, every lag \ell\geq1, and each r\in\{1,2\}, \sum_{A\in B_r}\widetilde f(A)=0. If 3\nmid b, then the uncentered mean in each true channel is also the global mean \frac1{|B_r|} \sum_{A\in B_r}T_{b,\ell}(A\bmod m)=-\frac12.
Proof. Suppose first that 3\nmid b. Then 3\nmid m and M=3m. The Chinese remainder theorem gives U_M\cong U_m\times U_3. For fixed r, projection from B_r onto U_m is a bijection. Hence \sum_{A\in B_r}\widetilde f(A) =\sum_{a\in U_m}f_{b,\ell}(a)=0 by (15). The same bijection and (9) prove (28).
Now suppose that 3\mid b. Then M=m. For each r, the set B_r is the union of the reduction fibers A_u for which u\equiv r\pmod3. Every one of those fibers has centered sum zero by (14). Their union therefore has centered sum zero. ◻
The two arithmetic cases have the same outcome for different reasons. When three does not divide the base, each true channel contains one copy of every collision class. When three divides the base, each true channel is a union of complete reduction fibers. Global centering settles the first case, and fiber centering settles the second.
Corollary 6 (No principal channel). For r\in\{1,2\}, define a function on U_M by h_r(A)= \begin{cases} \widetilde f(A),&A\in B_r,\\ 0,&A\notin B_r. \end{cases} The principal Dirichlet-character coefficient of h_r is zero.
Proof. The principal coefficient is \frac1{\varphi(M)}\sum_{A\in U_M}h_r(A) =\frac1{\varphi(M)}\sum_{A\in B_r}\widetilde f(A), which vanishes by Theorem 5. ◻
Each true channel therefore has zero principal prime term. The residue-three component that remains is nonprincipal.
Theorem 7 (Separate convergence at one). For each r\in\{1,2\}, the limit \lim_{X\to\infty} \sum_{\substack{M<p\leq X\\p\equiv r\pmod3}} \frac{f_{b,\ell}(p\bmod m)}p exists.
Proof. Mertens’ theorem in arithmetic progressions for the fixed modulus M [8, 9] gives, for every A\in U_M, \sum_{\substack{p\leq X\\p\equiv A\pmod M}}\frac1p =\frac1{\varphi(M)}\log\log X+C_M(A)+o(1). Multiply by \widetilde f(A) and sum over A\in B_r. The coefficient of \log\log X is zero by (27). Only a finite sum of constants and vanishing error terms remains. Omitting the finitely many primes at or below M does not affect convergence. ◻
Let X>M be finite, let s\in\mathbb C, and define \begin{aligned} F_X(s)&= \sum_{M<p\leq X}\frac{f_{b,\ell}(p\bmod m)}{p^s}, \\ F_{r,X}(s)&= \sum_{\substack{M<p\leq X\\p\equiv r\pmod3}} \frac{f_{b,\ell}(p\bmod m)}{p^s}, \\ G_X(s)&= \sum_{M<p\leq X} \frac{f_{b,\ell}(p\bmod m)\chi_3(p)}{p^s}, \end{aligned} where \chi_3 is the nontrivial character modulo three. Thus \chi_3(n) is 1 on the class one, -1 on the class two, and 0 on multiples of three.
Proposition 8 (Exact channel split). For every finite cutoff and every s, F_{r,X}(s)=\frac12 \bigl(F_X(s)+\chi_3(r)G_X(s)\bigr). Equivalently, F_X=F_{1,X}+F_{2,X}, \qquad G_X=F_{1,X}-F_{2,X}.
Proof. Every prime in the sums is greater than three, and \mathbf1_{p\equiv r\bmod3} =\frac12\bigl(1+\chi_3(r)\chi_3(p)\bigr). Multiplication by the collision coefficient and summation proves (35). The sum and difference identities follow. ◻
The split leaves a genuine mod-three object, but it is a twist rather than a mean. At s=1, Theorem 7 shows that the total and twisted sums both converge. The finite split remains exact for every s.
The finite calculation in Table 1 was performed with nfield [10] in base ten at lag one. It records the primes satisfying 10^6<p\leq10^7 and keeps their true residue modulo three. There are 292{,}963 primes in class one and 293{,}118 in class two. The two channel contributions have opposite signs at every displayed exponent. Their sum is the unsplit collision contribution, and their difference is the twist.
| s | class one | class two | total | twist |
|---|---|---|---|---|
| 1.0 | +0.0002534 | -0.0001884 | +0.0000650 | +0.0004418 |
| 0.9 | +0.0009126 | -0.0008658 | +0.0000469 | +0.0017784 |
| 0.8 | +0.0031182 | -0.0039858 | -0.0008676 | +0.0071040 |
| 0.7 | +0.0097059 | -0.0183773 | -0.0086714 | +0.0280832 |
| 0.6 | +0.0245861 | -0.0848446 | -0.0602585 | +0.1094307 |
| 0.5 | +0.0256653 | -0.3921645 | -0.3664992 | +0.4178298 |
For every base from two through forty at lag one, nfield rebuilds the finite table directly, then checks the fiber sums, reflection, and the two true residue channels. It also reproduces Table 1 from the declared prime window. The symbolic argument proves the identities, while the computation certifies these stated finite ranges and values.
Neutrality settles the principal term. It does not settle the size or sign of the remaining nonprincipal twist. In particular, the zero mean of each true residue channel does not imply ordinary convergence of (33) at any real exponent below one.
The finite opposed signs in Table 1 do not determine their long-range course. They may decay, persist through long ranges, or reverse. A proof below one requires control of the prime-weighted character fluctuations themselves. Vanishing of the principal coefficient supplies no such control.
Internal reflection and true prime neutrality live on different finite spaces. Reflection acts on canonical representatives modulo m. True residue-three channels live on the joint modulus \operatorname{lcm}(m,3). Once those coordinates are separated, each channel has zero principal coefficient on its own. The surviving object is the twisted collision fluctuation.
The transport from zero class means to reciprocal-prime convergence is classical Mertens theory. The content supplied here is the finite collision geometry that makes both true residue-three means vanish separately. The opposed signs in a finite window are therefore not two principal biases whose sum happens to be small. They are the two faces of a nonprincipal twist. A remaining structural question is whether the same independent neutrality persists for arbitrary auxiliary primes and for several congruence coordinates imposed at once.
[1]A. S. Petty, The Centered Collision Sum, research note, October 2023. https://doi.org/10.5281/zenodo.21852556
[2]A. S. Petty, The Collision Transform and the Critical Strip, research note, January 2024. https://doi.org/10.5281/zenodo.21853435
[3]A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inform. Theory 20 (1974), no. 1, 90–94. https://doi.org/10.1109/TIT.1974.1055169
[4]S. C. Kak and A. Chatterjee, On decimal sequences, IEEE Trans. Inform. Theory 27 (1981), no. 5, 647–652. https://doi.org/10.1109/TIT.1981.1056394
[5]K. Girstmair, Digit variance and Dedekind sums, J. Number Theory 65 (1997), no. 2, 197–205. https://doi.org/10.1006/jnth.1997.2149
[6]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
[7]A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9780511626265
[8]K. S. Williams, Mertens' theorem for arithmetic progressions, J. Number Theory 6 (1974), no. 5, 353–359. https://doi.org/10.1016/0022-314X(74)90032-8
[9]H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000.
[10]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
Discussion
Sign in to join the discussion.