The Fixed Diagonal

In base five, the two-digit strings 00,11,22,33,44 form a diagonal inside the twenty-five possible strings. Their values are 0,6,12,18,24. The changes in a character across these five positions determine a finite factor in the collision spectrum. An analytic lift of that factor lets us ask how the digit geometry enters a family of functions, and whether the original collision coefficient can be recovered from it.

The construction uses the classical periodic zeta function as its kernel. Its boundary values supply the Bernoulli factor in the finite coefficient. Throughout the open critical strip, its primitive odd character transform supplies a Dirichlet L-function. The digit diagonal stays fixed as the complex parameter changes.

Coordinate agreement under cyclic shift belongs to classical Hamming correlation theory [1]. Kak and Chatterjee studied Hamming distance and autocorrelation for decimal prime-reciprocal sequences and their cyclic shifts [2]. A neighboring literature studies digit values rather than equality indicators. Girstmair expressed full-period digit variance through a Dedekind sum [3], while Murty and Thangadurai used generalized Bernoulli numbers and Dirichlet L-functions for digit averages on multiplicative subgroups  [4].

The analytic machinery used here is also classical. Lerch’s functional equation and the periodic-zeta continuation are treated by Apostol [5]. The Hurwitz-zeta decomposition, Gauss sums, generalized Bernoulli special values, and the functional equation of a primitive Dirichlet L-function are standard [6, 7].

The result specific to this manuscript is their exact assembly around the leading-to-trailing digit diagonal. Endpoint pairing removes the singular kernel argument. Primitive conductor cancellation removes the nonunit fibers. The remaining character transform separates into the fixed oriented boundary D_\ell(\chi) and one Dirichlet L-function. At lag one, the finite collision coefficient is proved below to be its left boundary value divided by i\pi.

For a power of the base, the diagonal consists of the residues whose leading base-b digit equals their trailing digit. A Dirichlet character measures its oriented boundary through the differences \chi(n+1)-\chi(n).

The periodic zeta function supplies an analytic kernel for every slice of that boundary. The kernel is placed on the finite diagonal before the character transform is taken. Primitive character cancellation then removes the nonunit fibers, and the surviving transform factors into two pieces. One is the finite diagonal sum. The other is a Dirichlet L-function with its standard gamma and sine factors.

The factorization has an exact anchor. At lag one, the value at s=0 is the centered finite collision coefficient multiplied by i\pi. The equality preserves the full complex coefficient with its character index. The finite coefficient is therefore a boundary value of the analytic family, not a numerical resemblance to it.

The common gamma-sine prefactor never vanishes in the open strip. An active transform therefore has the same zero divisor as its Dirichlet channel. This conclusion follows from the factorization. It neither locates a zero nor supplies a new zero-free region.

The Finite Boundary

Fix an odd prime b and put m=b^2, \qquad U_m=(\mathbb Z/m\mathbb Z)^\times. Euler’s totient is denoted by \varphi. Dirichlet characters are extended by zero on nonunits. The lag-one digit diagonal is G_1 = \left\{0\le n<m: \left\lfloor\frac nb\right\rfloor=n\bmod b\right\} = \{r(b+1):0\le r\le b-1\}. For a character \chi modulo m, define its direct diagonal boundary D_1(\chi) = \sum_{n\in G_1} \bigl[\chi(n+1)-\chi(n)\bigr].

For a\in U_m and n\in G_1, put d_n(a) = \left\lfloor\frac{(n+1)a}{m}\right\rfloor - \left\lfloor\frac{na}{m}\right\rfloor. The finite collision function is S(a) = -1-\left\lfloor\frac ab\right\rfloor +\sum_{n\in G_1}d_n(a). For 1\le r\le b-1, let U_r=\{a\in U_m:a\equiv r\pmod b\}, \qquad \overline S_r=\frac1b\sum_{a\in U_r}S(a). Define S^\circ(a)=S(a)-\overline S_{a\bmod b} and \widehat S^\circ(\chi) = \frac1{\varphi(m)} \sum_{a\in U_m}S^\circ(a)\overline\chi(a). For any modulus q and any nonprincipal character \psi modulo q, the classical generalized-Bernoulli special-value formula [7] gives B_{1,\psi} = \frac1q\sum_{a=1}^{q}a\psi(a) = -L(0,\psi).

The Periodic Zeta Lift

Now let b be any prime, let \ell\ge 1, and put m=b^{\ell+1}. The leading-to-trailing digit diagonal is G_\ell = \left\{n\in\{0,\ldots,m-1\}: \left\lfloor\frac{n}{b^\ell}\right\rfloor=n\bmod b\right\}. Both 0 and m-1 belong to G_\ell. Define D_\ell(\chi) = \sum_{n\in G_\ell} \bigl[\chi(n+1)-\chi(n)\bigr].

For every integer r\not\equiv0\pmod m, let \rho_m(r) be the least positive representative of r modulo m. For 0<x<1, define K(x;s)=F(x,1-s), \qquad F(x,z)=\sum_{n=1}^{\infty}\frac{e^{2\pi i n x}}{n^z}. The series defining F begins in its half-plane of convergence and continues analytically. Substitution in the classical periodic-zeta connection formula [5][7] gives, first for \operatorname{Re}(s)<0 and then by analytic continuation, K(x;s) = \frac{\Gamma(s)}{(2\pi)^s} \left[ e^{i\pi s/2}\zeta(s,x) + e^{-i\pi s/2}\zeta(s,1-x) \right]. For 0<x<1, Dirichlet convergence makes K(x;s) regular at s=0. At s=1, the two Hurwitz residues cancel. The removable value is K(x;1) = -\frac12+\frac{i}{2}\cot(\pi x).

The Hurwitz representation is singular at x=0. The two endpoint slices are paired before the transform is taken.

Primitive Character Cancellation

The Analytic Transform

Recovery at the Left Boundary

The factor i\pi is universal across the primitive odd channels. The analytic family reaches the finite collision coefficient directly at the left boundary.

The Opposite Boundary

For a primitive character modulo m, define the Gauss sum \tau(\chi) = \sum_{r\,(\mathrm{mod}\,m)} \chi(r)e^{2\pi i r/m}.

The right-boundary value is exact, but it is not the finite collision coefficient. The Gauss sum and the opposite Bernoulli index record the functional equation at the other edge of the strip.

Active Channels

The theorem identifies the active analytic channels exactly at a fixed conductor. Their complete open-strip zero divisor is preserved with multiplicity. This preservation is algebraic after the transform has factored. The location of those zeros remains a separate problem.

Across Conductors

The periodic zeta function, its Hurwitz representation, character orthogonality, generalized Bernoulli values, Gauss sums, and the Dirichlet functional equation are inherited analytic machinery. The finite datum introduced here is the oriented boundary of a digit-equality diagonal. The main identity shows that this datum remains separate from the moving L-function factor throughout the strip.

At lag one, the left boundary recovers the full complex collision coefficient, not only its magnitude. At the opposite boundary, the functional equation supplies a different Bernoulli index and its Gauss twist. In the interior, equality of zero divisors is the exact consequence of multiplying by factors that do not vanish there. It offers no new constraint on those zeros.

The unresolved question concerns comparison across conductors. The diagonal D_\ell(\chi) changes with the base, lag, and character. A comparison of those finite factors would require estimates that are absent from the fixed-conductor identity. The theorem proved here establishes the carrier and its two boundary values. It leaves conductor-uniform control and zero location open.

References