The Two-Valued Selector

A prime Ramanujan sum takes only two values, yet those values can isolate one line from an entire two-dimensional spectrum. Applied to the digit partition of long division, the selected line counts exactly how many residues keep their digit under multiplication. A classical arithmetic kernel therefore becomes a finite gate for phase survival.

Long division puts two finite structures on the nonzero residues. Multiplication moves a residue through its orbit, while the first displayed digit places it in an interval. A collision occurs when the multiplication leaves that interval label unchanged.

The same-bin cross-spectrum and its multiplicative line slice are developed in The Autocorrelation Formula [1]. Here a Ramanujan sum detects the linear relation between those two Fourier coordinates. For a prime modulus it takes only two values. The total spectral mass vanishes at the excluded zero residue, so the two-valued kernel cancels everything away from one line and leaves the collision count.

Ramanujan sums also occur in classical Dirichlet series. For fixed n\ge1 and \operatorname{Re}(s)>1, one has \sum_{q=1}^{\infty}\frac{c_q(n)}{q^s} =\frac{\sigma_{1-s}(n)}{\zeta(s)} by the classical theory of Ramanujan and Hardy [2, 3]. Equation (1) gives the analytic context. In the collision identity below, the prime Ramanujan sum instead serves as a finite selector on a linear form in two frequency variables.

The Ramanujan Collision Identity

Let \mathbb F_p be the field of residues modulo p, represented by 0,1,\ldots,p-1, and put e_p(t)=\exp\!\left(\frac{2\pi i t}{p}\right). For x\in\{1,\ldots,p-1\} define the digit map \delta_{p,b}(x)=\left\lfloor\frac{bx}{p}\right\rfloor. For 0\le d<b, let B_d=\{x\in\mathbb F_p^*\mid \delta_{p,b}(x)=d\}, \qquad f_d=\mathbf 1_{B_d}, where each f_d is extended to \mathbb F_p by f_d(0)=0. Its additive Fourier transform is \widehat f_d(k)=\sum_{x\in\mathbb F_p}f_d(x)e_p(-kx).

The same-bin kernel and its two-dimensional transform are K(x,y)=\sum_{d=0}^{b-1}f_d(x)f_d(y) and G(k,k') =\sum_{x,y\in\mathbb F_p}K(x,y)e_p(-kx-k'y) =\sum_{d=0}^{b-1}\widehat f_d(k)\widehat f_d(k'). For a\in\mathbb F_p^* define the multiplicative collision count C_{p,b}(a) =\sum_{x\in\mathbb F_p^*}K(x,ax) =\bigl|\{x\in\mathbb F_p^*\mid \delta_{p,b}(x)=\delta_{p,b}(ax)\}\bigr|.

For m\in\mathbb F_p, the prime Ramanujan sum is c_p(m)=\sum_{x\in\mathbb F_p^*}e_p(mx) = \begin{cases} p-1 & m=0,\\ -1 & m\ne0. \end{cases}

Theorem 1 is the precise Ramanujan connection. The kernel acts on the linear form k+ak' in the full two-dimensional spectrum.

The Phase Gate

Put \Sigma_{p,b}(a)=\sum_{k\in\mathbb F_p}G(k,-a^{-1}k) and M_{p,b}(a) =\sum_{d=0}^{b-1}\sum_{k\in\mathbb F_p} |\widehat f_d(k)|\,|\widehat f_d(-a^{-1}k)|.

The denominator is positive. Indeed, the term k=0 contributes \sum_d|B_d|^2, and the bins partition p-1 nonzero residues.

The coefficient \Gamma_{p,b}(a) is the normalized fraction of unsigned spectral mass that survives cancellation along one slice. The spectral-slice identity ties that surviving mass exactly to the collision count.

Repetend Autocorrelation

Assume that b is a primitive root modulo p. The remainders in the long division of 1/p are 1,b,b^2,\ldots,b^{p-2}\pmod p. The corresponding digit word is d_j=\delta_{p,b}(b^j\bmod p), \qquad 0\le j<p-1. For a cyclic lag \ell\in\mathbb Z/(p-1)\mathbb Z define R_{p,b}(\ell) =\bigl|\{j\bmod(p-1)\mid d_j=d_{j+\ell}\}\bigr|.

Call p digit-partitioning in base b when \delta_{p,b} is injective on \mathbb F_p^*.

For singleton bins, the full transform is the Ramanujan kernel.

The Euclidean Bin Word

Write p=bq+r, \qquad 1\le r<b. The remainder is nonzero because p\nmid b. Put n_d=|B_d|.

The binary word (w_0,\ldots,w_{b-1}) will be called the excess-bin word. It depends only on b and the remainder r. Different remainders give different words because the number of 1 entries is r-1.

In base 10, primes different from 2 and 5 lie in four unit residue classes with the following coarse data.

For a general base b, every prime p\nmid b lies in one of the \varphi(b) unit residue classes modulo b. These classes label the coarse excess pattern; the quotient and modulus supply the remaining spectral data.

The Euclidean remainder fixes the coarse excess pattern. The quotient and the modulus remain visible in the Fourier data. A continued-fraction tail organizes the first layer of the geometry; the phase gate retains the full quotient and modulus data.

Collision-Free Multipliers

The zero set of the phase gate has a direct combinatorial form.

This converts spectral cancellation into a finite disjointness problem.

The Finite Boundary of the Gate

The collision count is a line selected from a two-dimensional spectrum. The prime Ramanujan sum performs that selection exactly. Its two values remove the off-line mass and leave the spectral slice whose normalized surviving mass is the phase gate.

The Euclidean division p=bq+r supplies a second exact layer. The remainder fixes the excess-bin word and the continued-fraction tail, while the quotient remains visible even at zero frequency. Together they separate the coarse Euclidean label from the full phase data.

The finite arithmetic is available for direct exploration in the nfield repository [5].

The boundary is concrete. For fixed b and p, determine the multipliers that move every digit bin away from itself.

References