The finite secondary problem

The lag-one digit-collision energy studied here is the square mass of a class-centered response on prime-square residues. Its leading order is cubic, while its first correction sits two powers lower and carries two logarithms. The definitions and exact finite identity are reconstructed below. The continuous response predicts the coefficient -1/\pi^2, but the prime-square carrier grid could still contribute at exactly the same scale. The question is whether the correction -\pi^{-2}p^2(\log p)^2 survives finite sampling unchanged.

The distinction is structural. A coefficient altered by the grid would belong partly to the sampling scheme. A coefficient that survives belongs to the collision law itself. The floor weights retain the entire discrepancy. Each downward rounding selects a prefix of an inverse-residue sequence. A single prefix can spike, and a single carrier can retain one sign. The proof keeps the additive phase within each descended carrier, then uses a bounded mean for the divisor burden across carriers.

The continuous response has a classical form. Put A(N)=\sum_{r,s\le N}\frac{(r,s)}{[r,s]}, where (r,s) and [r,s] denote the greatest common divisor and least common multiple. Since (r,s)/[r,s]=(r,s)^2/(rs), the Bernoulli square capacity below is A(N)/3. Hilberdink, Luca, and Tóth proved A(N)=3N+O\bigl((\log N)^2\bigr) in [3]. The continuous argument here determines the leading correction at that scale and gives A(N)=3N-\frac{3}{\pi^2}(\log N)^2 +O\bigl((\log N)^{5/3}\bigr). The finite question is then precise. One must show that sampling the Bernoulli response on the prime-square carrier grid does not create another quadratic logarithm. The strict all-prime saving proved below places the sampling defect beneath that scale and transfers the coefficient unchanged to the digit-collision energy.

The surrounding analytic tools are classical. Saffari and Vaughan provide the distribution input for fractional parts of N/k [8]. Rademacher and Grosswald supply the reciprocity framework [7]; Vardi and Conrey, Fransen, Klein, and Scott place Dedekind sums in Kloosterman and mean-value theory [9, 1]; Lemke Oliver and Soundararajan study the relevant additive Fourier transform at prime moduli [6]; and Korolev supplies the incomplete inverse-residue estimate used after modulus descent [5]. The results proved here are the explicit coefficient left unresolved by the published square-box error term, two exact descriptions of the carrier-grid defect, and a uniform saving that proves the grid does not change that coefficient.

Fix an odd prime p, put N=p-1 and m=p^2, and define the lag-one collision diagonal G_p=\{d(p+1):0\le d<p\}. For 0\le a<m, let C_p(a) =\sum_{n\in G_p} \left( \left\lfloor\frac{(n+1)a}{m}\right\rfloor -\left\lfloor\frac{na}{m}\right\rfloor \right), \qquad F_p(a)=C_p(a)-\frac ap. The finite collision energy is E_p=\sum_{a=0}^{m-1}F_p(a)^2. Define \mathop{\mathrm{saw}}(x)= \begin{cases} \{x\}-\tfrac12,&x\notin\mathbb{Z},\\ 0,&x\in\mathbb{Z}, \end{cases} \qquad \mathcal{B}_p(x)=2\sum_{r=1}^{N}\mathop{\mathrm{saw}}(rx).

To identify the finite object with the digit-collision energy, write a unit as a=pq+s with 1\le s<p. Its class-centered response is C_p(a)-q-\frac sp=C_p(a)-\frac ap=F_p(a). If a=pc, then the defining floor sum telescopes to C_p(pc) =\sum_{d=0}^{p-1} \left( \left\lfloor\frac{(d+1)c}{p}\right\rfloor -\left\lfloor\frac{dc}{p}\right\rfloor \right) =c, so F_p(pc)=0. Thus E_p is exactly the square mass of the class-centered finite response defined above.

The normalized lag-one energy is therefore exactly the carrier-grid average \mathcal{D}_p =\frac1m\sum_{A=0}^{m-1}\mathcal{B}_p(A/m)^2 =\frac{E_p}{p^2}. The L^2 Fourier series of the centered sawtooth gives \int_0^1\mathop{\mathrm{saw}}(rx)\mathop{\mathrm{saw}}(sx)\,dx =\frac{(r,s)^2}{12rs} \qquad (r,s\ge1). Summing this identity gives the continuous square capacity \mathcal{J}(N) =\int_0^1\mathcal{B}_p(x)^2\,dx =\frac13\sum_{1\le r,s\le N}\frac{(r,s)^2}{rs}. The GCD sum on the right defines \mathcal{J}(N) for every positive integer N. We study \mathcal{S}_p=\mathcal{J}(N)-\mathcal{D}_p.

The defect first appears in two exact forms. The geometric form uses the rational jump phases of the straightened boundary. The arithmetic form uses Rademacher reciprocity. Their equality identifies the quotient-shell reservoir with the sampling alias. The additive transform then opens every carrier into inverse-residue series and supplies the strict saving needed for finite transfer.

The continuous secondary coefficient

The leading estimate \mathcal{J}(N)=N+O((\log N)^2) is equivalent to the published GCD and LCM asymptotic of Hilberdink, Luca, and Tóth [3]. The argument below determines the first coefficient by retaining the fractional-floor term that their error bound leaves unresolved.

For k\ge2, put H_k^*=\sum_{\substack{1\le a<k\\(a,k)=1}}\frac1a, \qquad H_1^*=0.

Both terms in (8) are nonnegative. The tail is O(\log(2N)), since H_k^*\le h_{k-1}\ll\log(2k). The squared logarithm comes from the fractional-floor term.

The only external asymptotic input in this continuous calculation is the logarithmically weighted fractional-part theorem of Saffari and Vaughan [8]. For 2\le Y\le X, it gives, uniformly in 0\le\alpha\le1, \frac1{\log Y}\sum_{n\le Y}\frac1n \mathbf 1_{[0,\alpha)}\!\left(\left\{\frac Xn\right\}\right) =\alpha+O\left(\frac{(\log X)^{2/3}}{\log Y}\right).

The origin of the quadratic logarithmic coefficient is now explicit. The finite question is whether sampling on the p^2-carrier adds another term of that size. Theorem 27 proves that it does not.

The jump geometry of the straightened response

Write B_1(x)=\mathop{\mathrm{saw}}(x), \qquad B_2(x)=\{x\}^2-\{x\}+\frac16. Away from its jumps, \mathcal{B}_p has the constant slope L=2\sum_{r=1}^{N}r=N(N+1)=pN. Every nonzero jump has a unique reduced location x=\frac ak, \qquad 2\le k\le N, \qquad 1\le a<k, \qquad (a,k)=1. Put M_k=\left\lfloor\frac Nk\right\rfloor, \qquad s_k=N-kM_k, and define the cyclic prefix response T_{k,s}(a)=2\sum_{r=1}^{s}\mathop{\mathrm{saw}}\left(\frac{ra}{k}\right).

Exact carrier-grid aliasing

The carrier grid does not meet any nonzero jump. Indeed, if a reduced fraction a/k with k\le N<p were equal to A/p^2, then k would divide p^2, which is impossible for k\ge2.

We use the following periodic aliasing identity. The statement is specialized to the present piecewise-quadratic function in order to keep all endpoint conventions explicit.

Define \mathcal{A}_p =\sum_{k=2}^{N}M_k \sum_{\substack{1\le a<k\\(a,k)=1}} T_{k,s_k}(a)\mathop{\mathrm{saw}}\left(\frac{ma}{k}\right) and \mathcal{C}_p =\frac N6+ \sum_{k=2}^{N}M_k \sum_{\substack{1\le a<k\\(a,k)=1}} B_2\left(\frac{ma}{k}\right).

The final term is elementary and lower order.

Thus the nontrivial alias is the single signed jump-phase correlation \mathcal{A}_p. \mathcal{S}_p =\frac{(p-1)^2}{p^2} +\frac4{p^2}\mathcal{A}_p +O\left(\frac{\log p}{p}\right). This is the collision-native form of the sampling problem.

Exact Rademacher-reservoir bridge

We now identify the same residual in the reduced-ratio coordinates of Dedekind reciprocity. For coprime h and q, write s(h,q)=\sum_{x=1}^{q-1}\mathop{\mathrm{saw}}(x/q)\mathop{\mathrm{saw}}(hx/q) for the classical Dedekind sum.

We shall also use its inversion symmetry. If (h,q)=1, then s(h,q)=s(\overline h,q). Indeed, the substitution y=hx\bmod q in the defining sum exchanges the two sawtooth factors.

For a>k\ge1, (a,k)=1, put w_p(a)=\left\lfloor\frac Na\right\rfloor and define X_p= \sum_{\substack{2\le a\le N\\1\le k<a\\(a,k)=1}} w_p(a)\left(\frac ak+\frac ka\right), \mathfrak R_p(k) =\sum_{\substack{k<a\le N\\(a,k)=1}} w_p(a)\,s\bigl(m\overline a^{(k)},k\bigr), where \overline a^{(k)} is the inverse of a modulo k. Then R_p^{(k)}=-8\sum_{k=2}^{N-1}\mathfrak R_p(k).

The absolute-moment baseline

The exact bridge converts every improvement in the signed k-modulus reservoir directly into a sampling theorem.

Inverse-residue damping

The absolute moment discards the phase of each Dedekind sum. To go below the quadratic logarithmic scale, we keep that phase and open it additively. Classical work relates Dedekind sums to Kloosterman sums and studies their mean values [9, 1]. The estimate needed here must remain uniform after every composite carrier is descended to its lower moduli and the resulting divisor costs are summed. For prime modulus, Lemke Oliver and Soundararajan identify the normalized additive Fourier transform of the Dedekind sum with an inverse-residue sawtooth series and establish its limiting distribution and large-value behavior [6]. The bound required here has a different uniformity requirement. The carrier moduli are generally composite, and their Fourier arguments may be nonunits. The exact transform and numerator descent isolate the true lower moduli before the carrier family is summed. For q\ge2, put \mathrm{e}_q(x)=\exp(2\pi i x/q) and extend the Dedekind sum to every residue h\bmod q by \widetilde s(h,q) =\sum_{x\bmod q}\mathop{\mathrm{saw}}(x/q)\mathop{\mathrm{saw}}(hx/q). This agrees with s(h,q) when (h,q)=1. Define the unrestricted and unit-supported additive transforms \begin{aligned} F_q(t) &=\sum_{h\bmod q}\widetilde s(h,q)\mathrm{e}_q(-th), \\ D_q(t) &=\sum_{h\in(\mathbb{Z}/q\mathbb{Z})^\times}s(h,q)\mathrm{e}_q(-th). \end{aligned}

The transform is governed by one centered inverse-residue series. For q\ge2 and r\in\mathbb{Z}, let B_q(r) =\sum_{\substack{n\ge1\\(n,q)=1}} \frac1n\mathop{\mathrm{saw}}\left(\frac{r\overline n^{(q)}}q\right). The coefficient sequence is periodic and has mean zero. The series therefore converges by summation by parts.

Nonunit numerators in (50) descend to their true modulus before any analytic estimate is applied.

The completion estimate needed beyond Korolev’s range is recorded separately.

The logarithmic saving enters at the descended modulus.

The divisor cost can be large at selected carriers, but no fixed power of it has a growing mean.

The inverse-residue saving is obtained within each descended carrier, but a highly composite carrier may pay a large divisor cost after descent. Lemma 26 shows that these costs have bounded mean, so summing the carrierwise absolute bounds cannot rebuild a quadratic logarithm.

Small square-lock displacements cannot carry the secondary term

For 2\le k<p, let c_p(k) be the least signed representative of p^2 modulo k, chosen so that -\frac k2<c_p(k)\le\frac k2. We call c_p(k) the square-lock displacement of the carrier. It is an ordinary signed residue, not a character twist. It is nonzero because p is prime and k<p. Put \Delta_{k,p}(X) =\sum_{\substack{1\le a\le X\\(a,k)=1}} s\bigl(p^2\overline a^{(k)},k\bigr). Complete reduced residue blocks have sum zero. Expanding the floor weight in (37) therefore gives the exact collar decomposition \mathfrak R_p(k) =\sum_{h\le N/(k+1)} \Delta_{k,p}\left(\left\lfloor\frac Nh\right\rfloor\right). For an integer C with 1\le C\le p/2, define the normalized absolute collar mass \mathcal M_p(C) =\frac8{p^2} \sum_{\substack{2\le k<p\\0<|c_p(k)|\le C}} \sum_{h\le N/(k+1)} \left| \Delta_{k,p}\left(\left\lfloor\frac Nh\right\rfloor\right) \right|. The corresponding restricted part of the normalized reservoir satisfies \frac8{p^2} \left| \sum_{\substack{2\le k<p\\0<|c_p(k)|\le C}} \mathfrak R_p(k) \right| \le \mathcal M_p(C).

Large local spikes and global asymptotic weight are different. Fixed-displacement square locks can violate a uniform pointwise bound, but their total normalized collar mass is o(1). They cannot carry the quadratic logarithmic secondary term. More generally, the entire C_2(p) sector is too small at the coefficient scale. Only the large-displacement, short-cadence sector isolated by Corollary 29 survives this absolute localization. Theorem 27 controls that sector after all carrier phases are assembled.

The floor-state switching ledger

Square-lock localization identifies where the residual mass can live. The floor machine identifies how that mass moves. Fix a set of carriers \mathcal L\subseteq\{2,\ldots,N-1\}. Set B_{p,\mathcal L}(1)=0. For 2\le X\le N, define the state burden and its fresh increment by \begin{aligned} B_{p,\mathcal L}(X) &=\sum_{\substack{k\in\mathcal L\\k<X}}\Delta_{k,p}(X), \\ F_{p,\mathcal L}(X) &=B_{p,\mathcal L}(X)-B_{p,\mathcal L}(X-1). \end{aligned} Put H_X=\left\lfloor\frac NX\right\rfloor, \qquad d_X=H_X-H_{X+1}. The integer d_X counts the positive integers h for which \lfloor N/h\rfloor=X. The integer H_X counts those for which \lfloor N/h\rfloor\ge X.

For p\ge5, the energy ledger also separates coherent signed area from fluctuation. Put D=H_3, \qquad \overline B=\frac1D\sum_{X=3}^{N}d_XB(X). Then \frac1D\left(\sum_{X=3}^{N}d_XB(X)\right)^2 +\sum_{X=3}^{N}d_X\bigl(B(X)-\overline B\bigr)^2 =Q_{p,\mathcal L}. This is the exact point at which a tall local spike and a persistent signed area become different quantities.

Three-term reciprocity rewrites the full fresh sum on the prime-square carrier. Let \begin{aligned} \mathcal H^{\ast}(X) &=\sum_{\substack{1\le k<X\\(k,X)=1}}\frac1k, \\ \mathcal P_p(X) &=\sum_{\substack{1\le k<X/2\\(k,X)=1}} s\bigl(k\overline{(X-k)}^{(m)},m\bigr). \end{aligned} The packet \mathcal P_p(X) lies on the anti-diagonal k+(X-k)=X of the prime-square Dedekind table.

The anti-diagonal packet is the primitive part of an additive convolution. For integers a,b coprime to m, put D_m(a,b) =\sum_{y\bmod m}\mathop{\mathrm{saw}}(ay/m)\mathop{\mathrm{saw}}(by/m). Changing variables gives D_m(a,b)=s\bigl(a\overline b^{(m)},m\bigr).

The lifetime-weighted packets combine into one triangular sum. Put \mathcal T_m(N) =\sum_{\substack{a,b\ge1\\a+b\le N}}D_m(a,b) and let \mathcal E_p(X) =\frac{m}{12X}\mathcal H^{\ast}(X) +\frac{X\mathcal H^{\ast}(X)-\varphi(X)}{12m} -\frac{\varphi(X)}8. Then \boxed{ \sum_{k=2}^{N-1}\mathfrak R_p(k) =\sum_{X=3}^{N}H_X\mathcal E_p(X) -\frac12\mathcal T_m(N) +\frac{H_2}{2}s(1,m). } Indeed, \sum_{X=2}^{N}H_XC_m(X) =\sum_{n=2}^{N}A_m(n) =\mathcal T_m(N), because H_X counts the multiples of X not exceeding N. Now use C_m(2)=D_m(1,1)=s(1,m), Proposition 30, and (93).

One state X is one integer, and k runs through its two-part divisions X=k+(X-k). The kernel D_m measures the overlap of the two parts on the ambient prime-square carrier. Möbius inversion removes the common scale shared by the two parts and leaves their primitive divisions. Formula (97) then places the whole weighted fresh sum in one triangular overlap packet.

The difference between the explicit partition term and this prime-square packet, together with the displayed depth-two correction, is another exact coordinate for the residual. Reciprocity fixes its sign geometry. Complementary members map to inverse prime-square ratios, and Dedekind sums are invariant under inversion. They reinforce rather than cancel.

The half packet contains no opposite prime-square partner supplied by oddness. If two terms in (92) had opposite prime-square ratios, then m\mid X(k+k')-2kk'. For 1\le k,k'<X/2, the integer on the right lies strictly between 0 and X^2<m. This is impossible. The complementary-partition identity exposes the aggregate controlled by Theorem 27. The damping enters through the additive modes of the descended carriers.

One carrier can retain a single sign even at large displacement. At p=101 and k=14, the square-lock displacement is -5, while C_2(101)=4. The six floor levels have endpoint residues 2,\ 8,\ 5,\ 11,\ 6,\ 2 and exact collar values -\frac3{14},\ -\frac{13}{14},\ -\frac{13}{14},\ -\frac3{14},\ -\frac{13}{14},\ -\frac3{14}. They are all negative and sum to -24/7. Thus no universal carrierwise sign-reversal argument exists. The coefficient theorem assembles the additive modes within each carrier before taking absolute values, then controls the resulting divisor costs on average across carrier moduli.

The live sector and sharper targets

Let \mathcal L_p be the large-displacement sector \mathcal L_p =\{2\le k\le N-1:|c_p(k)|>C_2(p)\}. The all-prime damping theorem and square-lock localization give a proved common-state estimate on this sector.

This is lifetime-weighted cancellation of fresh collision increments. No pointwise bound on spike height is required. The next questions concern a stronger scale.

A stronger local estimate would remove a full logarithm from the sampling bound.

There is an equally native jump-correlation criterion. Put V_{k,s}(c) =\sum_{a\in(\mathbb{Z}/k\mathbb{Z})^\times} T_{k,s}(a)\mathop{\mathrm{saw}}(ca/k). The alias formula uses V_{k,s_k}(m\bmod k).

The finite secondary term

The exact decomposition is \frac{E_p}{p^2}=\mathcal{J}(p-1)-\mathcal{S}_p. Combining it with (14) and Theorem 27 closes the finite transfer.

The coefficient 1/\pi^2 enters through continuous capacity. The inverse-residue theorem proves that finite carrier sampling preserves it.

Exact computation

The continuous capacity is evaluated by the Jordan-totient identity 3\mathcal{J}(N) =\sum_{d\le N}\frac{J_2(d)}{d^2} H_{\lfloor N/d\rfloor}^{\,2}. The energy is computed by exact integer floor sums. The values below were obtained with nfield [11]. At 26 declared prime checkpoints from 3 through 10007, nfield evaluates the continuous capacity and the finite energy independently. At the 21 declared checkpoints through 2003, it also evaluates both exact bridge formulas. Their discrepancies are below 2\mathbin{\cdot}10^{-15} in long-double arithmetic. The tabulated values provide independent finite checks of the formulas.

The selected values in Table 1 show the arithmetic oscillation. They are consistent with bounded sampling and the endpoint limit (103). Both assertions remain open.

Relation to the endpoint reservoir

The finite secondary correction also has a lifted endpoint-reservoir form. Its quotient-shell and visible-minus-shadow coordinates lead to the same surviving lower-modulus carrier. Theorem 16 fixes its relation to the Bernoulli normal form. \boxed{ \begin{aligned} \text{carrier-grid alias} &=\text{regulated endpoint term}-\frac{R_p^{(k)}}{p^2}\\ &\quad+\text{explicit curvature}. \end{aligned} } Quotient-shell damping is a second coordinate for the same signed sum. The inverse-residue argument proves enough damping to preserve the coefficient. The quotient-shell coordinate remains useful for sharper estimates because it records where the cancellation occurs rather than only its final size.

This also explains the observed scale separation. The endpoint term tends to 1. The algebraic curvature is O(\log p/p^2) in the reciprocity coordinates. Any additional quadratic logarithm had to come from R_p^{(k)}. Theorem 27 rules that out by saving a factor of \log\log p. The stronger signed damping problem asks whether a full logarithm can be removed.

The coefficient survives the carrier grid

The finite sampling defect has two exact descriptions. The first is a collision-native correlation between cyclic boundary prefixes and carrier-grid jump phases. The second is the surviving lower-modulus Dedekind reservoir from Rademacher reciprocity. Each description includes its own explicit endpoint and curvature terms. Together they identify the same signed obstruction.

Opening the Dedekind carrier additively reveals the decisive cancellation. Every nonunit phase descends to its true modulus. At that modulus the centered inverse-residue series saves a factor of \log\log p. The divisor burden of lifting back through the carrier grid has bounded mean, so it cannot restore the lost factor. This proves \mathcal{S}_p=o((\log p)^2) for every odd prime. The continuous coefficient 1/\pi^2 therefore survives unchanged in the finite collision energy.

The square-lock and floor-state coordinates supply the local picture behind the global estimate. Small displacement carries too little absolute mass to reach the coefficient scale. Large displacement can produce a sustained one-sided spike inside a single carrier, so carrierwise sign reversal is unavailable. The required saving comes from additive cancellation within each descended carrier, together with the bounded mean of the divisor-lift cost. The spikes remain real local features, but the sum of the resulting carrierwise bounds is too small to alter the secondary term.

The finite secondary coefficient is settled. An O(\log p) sampling bound would remove a full logarithm, an O(1) bound would reach the endpoint scale, and the observed limit \mathcal{S}_p\to1 remains the strongest target. Those questions begin below the coefficient proved here. The surviving correction is -\pi^{-2}p^2(\log p)^2. This is The Secondary Term in Digit-Collision Energy.

References