The Collision Spectrum and the L-Function Landscape
Abstract
Let b be an odd prime and let the lag-one collision function be centered on the digit fibers of (\mathbb Z/b^2\mathbb Z)^\times. Every primitive odd Fourier coefficient factors exactly as \widehat S^{\circ}(\chi) = -\frac{B_{1,\overline\chi}\, \overline{S_G(\chi)}}{\varphi(b^2)}. The generalized Bernoulli number carries the value at one of the L-function. The diagonal character sum S_G(\chi) is finite collision geometry, and it reduces to twice a short initial character sum.
Parseval turns the coefficient factorization into an exact identity between the collision energy and a weighted second moment of the values L(1,\chi). At base five, a cyclotomic identity forces |S_G(\chi)|=\sqrt5\,|B_{1,\overline\chi}| for every primitive odd character modulo 25. The coefficient magnitude is then a fixed multiple of |L(1,\chi)|^2. The centered collision energy is 48, and \sum_{\substack{\chi\bmod25\\ \chi\ {\rm primitive\ odd}}}|L(1,\chi)|^4 =\frac{192\pi^4}{625}. An exact finite nfield enumeration [3] records the remaining relation between the two factors over every odd prime base from 5 through 71. The finite ledger supplies no asymptotic decay law.
Introduction
Long division produces a finite collision table. Centering removes the part already fixed by the final digit class. Fourier expansion then asks what remains in each multiplicative character channel.
The answer at lag one is an exact product. One factor is a generalized Bernoulli number and therefore the value at one of a Dirichlet L-function. The other is a signed sum over the diagonal of the digit table. The analytic value is not merely correlated with the collision coefficient. It occurs inside the coefficient as a multiplicative factor.
The construction begins with the finite digit table itself. All definitions and proof inputs needed below are given explicitly. No analytic continuation into the critical strip is used.
Base five is the first prime-square family with more than one conjugate pair. Here the two factor magnitudes differ by a fixed scalar. A symbolic cyclotomic calculation shows that the diagonal magnitude is exactly \sqrt5 times the Bernoulli magnitude. The resulting coefficient magnitude is quadratic in |L(1,\chi)|, and the full collision energy evaluates an exact fourth moment.
The Centered Collision Function
Fix an odd prime b and put m=b^2, \qquad U_m=(\mathbb Z/m\mathbb Z)^\times. Every residue in U_m will be represented by its unique integer a with 1\le a<m. Dirichlet characters are extended by zero on nonunits.
The lag-one diagonal is \begin{aligned} G &= \left\{0\le n<m: \left\lfloor\frac nb\right\rfloor=n\bmod b\right\}\\ &= \{r(b+1):0\le r\le b-1\}. \end{aligned} For n\in G, define the digit increment 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}d_n(a). This is the finite-determination form of the lag-one collision deviation. Primality of an original denominator has disappeared from (1). Only its residue a modulo b^2 remains.
For 1\le s\le b-1, let U_s=\{a\in U_m:a\equiv s\pmod b\}, \qquad \overline S_s=\frac1b\sum_{a\in U_s}S(a). Define S^\circ(a)=S(a)-\overline S_{a\bmod b}. The centered coefficient is \widehat S^\circ(\chi) = \frac1{\varphi(m)} \sum_{a\in U_m}S^\circ(a)\overline{\chi}(a).
Lemma 1 (Reflection). For every a\in U_m, S(a)+S(m-a)=-1. Consequently, \overline S_s+\overline S_{b-s}=-1 and S^\circ(m-a)=-S^\circ(a).
Proof. For 1\le n\le m-2, neither na/m nor (n+1)a/m is an integer. The floor identity \left\lfloor\frac{k(m-a)}m\right\rfloor = k-1-\left\lfloor\frac{ka}m\right\rfloor therefore gives d_n(a)+d_n(m-a)=1. The endpoint terms satisfy d_0(a)=d_0(m-a)=0, \qquad d_{m-1}(a)=d_{m-1}(m-a)=1. There are b-2 interior elements of G. Since a is not divisible by b, \left\lfloor\frac ab\right\rfloor + \left\lfloor\frac{m-a}b\right\rfloor =b-1. Substitution in (1) gives the first identity. The map a\mapsto m-a carries U_s bijectively onto U_{b-s}. Averaging the first identity over a fiber gives the second, and subtracting the paired means gives the third. ◻
Lemma 2 (Primitive fiber cancellation). If \chi is primitive modulo b^2, then \sum_{a\in U_s}\overline\chi(a)=0 for every 1\le s\le b-1. Hence \sum_{a\in U_m}\overline S_{a\bmod b}\, \overline\chi(a)=0.
Proof. The subgroup H=\{1+jb:0\le j\le b-1\} is the kernel of reduction from U_m to (\mathbb Z/b\mathbb Z)^\times. A primitive character modulo b^2 is nontrivial on H. Character orthogonality gives \sum_{u\in H}\overline\chi(u)=0. Every fiber U_s is a multiplicative coset of H, so its character sum also vanishes. The fiber mean is constant on that coset, which proves the second identity. ◻
Diagonal Reduction
For an odd character \chi modulo b^2, define S_G(\chi) = \sum_{n\in G} \bigl[\overline\chi(n+1)-\overline\chi(n)\bigr].
Lemma 3 (Diagonal reduction). For every odd character \chi modulo b^2, S_G(\chi) = -2\,\overline\chi(b+1) \sum_{k=1}^{b-1}\overline\chi(k). In particular, |S_G(\chi)| = 2\left|\sum_{k=1}^{b-1}\overline\chi(k)\right|.
Proof. Put P_\chi=\sum_{k=1}^{b-1}\overline\chi(k), \qquad \alpha=\overline\chi(b+1). The terms indexed by n=r(b+1) give S_G(\chi)=Q_\chi-\alpha P_\chi, where Q_\chi = \sum_{r=0}^{b-2} \overline\chi\bigl(r(b+1)+1\bigr). The reflection b^2-\bigl(r(b+1)+1\bigr) =(b-1-r)(b+1) maps the arguments in Q_\chi bijectively onto \{j(b+1):1\le j\le b-1\}. Oddness gives -Q_\chi = \alpha\sum_{j=1}^{b-1}\overline\chi(j) = \alpha P_\chi. Thus Q_\chi=-\alpha P_\chi, which proves the result. ◻
Bernoulli Factorization
For a character \chi modulo m, put B_{1,\overline\chi} = \frac1m\sum_{a\in U_m}a\,\overline\chi(a).
Lemma 4 (Fractional-part transform). Let \chi be a primitive character modulo m, and let n be coprime to m. Then \sum_{a\in U_m} \overline\chi(a) \left\{\frac{na}{m}\right\} = \chi(n)B_{1,\overline\chi}. Consequently, \sum_{a\in U_m} \left\lfloor\frac{na}{m}\right\rfloor \overline\chi(a) = \bigl(n-\chi(n)\bigr)B_{1,\overline\chi}.
Proof. Multiplication by n permutes U_m. Substitution by n^{-1}a gives \sum_{a\in U_m} \overline\chi(a) \left\{\frac{na}{m}\right\} = \chi(n)\sum_{a\in U_m} \overline\chi(a)\frac am. This is the first identity. Subtracting it from \frac nm\sum_{a\in U_m}a\,\overline\chi(a) gives the second. ◻
Theorem 5 (Collision spectrum factorization). Let b be an odd prime, let m=b^2, and let \chi be primitive and odd modulo m. Then \widehat S^\circ(\chi) = -\frac{B_{1,\overline\chi}\, \overline{S_G(\chi)}}{\varphi(m)}.
Proof. Write \varphi=\varphi(m). By Lemma 2, the centering term contributes zero. Hence \varphi\,\widehat S^\circ(\chi) = \sum_{a\in U_m}S(a)\overline\chi(a).
The constant term in (1) contributes zero. Also, \left\{\frac ab\right\} is constant on each fiber U_s. Primitive fiber cancellation gives \sum_{a\in U_m} \left\{\frac ab\right\}\overline\chi(a)=0. It follows that -\sum_{a\in U_m} \left\lfloor\frac ab\right\rfloor\overline\chi(a) = -\frac1b\sum_{a\in U_m}a\,\overline\chi(a) = -bB_{1,\overline\chi}.
At one endpoint, d_0(a)=0. At the other, d_{m-1}(a)=1. Every nontrivial character sums to zero, so both endpoint slices contribute zero. Every remaining diagonal index has the form n=r(b+1) with 1\le r\le b-2. Both n and n+1 are units modulo m, so Lemma 4 gives \sum_{a\in U_m}d_n(a)\overline\chi(a) = \bigl[1+\chi(n)-\chi(n+1)\bigr]B_{1,\overline\chi}.
The two endpoint terms in \overline{S_G(\chi)} both equal 1. Therefore the sum over the b-2 interior slices is B_{1,\overline\chi} \bigl[b-\overline{S_G(\chi)}\bigr]. Combining this with the floor contribution gives \varphi\,\widehat S^\circ(\chi) = -bB_{1,\overline\chi} + B_{1,\overline\chi} \bigl[b-\overline{S_G(\chi)}\bigr] = -B_{1,\overline\chi}\overline{S_G(\chi)}. ◻
Values at One
For a primitive odd character modulo b^2, the classical Bernoulli–L-value relation gives [1] |B_{1,\overline\chi}| = \frac b\pi |L(1,\chi)|.
Corollary 6 (L-value encoding). For every primitive odd character modulo b^2, |\widehat S^\circ(\chi)| = \frac{b}{\pi\varphi(b^2)} |L(1,\chi)|\,|S_G(\chi)|. Equivalently, |\widehat S^\circ(\chi)| = \frac{2b}{\pi\varphi(b^2)} |L(1,\chi)| \left|\sum_{k=1}^{b-1}\overline\chi(k)\right|.
The value at one and the finite diagonal sum play different roles. Neither is supplied by the other in general. Their product, after the sign and normalization in Theorem 5, is the exact collision coefficient.
Collision Energy
Define the centered collision energy E_b=\sum_{a\in U_{b^2}}|S^\circ(a)|^2.
Lemma 7 (Primitive odd population). The number of primitive odd characters modulo b^2 is \frac{(b-1)^2}{2}.
Proof. There are b(b-1)/2 odd characters modulo b^2. The imprimitive ones are induced from the (b-1)/2 odd characters modulo b. Subtraction gives the stated count. ◻
Theorem 8 (Exact special-value moment). For every odd prime b, \sum_{\substack{\chi\bmod b^2\\ \chi\ {\rm primitive\ odd}}} |L(1,\chi)|^2\,|S_G(\chi)|^2 = \frac{\pi^2\varphi(b^2)}{b^2}\,E_b.
Proof. Parseval with the normalization (2) gives \sum_{\chi\bmod m}|\widehat S^\circ(\chi)|^2 = \frac1{\varphi(m)} \sum_{a\in U_m}|S^\circ(a)|^2. Lemma 1 makes every even coefficient vanish. Every imprimitive character modulo b^2 is induced from modulus b and is constant on each fiber U_s. Fiber centering makes every imprimitive odd coefficient vanish as well. Only primitive odd characters remain.
Substitution of Corollary 6 gives \frac{b^2}{\pi^2\varphi(m)^2} \sum_{\chi\ {\rm primitive\ odd}} |L(1,\chi)|^2|S_G(\chi)|^2 = \frac{E_b}{\varphi(m)}. Rearranging proves the identity. ◻
This is an exact moment identity, not a triangle-inequality majorant. The finite collision energy evaluates a weighted second moment of the values L(1,\chi).
The Exact Base-Five Sector
Theorem 9 (Base-five cyclotomic identity). For every primitive odd character \chi modulo 25, \left| \sum_{k=1}^{4}\overline\chi(k) \right| = \frac{\sqrt5}{2} \left|B_{1,\overline\chi}\right|. Moreover, |S_G(\chi)| = \sqrt5\left|B_{1,\overline\chi}\right|.
Proof. Put \psi=\overline\chi and z=\psi(2). The residue 2 generates U_{25} and has order 20. Write z=e^{2\pi i j/20}. Oddness makes j odd, while primitivity modulo 25 gives 5\nmid j. Thus \gcd(j,20)=1, and z is a primitive twentieth root of unity.
Since 3\equiv2^7\pmod{25}, \qquad 4\equiv2^2\pmod{25}, the short character sum is P_5(\chi) = \sum_{k=1}^{4}\psi(k) = p(z), \qquad p(x)=1+x+x^2+x^7. Lemma 3 gives |S_G(\chi)|=2|p(z)|.
Let a_r be the least positive residue of 2^r modulo 25. For 0\le r\le9, (a_0,\ldots,a_9) = (1,2,4,8,16,7,14,3,6,12). The relations a_{r+10}=25-a_r, \qquad z^{r+10}=-z^r pair the Bernoulli sum into 25B_{1,\psi} = \sum_{r=0}^{9}(2a_r-25)z^r. Reduction by \Phi_{20}(z)=z^8-z^6+z^4-z^2+1=0 gives 25B_{1,\psi}=-10q(z), where q(x) = 1+2x+3x^2+x^3-2x^4+x^5+x^6+2x^7.
The Laurent polynomial identity \begin{aligned} x^7\bigl( q(x)q(x^{-1})-5p(x)p(x^{-1}) \bigr) ={}& \Phi_{20}(x)\\ &\cdot \bigl(-3+x^2+5x^3+x^4-3x^6\bigr) \end{aligned} is exact. Setting x=z gives |q(z)|^2=5|p(z)|^2. Since B_{1,\psi}=-(2/5)q(z), |B_{1,\psi}|^2 = \frac45|p(z)|^2. The two conclusions follow. ◻
Corollary 10 (Exact base-five coefficient). For every primitive odd character \chi modulo 25, |\widehat S^\circ(\chi)| = \frac{5\sqrt5}{4\pi^2}|L(1,\chi)|^2.
Proof. At modulus 25, |B_{1,\overline\chi}|=\frac5\pi|L(1,\chi)|. Theorem 9 gives |S_G(\chi)| = \frac{5\sqrt5}{\pi}|L(1,\chi)|. Substitution in Theorem 5 proves the result. ◻
Theorem 11 (Exact collision energy and fourth moment). The centered base-five collision table satisfies E_5=48. Moreover, \sum_{\substack{\chi\bmod25\\ \chi\ {\rm primitive\ odd}}} |L(1,\chi)|^4 = \frac{4\pi^4}{625}E_5 = \frac{192\pi^4}{625}.
Proof. Let \zeta=e^{2\pi i/20}. The primitive odd characters modulo 25 correspond to the eight exponents j\in(\mathbb Z/20\mathbb Z)^\times. For p(x)=1+x+x^2+x^7, put d(x)=p(x)p(x^{-1}). Its exact Laurent expansion is \begin{aligned} d(x) ={}& 4+2(x+x^{-1})+(x^2+x^{-2})\\ &+(x^5+x^{-5})+(x^6+x^{-6})+(x^7+x^{-7}). \end{aligned}
Write c_{20}(n) = \sum_{j\in(\mathbb Z/20\mathbb Z)^\times}\zeta^{jn} for the Ramanujan sum. The even-power coefficients of d(x)^2, reduced modulo x^{20}-1, are \begin{aligned} 32 &+14(x^2+x^{-2})+7(x^4+x^{-4})\\ &+17(x^6+x^{-6})+9(x^8+x^{-8})+2x^{10}. \end{aligned} Terms with odd exponent do not contribute because c_{20}(n)=0 for odd n. The standard Ramanujan-sum formula [2] gives \begin{array}{c|rrrrrr} n&0&2&4&6&8&10\\ c_{20}(n)&8&2&-2&2&-2&-8. \end{array} Therefore \begin{aligned} \sum_{j\in(\mathbb Z/20\mathbb Z)^\times} |p(\zeta^j)|^4 ={}& 32(8)+28(2)+14(-2)\\ &+34(2)+18(-2)+2(-8)\\ ={}&300. \end{aligned} Since |S_G(\chi)|=2|p(\zeta^j)|, \sum_{\chi\ {\rm primitive\ odd}}|S_G(\chi)|^4=4800.
The cyclotomic identity gives |B_{1,\overline\chi}|^2 = \frac15|S_G(\chi)|^2. Theorem 5 now yields \sum_{\chi\ {\rm primitive\ odd}} |\widehat S^\circ(\chi)|^2 = \frac1{2000} \sum_{\chi\ {\rm primitive\ odd}}|S_G(\chi)|^4 = \frac{12}{5}. Parseval gives \frac{E_5}{20}=\frac{12}{5}, so E_5=48.
Finally, Corollary 10 and Parseval give \sum_{\chi\ {\rm primitive\ odd}}|L(1,\chi)|^4 = \frac{4\pi^4}{625}E_5. Substitution of E_5=48 finishes the proof. ◻
Finite Correlation
For each primitive odd character modulo b^2, put X_\chi = \left|\sum_{k=1}^{b-1}\overline\chi(k)\right|, \qquad Y_\chi = |B_{1,\overline\chi}|. At fixed b, the Bernoulli–L-value relation multiplies every Y_\chi by the same constant. The Pearson correlation between X_\chi and Y_\chi is therefore exactly the correlation between X_\chi and |L(1,\chi)|.
The nfield enumeration [3] checks every primitive odd character for every odd prime base from 5 through 71. Both members of each conjugate pair are retained. Let r_b denote the Pearson correlation between X_\chi and Y_\chi at fixed base b.
| b | characters | r_b | b | characters | r_b |
|---|---|---|---|---|---|
| 5 | 8 | 1.0000 | 37 | 648 | 0.7080 |
| 7 | 18 | 0.8440 | 41 | 800 | 0.7111 |
| 11 | 50 | 0.8042 | 43 | 882 | 0.6987 |
| 13 | 72 | 0.8004 | 47 | 1058 | 0.6875 |
| 17 | 128 | 0.7925 | 53 | 1352 | 0.6856 |
| 19 | 162 | 0.7630 | 59 | 1682 | 0.6759 |
| 23 | 242 | 0.7517 | 61 | 1800 | 0.6790 |
| 29 | 392 | 0.7203 | 67 | 2178 | 0.6690 |
| 31 | 450 | 0.7267 | 71 | 2450 | 0.6753 |
The calculation uses the finite Bernoulli sum for Y_\chi. No Euler product or truncated prime sum enters the ledger. The values in Table 1 are rounded to four decimal places.
The finite ledger is exhaustive over the declared bases and character families. It uses no auxiliary prime cutoff to approximate an L-value. It does not determine a limiting correlation or a decay law.
The Finite Spectrum
The coefficient factorization is exact at s=1. The finite collision geometry contributes the short diagonal sum. The generalized Bernoulli number contributes the value at one. Their product, after the stated sign and normalization, is the collision coefficient.
Base five is the first prime-square family with more than one conjugate pair. Here the two factors become the same magnitude up to a fixed scalar. The resulting square law and fourth moment are exact. At the listed bases beyond five, the finite ledger shows a positive relation between the factors, but no uniform law has been proved.
The remaining analytic problem is to explain the relation between the short initial character sum and the Bernoulli magnitude as the prime base grows. Nothing above moves the finite observable into the critical strip, measures distance to a zero, or locates zeros. Those questions require a different analytic construction.
References
[1]H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000.
[2]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., revised by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, 2008.
[3]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield