Let b be an odd prime. Center the lag-one digit-collision function on the fibers of (\mathbb Z/b^2\mathbb Z)^\times. For every primitive odd character \chi modulo b^2, a generalized Bernoulli number and a finite sum across digit boundaries give the exact coefficient \widehat S^{\circ}(\chi) = -\frac{B_{1,\overline\chi}\, \overline{S_G(\chi)}}{\varphi(b^2)}. Thus Parseval identifies the finite collision energy with a diagonally weighted second moment of |L(1,\chi)|. At base five, an exact cyclotomic identity forces |S_G(\chi)|=\sqrt5\,|B_{1,\overline\chi}| for every primitive odd character modulo 25. Consequently, the coefficient magnitude is 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}. This is a fixed-modulus evaluation, not an asymptotic moment theorem. An exhaustive nfield computation records the two factors over every odd prime base from 5 through 71. Their finite correlations are positive, while a base-uniform law remains open.
A coefficient built from floor jumps contains the value at one of a Dirichlet L-function. The product formula identifies each surviving coefficient, including its complex phase, and Parseval converts it into a weighted special-value moment. The weight comes from a finite character sum across the digit-equality diagonal.
The finite table is developed in The Centered Collision Sum [1]. Fiber centering removes the contribution fixed by the final digit class. The Spectral Repulsion [2] places the surviving Fourier mass in the primitive odd sector. Factoring those coefficients separates the classical Bernoulli contribution from the finite diagonal boundary.
Coordinate agreement under cyclic shift belongs to classical Hamming correlation theory [3]. For prime-reciprocal decimal sequences, Kak and Chatterjee obtained Hamming-distance and autocorrelation bounds [4].
A different literature weights digit values rather than testing equality. Girstmair related reciprocal digits to class-number factors and expressed full-period digit variance through a Dedekind sum [5, 6]. Murty and Thangadurai study digit averages for bases of prescribed order. Girstmair’s 2026 paper treats half-period variance. Generalized Bernoulli numbers, Dirichlet L-functions, and class numbers enter those two settings [7, 8].
None of those classical ingredients is claimed here as new. The object below is the equality indicator on the complete unit table modulo b^2, followed by centering on its residue fibers. Finite Fourier analysis and Parseval are standard [10]. The result specific to this collision table is the exact factorization of each surviving coefficient into its Bernoulli factor and its diagonal boundary factor. The collision energy consequently becomes a particular weighted moment of values at one. Existing work on moments of L(1,\chi) supplies the neighboring analytic setting [9], while the weight in Theorem 8 is supplied by the collision diagonal itself.
Base five is the first prime-square family with more than one conjugate pair. Here an exact cyclotomic calculation makes the diagonal magnitude \sqrt5 times the Bernoulli magnitude. The coefficient magnitude is therefore quadratic in |L(1,\chi)|, and the full collision energy evaluates the corresponding fourth moment over the eight primitive odd characters modulo 25. This fixed family supplies an explicit fourth moment.
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. ◻
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. ◻
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)}. ◻
For a primitive odd character modulo b^2, the classical identity relating the generalized Bernoulli number to the L-value gives [12] |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.
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. Mean values of |L(1,\chi)|^2 over character families and subgroups have an established theory [9]. The statement here is different in one precise respect. Its weight |S_G(\chi)|^2 is the finite diagonal boundary of the collision table.
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 [13] 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. ◻
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 identity between the Bernoulli number and the L-value multiplies every Y_\chi by the same constant. The Pearson correlation [11] between X_\chi and Y_\chi is therefore exactly the correlation between X_\chi and |L(1,\chi)|.
The nfield enumeration [14] 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 ledger exhausts the declared bases and character families. It uses the finite Bernoulli sum itself, without an auxiliary prime cutoff. A law in the base would require control of the short initial character sum as b grows.
The finite collision table contains more than a numerical correlation. After fiber centering, every active coefficient is exactly the product of a classical analytic boundary value and a collision-specific diagonal sum. Parseval preserves that separation at the energy level. At base five, the two factors share one cyclotomic magnitude, which gives the exact energy 48 and the fourth moment 192\pi^4/625.
The generalized Bernoulli and L-value identities, character orthogonality, Parseval, Ramanujan sums, and Pearson correlation are classical. What is proved here is their exact assembly around this fiber-centered equality indicator, including the short diagonal reduction and the base-five cyclotomic identity. The correlation ledger covers every primitive odd character at the stated prime bases from 5 through 71. Those finite correlations leave the dependence on a growing base open.
The unresolved finite problem is to control the short initial character sum against the Bernoulli magnitude as b grows. A separate analytic problem is to transport both factors away from s=1 without hiding the finite diagonal. Either step requires a new theorem. At the present boundary, the exact factorization shows how a digit equality, a character sum, and an L-function value occupy the same coefficient.
[1]A. S. Petty, The Centered Collision Sum, research note, October 2023, revised September 2026. https://doi.org/10.5281/zenodo.21852556
[2]A. S. Petty, The Spectral Repulsion, research note, January 2025, revised September 2026. https://doi.org/10.5281/zenodo.21855227
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[5]K. Girstmair, The digits of 1/p in connection with class number factors, Acta Arith. 67 (1994), no. 4, 381–386. https://doi.org/10.4064/aa-67-4-381-386
[6]K. Girstmair, Digit variance and Dedekind sums, J. Number Theory 65 (1997), no. 2, 197–205. https://doi.org/10.1006/jnth.1997.2149
[7]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
[8]K. Girstmair, On the variance of the digits of 1/p, Ramanujan J. 70 (2026), no. 2, article 27. https://doi.org/10.1007/s11139-026-01406-5
[9]S. R. Louboutin and M. Munsch, Mean square values of L-functions over subgroups for nonprimitive characters, Dedekind sums and bounds on relative class numbers, Canad. J. Math. 75 (2023), no. 5, 1711–1743. https://doi.org/10.4153/S0008414X2300010X
[10]A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9780511626265
[11]K. Pearson, Mathematical contributions to the theory of evolution. III. Regression, heredity, and panmixia, Philos. Trans. Roy. Soc. London Ser. A 187 (1896), 253–318. https://doi.org/10.1098/rsta.1896.0007
[12]H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000.
[13]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.
[14]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
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