Let b be an odd prime and fiber-center the lag-one collision function on (\mathbb Z/b^2\mathbb Z)^\times. Every primitive odd Fourier coefficient factors into a generalized Bernoulli number and a finite diagonal character sum. Retaining phase gives \widehat S^\circ(\chi) = \frac{2}{\pi(b-1)} \varepsilon(\overline\chi)L(1,\chi)\overline{A(\chi)}. This separates the classical root-number and L-value factor from the collision diagonal. Primitive odd orthogonality proves \sum_{\substack{\chi\bmod b^2\\ \chi\ {\rm primitive\ odd}}}|S_G(\chi)|^2 =2b(b-1)^2. The diagonal squares have mean 4b and normalize to probability weights. Parseval identifies the collision energy with their weighted second moment of |L(1,\chi)|.
At base five, a cyclotomic identity makes the two factor magnitudes proportional, gives centered collision energy 48, and yields \sum_{\substack{\chi\bmod25\\ \chi\ {\rm primitive\ odd}}}|L(1,\chi)|^4 =\frac{192\pi^4}{625}. An exhaustive finite ledger covers all 14{,}372 primitive odd characters at the odd prime bases from 5 through 71. It proves no asymptotic law. The co-variation of diagonal weight and |L(1,\chi)| as the base grows remains open.
A digit equality can retain an analytic special value. Long division gives a finite equality table. Centering removes the part already fixed by the final digit class, and Fourier expansion isolates what remains in each multiplicative character channel. The resulting coefficient is the product of a generalized Bernoulli number and a short character sum along the equality diagonal. The total mass of these diagonal factors is exact. It fixes the normalization of a collision-weighted mean of Dirichlet L-values at one.
Coordinate agreement under cyclic shift belongs to classical Hamming correlation theory [1]. Kak and Chatterjee studied decimal prime-reciprocal sequences as communication codes and obtained bounds for Hamming distance from cyclic shifts and for autocorrelation [2]. In the full-reptend case, the underlying collision observable is exactly cyclic shift agreement. The complete-unit formulation below does not require a primitive-root hypothesis. The Hamming observable itself is not new.
A neighboring 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 [3, 4]. Murty and Thangadurai studied digit averages for bases of prescribed order, and Girstmair later treated half-period variance [5, 6]. Generalized Bernoulli numbers, Dirichlet L-values, and class numbers enter those settings.
Finite Fourier analysis, character orthogonality, and Parseval are standard [8]. The generalized Bernoulli and L(1,\chi) identity is classical [10], as is the surrounding theory of second moments at one [7]. The result specific to the collision table is the exact coefficient factorization, the exact diagonal second moment, and the resulting collision-weighted special-value identity.
At base five, a symbolic cyclotomic identity makes the two factor magnitudes proportional in every primitive odd channel. The finite collision energy then evaluates a fourth moment over the eight primitive odd characters modulo 25. This is a fixed-modulus identity, not a fourth-moment asymptotic. The finite ledger through base 71 records the channelwise relation while leaving its base asymptotics open.
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. ◻
Write A(\chi) = \sum_{r=0}^{b-2} \overline\chi\bigl(r(b+1)+1\bigr). The proof gives the phase-sensitive identity S_G(\chi)=2A(\chi).
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 m=b^2, put \tau(\overline\chi) = \sum_{a\bmod m}\overline\chi(a)e^{2\pi ia/m}, \qquad \varepsilon(\overline\chi) = \frac{\tau(\overline\chi)}{ib}. The root number \varepsilon(\overline\chi) has absolute value one.
Corollary 6 (Phase-separated coefficient). For every primitive odd character modulo b^2, B_{1,\overline\chi} = -\frac b\pi\, \varepsilon(\overline\chi)L(1,\chi), and \widehat S^\circ(\chi) = \frac{2}{\pi(b-1)} \varepsilon(\overline\chi)L(1,\chi)\overline{A(\chi)}. Consequently, |\widehat S^\circ(\chi)| = \frac{b}{\pi\varphi(b^2)} |L(1,\chi)|\,|S_G(\chi)| = \frac{2b}{\pi\varphi(b^2)} |L(1,\chi)| \left|\sum_{k=1}^{b-1}\overline\chi(k)\right|.
Proof. The functional equation for a primitive odd character gives B_{1,\overline\chi} = -\frac b\pi\, \varepsilon(\overline\chi)L(1,\chi) with the stated Gauss-sum convention [10]. Substitute this identity and (3) into Theorem 5. Taking absolute values and applying Lemma 3 gives the last line. ◻
The value L(1,\chi) lies at one, not at the central point. Formula (4) retains the phase and separates three objects. The root number records the functional-equation phase. The L-value is analytic. The incomplete sum A(\chi) comes from the finite collision diagonal.
Let \mathcal P_b denote the primitive odd characters modulo b^2.
Lemma 7 (Primitive odd kernel). For every unit y modulo b^2, \begin{aligned} \sum_{\chi\in\mathcal P_b}\chi(y) ={}& \frac{\varphi(b^2)}2 \left( \mathbf 1_{y\equiv1\;(\bmod b^2)} - \mathbf 1_{y\equiv-1\;(\bmod b^2)} \right)\\ &- \frac{\varphi(b)}2 \left( \mathbf 1_{y\equiv1\;(\bmod b)} - \mathbf 1_{y\equiv-1\;(\bmod b)} \right). \end{aligned} In particular, |\mathcal P_b|=\frac{(b-1)^2}{2}.
Proof. For either odd modulus q=b or q=b^2, character orthogonality followed by the odd projector gives \sum_{\substack{\chi\bmod q\\\chi(-1)=-1}}\chi(y) = \frac{\varphi(q)}2 \left( \mathbf 1_{y\equiv1\;(\bmod q)} - \mathbf 1_{y\equiv-1\;(\bmod q)} \right). The imprimitive odd characters modulo b^2 are precisely the lifts of the odd characters modulo b. Subtraction proves the kernel formula. Setting y=1 gives the population. ◻
Theorem 8 (Exact diagonal moment). For every odd prime b, \sum_{\chi\in\mathcal P_b}|S_G(\chi)|^2 = 2b(b-1)^2. Equivalently, \frac1{|\mathcal P_b|} \sum_{\chi\in\mathcal P_b}|S_G(\chi)|^2 = 4b.
Proof. Write x_r=r(b+1)+1, \qquad 0\le r\le b-2. These are distinct units modulo b^2, and their reductions modulo b run once through the nonzero residue classes. Equation (3) reduces the theorem to \sum_{\chi\in\mathcal P_b}|A(\chi)|^2 = b|\mathcal P_b|.
Expand the left side and apply Lemma 7 with y=x_kx_j^{-1}. The positive congruence modulo b^2 holds only when j=k. Its contribution is \frac{\varphi(b^2)}2(b-1) = \frac b2(b-1)^2. The negative congruence modulo b^2 would require (j+k)(b+1)\equiv-2\pmod{b^2}. Since (b-1)(b+1)\equiv-1\pmod{b^2}, this forces j+k\equiv2b-2\pmod{b^2}. The range 0\le j+k\le2b-4 makes that impossible.
Modulo b, the positive congruence again holds on the b-1 diagonal pairs. The negative congruence is j+k=b-2, which also has b-1 ordered solutions. Their two kernel contributions have equal magnitude and opposite sign. They cancel. The surviving term is \sum_{\chi\in\mathcal P_b}|A(\chi)|^2 = \frac b2(b-1)^2 = b|\mathcal P_b|. Multiplication by four through (3) proves the theorem. ◻
The theorem removes aggregate uncertainty from the finite factor. Any remaining base dependence in the special-value moment must come from the values at one and their channelwise relation with the collision geometry.
Define the centered collision energy E_b=\sum_{a\in U_{b^2}}|S^\circ(a)|^2.
Theorem 9 (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. ◻
The finite collision energy therefore evaluates a weighted second moment of the values L(1,\chi) exactly.
Corollary 10 (Normalized collision weights). For \chi\in\mathcal P_b, define w_b(\chi) = \frac{|S_G(\chi)|^2}{2b(b-1)^2}. Then \sum_{\chi\in\mathcal P_b}w_b(\chi)=1 and \sum_{\chi\in\mathcal P_b} w_b(\chi)|L(1,\chi)|^2 = \frac{\pi^2E_b}{2b^2(b-1)}.
Proof. The first identity is Theorem 8. Divide Theorem 9 by 2b(b-1)^2 and use \varphi(b^2)=b(b-1). ◻
The normalized squares divide one unit of collision weight among the primitive odd channels. Equation (5) is therefore an exact expectation of |L(1,\chi)|^2 under a distribution supplied by the digit geometry.
Theorem 11 (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 12 (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 11 gives |S_G(\chi)| = \frac{5\sqrt5}{\pi}|L(1,\chi)|. Substitution in Theorem 5 proves the result. ◻
Theorem 13 (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 [11] 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 12 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 Bernoulli-to-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)| [9].
The nfield enumeration [12] checks every primitive odd character for every odd prime base from 5 through 71. This is an exhaustive ledger of 14{,}372 channels. 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 evaluates Y_\chi through its finite Bernoulli sum at every declared character, so no auxiliary prime cutoff enters the ledger. The values in Table 1 are rounded to four decimal places. The ledger is exhaustive over the declared bases and character families. Its behavior as the base grows remains open.
The underlying Hamming observable, generalized Bernoulli and L-value identity, character orthogonality, Parseval, Ramanujan sums, and Pearson correlation are classical. Their role here is exact and visible. The collision coefficient separates into a classical analytic boundary value and a short sum supplied by the equality diagonal.
The new finite statement is that the diagonal factor has a completely determined aggregate scale. Its squared magnitudes sum to 2b(b-1)^2 and therefore divide one unit of weight among the primitive odd channels. Parseval then turns the centered collision energy into the corresponding weighted mean of |L(1,\chi)|^2.
Base five closes the channelwise relation as well. The cyclotomic identity, collision energy 48, and fourth moment 192\pi^4/625 are exact identities for the eight primitive odd characters modulo 25, not asymptotic moment claims. Beyond base five, the exhaustive finite ledger records a positive relation but proves no limiting law. The unresolved problem is to control the short diagonal sum against the Bernoulli magnitude as the prime base grows. That boundary is where the finite collision spectrum ends.
[1]A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inform. Theory 20 (1974), no. 1, 90–94. https://doi.org/10.1109/TIT.1974.1055169
[2]S. C. Kak and A. Chatterjee, On decimal sequences, IEEE Trans. Inform. Theory 27 (1981), no. 5, 647–652. https://doi.org/10.1109/TIT.1981.1056394
[3]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
[4]K. Girstmair, Digit variance and Dedekind sums, J. Number Theory 65 (1997), no. 2, 197–205. https://doi.org/10.1006/jnth.1997.2149
[5]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
[6]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
[7]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
[8]A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9780511626265
[9]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
[10]H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000.
[11]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.
[12]A. S. Petty, nfield, software repository. https://github.com/alexspetty/nfield
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