arXiv subjects

David Niedbala Giraudin

Publications and source records attributed to David Niedbala Giraudin.

9 recordsLinked to original sources

arXiv2026

New bounds for the integer Chebyshev constant of [0,1]

The integer Chebyshev constant of [0,1] has been bracketed by 0.4213 <= t_Z([0,1]) <= 0.422685 since the work of Pritsker (2005) and of Flammang (2014). We prove 0.4222286000 <= t_Z([0,1]) <= 0.4226846976, narrowing the interval from 1.39e-3 to 4.56e-4. Both bounds are established by explicit certificates. The upper bound is carried by an integer polynomial with 92 factors and explicit integer exponents; the lower bound by a discrete measure on 947 cells together with a library of 206 irreducible polynomials, through an inequality in which the terms paying for divisibility are made explicit. Two self-contained scripts, included as ancillary files, re-derive both bounds from the certificates alone.

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arXiv2026

A closed-form law for the Salikhov-Zeilberger-Zudilin-Bai family, and computational evidence that Bai's point is optimal

Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) bound the irrationality measure of pi with the same shape of complex integral. This note gives one closed-form law mu(a0,a1,b) for the whole family, shows that it reproduces all three published records, proves that it is exactly scale-invariant, and gives evidence that Bai's parameter point is the global optimum of the family - a statement Bai explicitly declines to make about her own result. The admissible configurations are forced by a Machin condition: zeros at 0, +-w, +-wbar and poles at +-P give a linear form in 1 and pi if and only if P^2 - N(w) = 2P Im(w). The law depends only on the two ratios a0/a1 and b/a1, so the infinite three-parameter lattice is exactly two-dimensional. The 2-adic saving is exactly 5(2a1-b)/2 per n, in two regimes, and no further saving exists. The decay rate is governed by the min-max value of the steepest-descent method, the lowest level at which the two endpoints of the integral become connected within a sublevel set of the integrand; the value obtained from the least critical level is a lower bound for mu and serves as a screening. Among the sixteen admissible configurations with P < 100, exactly one carries a triple that yields a bound, namely Salikhov's (P = 5, w = 1+2i), and the least mu over all sixteen and over all primitive triples with a0, a1 <= 10 and b <= 22 is 7.103205334137 at (2,2,3). Within that configuration the minimum is 7.101862832356 at (1857,1857,2785), which is Bai's point, the vertex sitting exactly at Q = 2a/(3a-2b) = 3714. Each statement carries an explicit label, PROVED or VERIFIED or CONJECTURE, and the reservations of scope are stated in the note. An ancillary file re-derives every numerical claim from scratch, with no external dependency.

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arXiv2026

A new upper bound for the irrationality exponent of $ζ(2)$

We prove that the irrationality exponent of $ζ(2)=π^2/6$ satisfies $μ(ζ(2))<5.0193784$. The best published bound is 5.095412 (Zudilin, 2014); a bound 5.0495243 with a machine-checked proof was made public in September 2026 by J. Kleid. We use Zudilin's two hypergeometric constructions at a new choice of parameters. The two families of linear forms in 1 and $ζ(2)$ so obtained coincide for every sufficiently large index: this coincidence, an instance of an identity conjectured by Zudilin, is established by comparing an arithmetic bound for the denominators of the difference with an analytic bound for its size, without creative telescoping. All numerical constants are certified in ball arithmetic.

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arXiv2026

The proportion of polynomials over F_2 with positive orthogonal multiplicity

Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d >= 1 is the number of g in A with deg g < d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f) > 0 behaves as d tends to infinity. We attach to each prime divisor P of f a residue psi_P(f) in F_2 of the differential dT/(T(T+1)f), and prove two unconditional facts about this family: a reciprocity relation, namely that the psi_P(f) sum to zero whenever f is prime to T(T+1), and an exact local distribution, namely that psi_P vanishes for a proportion exactly (|P|-2)/(2(|P|-1)) of the admissible completions when P exactly divides f, and for exactly half of them when P^e exactly divides f with e >= 2. Granting that these local conditions pool independently, we obtain p(d) ~ L d^(-1/2) with L = (9/(4 sqrt(pi))) times the product over the monic irreducibles P of A of degree at least two of (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2), equal to 1.2116120743... The same Euler product evaluated at a second parameter returns the exact identity that the m(f) of a given degree sum to 2^d, which fixes its normalisation. We give the exact value of 2^d p(d) for 1 <= d <= 33; the predicted values agree with these to within 5 10^(-5) for 27 <= d <= 33.

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arXiv2026

An explicit family of twists of the Carlitz module with unbounded analytic rank

Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q >= 3. For q = p^s and n >= 1 put P_{q,n} = (theta^(q^n) - theta)^(q-2). We show that the directed graph of the matrix M(P_{q,n},1,k) of Grishkov-Logachev has a completely explicit set of circuits: they are pairwise disjoint, none of them meets an entry involving t, and they are in bijection with the orbits of E^n under cyclic rotation, where E is the set of l in [0,q-2] with p not dividing l+1. Consequently L(C_P, t, T) does not depend on t, it factors as a product of one term 1 - wt(nu) T^per(nu) per orbit, and the analytic rank equals the sum of p^(v_p(per(nu))) over the orbits of weight 1. When n is a power of p this equals (q/p)^n (p-1)^(n-1), which is unbounded. The argument also answers questions 3.6.1 to 3.6.4 of the survey for this family.

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arXiv2026

An infinite family of counterexamples to Goss's conjecture on L-functions of cyclotomic function fields

Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) <= 1 for every q-magic index i; this is an analogue for function fields of Vandiver's conjecture, and was raised as an open problem by Angles. We disprove it. For every a in F_p^*, put P_a = T^p - T - a, irreducible of degree p, and let i = p^n - 1 with 0 <= n <= p-1, a q-magic index. We prove an exact congruence showing that the L-polynomial reduces to (1-X)^n modulo P_a, whence deg_X g = n-1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting Goss's conjecture; moreover deg_X g = n-1 is unbounded, attaining p-2.

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arXiv2026

A new lower bound for the Schur-Siegel-Smyth trace problem

We prove that lambda_SSS >= 1.80220 for the smallest limiting trace-to-degree ratio of totally positive algebraic integers, improving the bound 1.80203 obtained by Orloski, Talebizadeh Sardari and Smith. The proof exhibits an explicit probability measure on [0,8] together with eighteen integer polynomials, and verifies their dual inequality on the whole of [0,infinity) by interval bisection. The certificate assumes nothing about the data it is built from: an error in that data can only weaken the bound, never invalidate it.

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arXiv2026

A new upper bound for the irrationality exponent of zeta(3)

We prove that the irrationality exponent of $ζ(3)$ satisfies $μ(ζ(3)) < 5.5138800$, improving the bound $5.513891$ obtained by Rhin and Viola in 2001, which has been the best known for twenty-five years. The proof is an application of their Theorem 5.1 at a new integral direction. That direction is not reached by enlarging the search range: its eight parameters are 52000 times theirs, shifted by at most 23. The mechanism is the one recently used by Bai for $π$. In particular the Rhin-Viola point, optimal in every region covered by enumeration, is not optimal.

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arXiv2026

A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes

Let A = F_q[T] with q a power of an odd prime p, let [n] = T^(q^n) - T, and let rho be the Carlitz module. A monic prime P of A is a c-Wieferich prime (to base 1) if rho_P(1) = 1 mod P^2. Thakur suggested in 2015, on the basis of limited data and of proofs in degrees 2 and 3, that in odd characteristic every c-Wieferich prime has degree divisible by p; the question was restated as open in 2024, and Bamunoba and Bergstrom, after extensive computations, expressed the belief that the statement holds in odd characteristic. We show that it is false: an explicit irreducible c-Wieferich prime of degree 5 over F_{19^3} is exhibited, with 19 not dividing 5. We further give a closed form for the resulting common factor of [5] and M_5: it equals mu(T^q - T) for an explicit quintic mu with coefficients in the prime field F_19, squarefree of degree 5*19^3, and it divides gcd([5], M_5); we conjecture equality, and verify it for the part of low degree over the prime field. Degree 5 is the least possible degree of such a counterexample, and exhaustive computations show that no counterexample exists over the prime fields F_p in a substantial range of degrees and characteristics. The proof that degree 5 is minimal, and the method by which the example was found, appear in a companion paper.

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