Geometric Moments and Rational Lambert Values: Proofs and Further Results

Introduction

For $t>1$, the Lambert series

has two useful descriptions: as a sum of simple fractions and as a divisor generating series. Here $\tau (n)$ counts the positive divisors of $n$. The equality follows by expanding each fraction, since all terms are nonnegative; the bound $\tau (n)\le n$ ensures convergence. Chowla conjectured that $F(t)$ is irrational for every rational $t>1$, as recorded by Erdős [2] and in Bloom’s catalogue [26]. Erdős proved the integer case in 1948 [1].

Two constructions of Zudilin enter, for different reasons. The 2016 approximation [9] supplies positive moments, whose Hankel determinants we evaluate asymptotically. The 2004 forms [8] supply polynomial coefficients whose cancellation and rational evaluation give an irrationality region. A denominator estimate for the second construction is not an estimate for the first; each coefficient family needs its own arithmetic analysis. The shorter paper, Hankel Determinants of Geometric Moments and Rational Lambert Values, begins with the determinant result. Here the polynomial forms come first, making their exact arithmetic available for the later comparisons. Readers following only the determinant argument may start with Section 3: its moment calculation does not use the rational-base transfer.

For positive weights with fixed-shift ratios tending to $1$ and a uniform polynomial bound on those ratios, we prove

where $P=(q;q{)}_{\mathrm{\infty }}$, $B_N=\sum _{j<N}{j}^2$ and $\mathcal{M}(q)=\prod _{d\ge 1}(1-q^d{)}^{-d}$. Heine’s identity expands the determinant into squared Vandermonde products. We divide by the contribution from the first $N$ nodes and index the remaining tuples by partitions. A bound summable uniformly in $N$ reduces their limiting sum to the unit-weight Cauchy determinant. For Zudilin’s moments, two finite convolutions then give the correction $N^{-8F(1/q)}$ and its constant. We also determine the first formal term $C_N{q}^{B_N}$, where $C_N=(N!{)}^2(N+1)!/2^N$.

The formal first term fixes the rank and expands at $q=0$; the analytic asymptotic fixes $q$ and lets the rank grow. Neither controls the coefficient denominators of an irrationality form. The ratio-limit remark extends the moment calculation to limiting adjacent-weight ratios, while Section 3.3 constructs a rational-value model with the same Hankel features. The latter shows why those features alone cannot imply irrationality. Both arguments are ordinary proofs without Lean proofs in the supplied record, and neither is used for the checked determinant asymptotic.

For the 2004 forms, we calculate the degree after cancelling the common polynomial factors. At a reduced rational base $a/b>1$, the resulting polynomials $U_n,V_n$ have degrees $W_n,W_n-1$ and satisfy

The form inside the logarithm is positive. Thus it tends to zero when $\log b/\log a<{\theta }^{\ast }=C_0/C_1=0.4056830213840605\dots$. Rationality of $F(a/b)$ would make a fixed integer multiple of every such value a positive integer, giving a contradiction. The constants and parameter direction are Zudilin’s; the cancellation and rational evaluation give the region used here. It contains $31/4$ and all its positive integral powers. Bundschuh and Väänänen already treated a smaller rational-base region containing $7/2$ [6]; Section 8 checks that specialisation. The value $3/2$ lies outside both regions and remains unresolved by these arguments. It has the smallest height among reduced noninteger rationals greater than $1$, where the height of $r/s$ is $max(r,s)$.

Section 2 gives the rational-base proof and its degree restrictions. Section 3 contains the determinant arguments and the coefficient-moment calculations. The coefficient pencil is treated for sizes $N\le 8$ and every $p>1$; the computational ranges for shifted minors and coefficientwise positivity are specified there. Positivity for every $N$ and convergence of the largest root remain open. Sections 4–5 apply rescaling and additive congruences to divided forms, including a quantitative bounded-fibre criterion. Sections 6–9 give the clearing, tail and exponent calculations, and Section 10 distinguishes unrestricted reformulations of irrationality from questions about specified approximation families.

All logarithms are natural, and empty products and determinants equal $1$. Constants in $O_x(\cdot )$ may depend on the fixed base. We introduce the notation for each construction as it is needed. Appendix A identifies the formal sources and the separate finite computations.

Keywords. irrationality; Lambert series; rational base; Padé approximation; Lean 4. MSC 2020. 11J72 (primary); 11J82, 68V20 (secondary).