Geometric Moments and Rational Lambert Values: Proofs and Further Results

Functional nonrationality is not value irrationality.

At the level of functions, Rivin proves that if a sequence $\gamma$ and its divisor-sum sequence are both eventually linearly recurrent, then $\gamma$ is finitely supported [24]. His periodic-coefficient corollary therefore shows that

is not a rational function [24]. This is an exact structural statement about the function underlying Problem #1049, but it gives no irrationality statement for a special value at $z=1/t$: a nonrational function may take rational values at particular rational points.

Padé and orthogonal-polynomial constructions.

In 1991 Borwein proved the irrationality of shifted series $\sum _{n\ge 1}(t^n+w{)}^{-1}$ at integer bases $t\ge 2$, for every nonzero rational $w$ with $w\ne -t^m$ for all $m\ge 1$, by Padé approximation rather than by digit clearing; his estimates also show that these values are not Liouville numbers [3]. In 2001 Van Assche recovered the integer-base irrationality and the bound ${\mu }_{\mathrm{i}\mathrm{r}\mathrm{r}}(F(p))\le 2{\pi }^2/({\pi }^2-2)=2.50828\dots$ using little $q$-Legendre Padé approximants  [18]. His more general Theorem 3 proves irrationality of $\sum _{k\ge 1}(c{p}^k-1{)}^{-1}$ for an integer $p>1$ and a fixed nonzero rational $c$ with $c{p}^k\ne 1$ for every $k\ge 1$  [18]; it does not cover a rational noninteger base $t$ in $F(t)$, because the multiplier needed to write $t^k$ over an integer base varies with $k$.

Related series and their different recurrences.

The same Lambert value was already the target of Amdeberhan and Zeilberger’s $q$-WZ construction [5]. Here $p>1$ is the integer-base parameter, $q=p^{-1}$, and $P_n(x\mid q)$ denotes the little $q$-Legendre polynomial used by these authors, normalised by $P_n(0\mid q)=1$. The two constructions share that bivariate little-$q$-Legendre Padé kernel, but Van Assche’s diagonal does not satisfy the Amdeberhan–Zeilberger scalar recurrence. Their Theorems 1 and 2 do prove, respectively, the irrationality of the non-alternating $h_p(1)$ and alternating $q$-logarithm values for their stated integer parameters, with reported irrationality measure $4.80$; those are external integer-parameter results and do not transfer to the rational noninteger base $3/2$ studied here.

Amdeberhan–Zeilberger use the moving diagonal $P_n(p^{n+1}\mid p^{-1})$, whereas Van Assche uses $P_n(p^n\mid p^{-1})$. Van Assche also records, citing Borwein’s 1992 Lemma 2, the neighbouring evaluation $P_{n-1}(c{p}^{n+1}\mid p^{-1})$ [18]; see [4]. The shared kernel therefore does not license transfer of recurrence, endpoint, lattice, or valuation claims between the diagonals. Indeed, if $A_n(p)=P_n(p^n\mid p^{-1})$, exact substitution at $n=0$ into the Amdeberhan–Zeilberger operator [5] leaves

which is nonzero for every real $p>1$. One nonzero residual is decisive for non-transfer of that recurrence.

Lean checks the exact residual factorisation and its nonvanishing for $p>1$. These are finite statements at $n=0$; they do not supply a recurrence for either diagonal. No recurrence, endpoint, lattice, or valuation statement for one diagonal is used for the other; the displayed exact residual is the sole claim made here about their incompatibility.

Coussement and Smet use little $q$-Jacobi polynomials for further Lambert series whose base parameter is the reciprocal of an integer [19], and Koizumi and Yokoi identify one of their three-parameter Apéry-type approximations with the Coussement–Smet Padé approximants [20]. That identification includes the source’s moving evaluation point $z_n=p_2^{n+1}$ and its normalization; it is not an identification of all little-$q$-Jacobi diagonals or their scalar recurrences.

At a noninteger rational base, convergence of the rational approximants is not enough. If $P_n/Q_n\to \xi$, the cleared error is $Q_n\xi -P_n=Q_n(\xi -P_n/Q_n)$, whose size also depends on $Q_n$. For example, $1/n\to 0$, but the corresponding cleared error is always $-1$. Thus convergence alone does not give the small integer linear forms required for irrationality.

In 2013 Vandehey proved that $\sum _{n\ge 1}d(n)a_n/b^n$ is irrational whenever $b>1$ is an integer and $(a_n)$ ranges in a finite integer alphabet excluding zero; taking $a_n=(-1{)}^n$ completes the elementary digit method for integer bases $b\le -2$ [21]. His companion theorem permits a finite alphabet of nonnegative integers containing zero, provided the coefficient sequence is not eventually zero [21]. All of these statements require an integer base. Luca and Tachiya’s periodic-coefficient theorem likewise proves irrationality at every integer base of absolute value greater than one; their full-support example strengthens this to joint linear independence for finite families of iterated divisor functions [22]. Duverney and Tachiya refine the Chowla–Erdős method to linear independence results for Lambert series at an integer base $q$ with $|q|>1$ [23]; their Corollary 1.1 generalises Vandehey’s Theorem 1.2 [23]. Each of these results assumes an integer base. For the underlying small-integral-tail criterion, see also Duverney [35]. We use that elementary criterion only as methodological context, not his separate square-index linear-independence argument.

In 2004 Zudilin obtained the uniform bound ${\mu }_{\mathrm{i}\mathrm{r}\mathrm{r}}(F(p))\le 2.46497868\dots$ for every integer $p\ge 2$, by way of Heine’s basic transform and a permutation group  [8]. His source theorem also covers negative integer inverse bases; our notation $F(t)$ and positive-remainder arguments are restricted to $t>1$. The ordinary-hypergeometric antecedent is Rhin and Viola’s $S_5$ action and twelve-coset denominator reduction for rational forms in $\zeta (2)$ [17].

These sources suggest looking for common polynomial factors before specialisation. The congruence lemmas in Section 5 describe what the first and last coefficients can imply at $3/2$. Their hypotheses have not been proved there for either cited coefficient family.

The neighbouring problem of Lambert subseries $\sum _{n\in A}(t^n-1{)}^{-1}$ over a restricted index set $A$ is treated by Kovač and Tao [25].

The rational non-integer progress relevant here is instead the height criterion of Bundschuh and Väänänen [6], which gives the strongest explicit numerical threshold for this value among the previously published sources compared here; Theorem 2.2 enlarges the region that criterion covers. Zudilin’s later Padé-and-Hankel treatment of the generalized $q$-logarithm also comments on non-integral rational bases: it remarks that its results can be given for non-integer $p=r/s$ with $|p|>1$, under an assumption $\log |r|>c\log |s|$ for some computable $c>0$ [9]. No value of $c$ is computed there. That remark announces the shape of a rational-base extension for the generalized $q$-logarithm of that paper. Section 2 proves an explicit extension for $F$ using the different, 2004 construction. Its polynomial cancellation and denominator calculation show that $c=\mu =2.464978683574975\dots$, the 2004 bound quoted above, is admissible, and gives the resulting region and its application to $31/4$.

A third published rational-base region for this same series is Duverney’s Théorème 2, whose hypothesis is $\log |s|/\log |r|<\frac{1}{3}(1-3/{\pi }^2)=0.2320\dots$ [14]; his Théorème 1 for general numerators is weaker still. Both cutoffs lie strictly inside the Bundschuh and Väänänen region. Matala-aho, Väänänen and Zudilin’s combined treatment of $q$-logarithms keeps the integer hypothesis $p=1/q\in \mathbb{Z}\mathrm{\backslash }\{0,\pm 1\}$ throughout, and states in print that its methods do not sharpen the $q$-harmonic case of [8] [15]. So none of the printed regions for $F$ compared here reaches $\log 4/\log 31$.