IRRATIONALITY MEASURES FOR q-ZETA VALUES^{1}

Wadim Zudilin (Moscow)

1. q-Introduction. As usual, quantities depending on a number q and becoming classical objects as q → 1 (at least formally) are regarded as q-analogues or q-extensions. A possible way to q-extend the values of Riemann’s zeta function reads as follows (here q ∈ ℂ, |q| < 1):

ζ_{q}(k) = ∑_{n=0}^{∞} σ_{k−1}(n)q^{n} = ∑_{ν=0}^{∞} \frac{q^{ν}ρ_{k}(q^{ν})}{(1 − q^{ν})^{k}} ,  k = 1, 2, …,  (1)

where σ_{k−1}(n) = ∑_{d|n} d^{k−1} is the sum of powers of divisors and the polynomials ρ_{k}(x) ∈ ℤ[x] can be determined recursively by the formulae ρ_{1} = 1 and ρ_{k+1} = (1 + (k − 1)x)ρ_{k} + x(1 − x)ρ′_{k} for k = 1, 2, …. Then we obtain

lim_{\begin{matrix}q→1 \\ |q|<1 \end{matrix}}(1 − q)^{k}ζ_{q}(k) = ρ_{k}(1) · ζ(k),  k = 2, 3, ….  (2)

The above q-zeta values (1) present several problems in transcendence number theory that are extensions of the corresponding problems for ordinary zeta values; we state some of these problems at the end of the talk. Here we would like to explain how some recent contributions to the arithmetic study of ζ(k), k = 2, 3, …, successfully work for q-zeta values. Namely, we mean the hypergeometric construction of linear forms (due to Nikishin, Gutnik, Nesterenko) and the arithmetic approach (due to Chudnovsky, Rukhadze, Hata) accompanied with the group-structure scheme (due to Rhin, Viola). We consider the quantities ζ_{q}(1) and ζ_{q}(2) for q^{−1} = p ∈ ℤ \ {0, ±1} and start with the following table illustrating a connection of some standard ordinary and q-objects.

ordinary objects  q-extensions, p = 1/q ∈ ℤ \ {0, ±1}
numbers n ∈ ℤ  ‘numbers’ [n]_{p} = \frac{p^{n} − 1}{p − 1} ∈ ℤ[p]

primes l ∈ {2, 3, 5, 7, … } ∈ ℤ

irreducible reciprocal polynomials

Φ_{l}(p) = ∏_{\begin{matrix}k=1 \\ (k,l)=1 \end{matrix}}^{l}(p − e^{2πik/l}) ∈ ℤ[p]

Euler’s gamma function Γ(t)

Jackson’s q-gamma function

Γ_{q}(t) = \frac{∏_{ν=1}^{∞}(1 − q^{ν})}{∏_{ν=1}^{∞}(1 − q^{t+ν−1})} (1 − q)^{1−t}

the factorial n! = Γ(n + 1)

n! = ∏_{ν=1}^{n} ν ∈ ℤ

the q-factorial [n]_{q}! = Γ_{q}(n + 1)

[n]_{p}! = ∏_{ν=1}^{n} \frac{p^{ν} − 1}{p − 1} = p^{n(n−1)/2}[n]_{q}! ∈ ℤ[p]

ord_{l} n! = ⌊\frac{n}{l}⌋ + ⌊\frac{n}{l^{2}} ⌋ + ⋯  ord_{Φ_{l}(p)}[n]_{p}! = ⌊\frac{n}{l}⌋, l = 2, 3, 4, …

1An extension of the talk given at Luminy’s conference Problèmes Diophantiens (CIRM, 6–10 May 2002).

2 IRRATIONALITY MEASURES FOR q-ZETA VALUES

ordinary objects  q-extensions, p = 1/q ∈ ℤ \ {0, ±1}
D_{n} = l.c.m.(1, …, n)
= ∏_{primes l⩽n} l^{⌊log n/ log l⌋} ∈ ℤ

\begin{aligned}D_{n}(p) & = l.c.m.([1]_{p}, …, [n]_{p}) \\ & = ∏_{l=1}^{n} Φ_{l}(p) ∈ ℤ[p] \end{aligned}

the prime number theorem  Mertens’ formula
lim_{n→∞} \frac{log D_{n}}{n} = 1  lim_{n→∞} \frac{log |D_{n}(p)|}{n^{2} log |p|} = \frac{3}{π^{2}}

If ψ(x) is the logarithmic derivative of Euler’s gamma function and {x} = x − ⌊x⌋ is the fractional part of a number x, then, for each demi-interval [u, v) ⊂ (0, 1), Mertens’ formula yields the limit relation

lim_{n→∞} \frac{1}{n^{2} log |p|} ∑_{l:\{n/l\}∈[u,v)} log |Φ_{l}(p)| = \frac{3}{π^{2}} (ψ′(u) − ψ′(v)) = \frac{3}{π^{2}} ∫_{u}^{v} d(−ψ′(x)),  (3)

which can be regarded as a q-extension of the formula

lim_{n→∞} \frac{1}{n} ∑_{\begin{matrix}l:\{n/l\}∈[u,v) \\ primes l>\sqrt{Cn} \end{matrix}} log l = ψ(v) − ψ(u) = ∫_{u}^{v} dψ(x)

in the arithmetic approach.

2. Rational approximations to q-zeta values and basic transformations. Let a_{0}, a_{1}, a_{2}, and b be positive integers satisfying the condition a_{1} + a_{2} ⩽ b. Then, Heine’s series

F (a, b) = \frac{Γ_{q}(b − a_{2})}{(1 − q)Γ_{q}(a_{1})} ∑_{t=0}^{∞} \frac{Γ_{q}(t + a_{1}) Γ_{q}(t + a_{2})}{Γ_{q}(t + 1) Γ_{q}(t + b)} q^{a_{0}t}

becomes a ℚ(p)-linear form F (a, b) = Aζ_{q}(1) − B with the property

p^{−M}D_{m}(p) · F (a, b) ∈ ℤ[p]ζ_{q}(1) + ℤ[p].  (4)

Here M = M(a, b) is some (explicit) integer and m is the successive maximum of the 6-element set

\begin{matrix}c_{00} = a_{0} + a_{1} + a_{2} − b − 1,  c_{01} = a_{0} − 1,  c_{11} = a_{1} − 1,  c_{21} = a_{2} − 1, \\ c_{12} = b − a_{1} − 1,  c_{22} = b − a_{2} − 1. \end{matrix}

Taking H(c) = F (a, b) and using the stability of the quantity

\frac{F (a_{0}, a_{1}, a_{2}, b)}{Γ_{q}(a_{0}) Γ_{q}(a_{2}) Γ_{q}(b − a_{2})} = \frac{H(c)}{Π_{q}(c)}, where Π_{q}(c) = [c_{01}]_{q}! [c_{21}]_{q}! [c_{22}]_{q}! = p^{−N(c)}Π_{p}(c),

under the actions of

τ = (c_{22} c_{21} c_{01} c_{11} c_{12} c_{00}): (a_{0}, a_{1}, a_{2}, b) ↦ (a_{1}, b − a_{1}, a_{0}, a_{0} + a_{2}),
σ = (c_{11} c_{21})(c_{12} c_{22}):  (a_{0}, a_{1}, a_{2}, b) ↦ (a_{0}, a_{2}, a_{1}, b)

we arrive at the better than (4) inclusions

p^{−M}D_{m}(p)Ω^{−1}(p) · F (a, b) ∈ ℤ[p]ζ_{q}(1) + ℤ[p]  (5)

with

Ω(p) = ∏_{l=1}^{m} Φ_{l}^{ν_{l}}(p),  ν_{l} = max_{𝔤∈⟨τ^{2},σ⟩} ord_{Φ_{l}(p)} \frac{Π_{p}(c)}{Π_{p}(𝔤c)}.  (6)

In addition, trivial estimates for F (a, b) and explicit formulae for the coefficient A imply that

|F (a, b)| = |p|^{O(b)},  |A| ⩽ |p|^{(a_{0}+a_{1}+a_{2})b−(a_{1}^{2}+a_{2}^{2}+b^{2})/2+O(b)}  (7)

with some absolute constant in O(b).

The basic transform τ of order 6 was proved by Heine more than 150 years ago. The group 𝔊 = ⟨τ, σ⟩ of order 12 has no ordinary analogue since corresponding (in limit q → 1) Gauß’s hypergeometric series is divergent. We use the group ⟨τ^{2}, σ⟩ of order 6 instead of the total available group 𝔊 to ensure the required condition a_{1} + a_{2} ⩽ b. Now, choosing a_{0} = a_{2} = 8n + 1, a_{1} = 6n + 1, and b = 15n + 1, and taking in mind (5), (7), and (3) we conclude that the irrationality exponent of ζ_{q}(1) satisfies the estimate

μ(ζ_{q}(1)) ⩽ 2.42343562 …

that can be compared with the previous result μ(ζ_{q}(1)) ⩽ 2π^{2}/(π^{2} − 2) = 2.50828476 … of Bundschuh and Väänänen (corresponding to the choice a_{0} = a_{1} = a_{2} = n + 1 and b = 2n + 2).

Similar arguments with a simpler group ⟨σ⟩ of order 2 can be put forward to improve Van Assche’s estimate μ(log_{q}(2)) ⩽ 3.36295386 … for the following q-extension of log(2):

log_{q}(2) = ∑_{ν=1}^{∞} \frac{(−1)^{ν−1}q^{ν}}{1 − q^{ν}} = ∑_{ν=1}^{∞} \frac{q^{ν}}{1 + q^{ν}} .

Namely, we are able to prove the estimate μ(log_{q}(2)) ⩽ 3.29727451 … for q^{−1} = p ∈ ℤ \ {0, ±1}.

In the case of ζ_{q}(2), we consider the positive parameters (a, b) = (a_{1}, a_{2}, a_{3}, b_{2}, b_{3}) satisfying the conditions a_{j} < b_{k}, a_{1} + a_{2} + a_{3} < b_{2} + b_{3} and the q-basic hypergeometric series

F̃(a, b) = \frac{Γ_{q}(b_{2} − a_{2}) Γ_{q}(b_{3} − a_{3})}{(1 − q)^{2}Γ_{q}(a_{1})} ∑_{t=0}^{∞} \frac{Γ_{q}(t + a_{1}) Γ_{q}(t + a_{2}) Γ_{q}(t + a_{3})}{Γ_{q}(t + 1) Γ_{q}(t + b_{2}) Γ_{q}(t + b_{3})} q^{(b_{2}+b_{3}−a_{1}−a_{2}−a_{3})t}
= Ãζ_{q}(2) − B̃.

Then p^{−M}D_{m_{1}}(p)D_{m_{2}}(p) · F̃(a, b) ∈ ℤ[p]ζ_{q}(2) + ℤ[p], where m_{1}, m_{2} are the two successive maxima of the 10-element set

c_{00} = (b_{2} + b_{3}) − (a_{1} + a_{2} + a_{3}) − 1, c_{jk} = \{ \begin{matrix}a_{j} − 1 & if k = 1, \\ b_{k} − a_{j} − 1 & if k = 2, 3, \end{matrix} j = 1, 2, 3,

and, in addition,

|F̃(a, b)| = |p|^{O(max{b_{2},b_{3}})}, |Ã| ⩽ |p|^{b_{2}b_{3}−(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})/2+O(max{b_{2},b_{3}})}.

The c-permutation group 𝔊 ⊂ 𝔖_{10} generated by all permutations of a_{1}, a_{2}, a_{3}, the permutation of b_{2}, b_{3}, and the permutation (c_{00} c_{22})(c_{11} c_{33})(c_{13} c_{31}) has order 120 and is known in connection with the Rhin–Viola proof of the new irrationality measure for ζ(2). In notation H̃(c) = F̃(a, b), the quantity

\frac{H̃(c)}{[c_{00}]_{q}! [c_{21}]_{q}! [c_{22}]_{q}! [c_{33}]_{q}! [c_{31}]_{q}!}

is stable under the action of the group 𝔊. This stability yields the inclusions

p^{−M}D_{m_{1}}(p)D_{m_{2}}(p)Ω̃^{−1}(p) · F̃(a, b) ∈ ℤ[p]ζ_{q}(2) + ℤ[p]

with a quantity Ω̃(p) defined like in (6). Finally, choosing a_{1} = 5n + 1, a_{2} = 6n + 1, a_{3} = 7n + 1, and b_{2} = 14n + 2, b_{3} = 15n + 2 we deduce the estimate

μ(ζ_{q}(2)) ⩽ 4.07869374 …

that cannot be compared with previous results, although the transcendence of ζ_{q}(2) for algebraic q with 0 < |q| < 1 follows from Nesterenko’s theorem.

It is nice to mention that the simpler choice a_{1} = a_{2} = a_{3} = n + 1, b_{2} = b_{3} = 2n + 2 also proves the irrationality of ζ_{q}(2) for q^{−1} ∈ ℤ \ {0, ±1} and the limit q → 1 produces Apéry’s original sequence of rational approximations to ζ(2).

We would like to stress that the both series F (a, b) and F̃(a, b) possess (multiple) q-integral representations as those considered by Rhin and Viola in their arithmetic study of ζ(2) and ζ(3). Inspite of this fact, there exists no general pattern to change the variable of q-integration, hence we do not expect a multiple-integral explanation of the above construction.

3. General problems for q-zeta values. We start this part with mentioning that, for an even integer k ⩾ 2, the series E_{k}(q) = 1 − 2kζ_{q}(k)/B_{k}, where B_{k} ∈ ℚ are Bernoulli numbers, is known to be the Eisenstein series. Therefore the modular origin (with respect to the parameter τ = \frac{log q}{2πi}) of E_{4}, E_{6}, E_{8}, … gives the algebraic independence of the functions ζ_{q}(2), ζ_{q}(4), ζ_{q}(6) over ℚ[q], while all other even q-zeta values are polynomials in ζ_{q}(4) and ζ_{q}(6). In this sence, the consequence of Nesterenko’s theorem “the numbers ζ_{q}(2), ζ_{q}(4), ζ_{q}(6) are algebraically independent over ℚ for algebraic q, 0 < |q| < 1” reads as a complete q-extension of the consequence of Lindemann’s theorem “ζ(2) is transcendental”.

The limit relations (2) as well as the expected algebraic structure of the ordinary zeta values motivate the following questions (we also regard ζ_{q}(1) to be an odd q-zeta value, although the corresponding ordinary harmonic series is divergent).

Conjecture 1. The q-zeta values ζ_{q}(1), ζ_{q}(2), ζ_{q}(3), … as functions of q are linearly independent over ℂ(q).

Conjecture 2. The q-functional set involving the three even q-zeta values ζ_{q}(2), ζ_{q}(4), ζ_{q}(6) and all odd q-zeta values ζ_{q}(1), ζ_{q}(3), ζ_{q}(5), … consists of functions that are algebraically independent over ℂ(q).

The associated diophantine problems are to prove the corresponding linear and algebraic independence over the algebraic closure of ℚ for algebraic q with 0 < |q| < 1. Even irrationality and ℚ-linear independence results at the point q ∈ ℚ with q^{−1} ∈ ℤ \ {0, ±1} look interesting problems.

Moscow Lomonosov State University

Department of Mechanics and Mathematics

Vorobiovy Gory, GSP-2, 119992 Moscow, RUSSIA

   URL: http://wain.mi.ras.ru/index.html
   E-mail address: wadim@ips.ras.ru