Linear Independence of 𝑞-Logarithms over the Eisenstein Integers

Abstract

For fixed complex 𝑞 with |𝑞|>1, the 𝑞-logarithm 𝐿_𝑞 is the meromorphic continuation of the series ∑_𝑛>0𝑧^𝑛/(𝑞^𝑛−1),|𝑧|<|𝑞|, into the whole complex plane. If 𝐾 is an algebraic number field, one may ask if 1,𝐿_𝑞(1),𝐿_𝑞(𝑐) are linearly independent over 𝐾 for 𝑞,𝑐∈𝐾^× satisfying |𝑞|>1,𝑐≠𝑞,𝑞²,𝑞³,…. In 2004, Tachiya showed that this is true in the Subcase 𝐾=ℚ, 𝑞∈ℤ, 𝑐=−1, and the present authors extended this result to arbitrary integer 𝑞 from an imaginary quadratic number field 𝐾, and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if 𝐾 is the Eisenstein number field ℚ(√−3), 𝑞 an integer from 𝐾, and 𝑐 a primitive third root of unity. Under these conditions, the linear independence holds also for 1,𝐿_𝑞(𝑐),𝐿_𝑞(𝑐⁻¹), and both results are quantitative.

1. Introduction and Results

For fixed complex 𝑞 of absolute value greater than 1, the 𝑞-logarithm 𝐿_𝑞 is defined by the power series𝐿_𝑞(𝑧)∶=^∞_𝑛=1𝑧^𝑛𝑞^𝑛,−1 (1.1) which converges in |𝑧|<|𝑞| and has the meromorphic continuation𝑧^∞_𝑛=11𝑞^𝑛−𝑧 (1.2) into the whole complex plane. In the early 1990s, the irrationality investigations on this 𝑞-logarithm got a fresh impetus by two papers of Borwein [1, 2], where he introduced new analytic tools to demonstrate quantitative versions of the following. If 𝑞∈ℤ⧵{0,±1} and if 𝑐∈ℚ^×⧵𝑞^ℕ, then both numbers 𝐿_𝑞(𝑐) and 𝐿_𝑞2(𝑐)−𝐿_𝑞2(𝑐𝑞) are irrational, with the second result appearing only in [2].

The next important step was made by Tachiya [3], who succeeded in proving, for 𝑞∈ℤ⧵{0,±1}, the linear independence of 1,𝐿_𝑞(1),𝐿_𝑞(−1) over ℚ using Borwein's function theoretic method from [2]. Shortly later, quantitative refinements of this result and also of the linear independence of 1,𝐿_𝑞(1),𝐿^′_𝑞(1) were obtained independently by Zudilin [4] and by the present authors [5]; here the dash indicates differentiation with respect to 𝑧. Somehow related to Tachiya's above-mentioned theorem is the linear independence over ℚ of 1,𝐿_𝑞(√𝑞),𝐿_𝑞(−√𝑞) for squares 𝑞∈ℤ⧵{0,1}, which was established in [6]. Another result, proved in [7], is the linear independence of 1,𝐿_𝑞(1),𝐿_−𝑞(1) for any 𝑞∈ℤ⧵{0,1}. It should be noted that all these linear independence statements remain true if one replaces ℚ by an arbitrary imaginary quadratic number field and if one supposes 𝑞 to be in its ring of integers.

One starting point of our present work was the question whether we can replace in Tachiya's result the primitive second root of unity (−1, of course) by a primitive third root of unity. As we will see in Theorem 1.2 below, this is indeed true if we study linear independence over the particular quadratic number field ℚ(√−3). The parameter 𝑞 has to be from its ring of integers, which is sometimes called ring of Eisenstein integers since Eisenstein (1844) was the first to thoroughly investigate its algebraic properties in the course of his proof of a cubic reciprocity law.

Another interesting question concerns the linear independence of 1 and the values of 𝐿_𝑞 at both primitive third roots of unity is to be answered quantitatively as follows.

Theorem 1.1. Let 𝑐 be a primitive third root of unity, and let 𝑂_𝐾 denote the ring of integers of 𝐾∶=ℚ(𝑐). Then, for any 𝑞∈𝑂_𝐾 with |𝑞|>1, the numbers 1,𝐿_𝑞(𝑐),𝐿_𝑞(𝑐⁻¹) are linearly independent over 𝐾. Moreover, there exists a constant 𝛾∈ℝ₊ depending at most on |𝑞| such that, for any 𝒬=(𝑄₀,𝑄₁,𝑄₂)∈𝑂³_𝐾 with |𝒬|∶=max(|𝑄₁|,|𝑄₂|) large enough, the inequality |𝑄₀+𝑄₁𝐿_𝑞(𝑐)+𝑄₂𝐿_𝑞(𝑐⁻¹)|≥|𝒬|^{−𝜂−𝛾(loglog|𝒬|)/(log|𝒬|)1/2} (1.3) holds with 𝜂∶=39.9475….

This question arose when preparing our recent work [8], where, as a very particular application, we obtained a quantitative version of the irrationality of the series𝑆_𝑞(𝑎)∶=^∞_𝑛=1𝑞^𝑛𝑞^2𝑛+𝑎𝑞^𝑛+1 (1.4) for 𝑞∈ℤ⧵{0,±1} and rational 𝑎 (with some necessary exceptions). Namely, it is easily seen that𝑆_𝑞𝐿(1)=^𝑞(𝑐)−𝐿_𝑞(𝑐⁻¹)𝑐−𝑐⁻¹ (1.5) holds for 𝑐 as in Theorem 1.1.

Next, we formulate our analogue of Tachiya's result for third roots of unity.

Theorem 1.2. Under the hypotheses of Theorem 1.1, the numbers 1,𝐿_𝑞(1),𝐿_𝑞(𝑐) are linearly independent over 𝐾. Moreover, there exists a constant 𝛾∈ℝ₊ depending at most on |𝑞| such that, for any 𝒬=(𝑄₀,𝑄₁,𝑄₂)∈𝑂³_𝐾 with |𝒬|∶=max(|𝑄₁|,|𝑄₂|) large enough, the inequality |𝑄₀+𝑄₁𝐿_𝑞(1)+𝑄₂𝐿_𝑞(𝑐)|≥|𝒬|^{−𝜂−𝛾(loglog|𝒬|)/(log|𝒬|)1/2} (1.6) holds with 𝜂∶=56.6026….

It should be noted that here the value of 𝜂 can be slightly decreased using more involved considerations, on which we will briefly comment at the end of Section 4 (see Remarks 4.1 and 4.2).

Of course, Theorems 1.1 and 1.2 together suggest the following problem. Is it true that the four numbers 1,𝐿_𝑞(1),𝐿_𝑞(𝑐),𝐿_𝑞(𝑐⁻¹) are linearly independent over 𝐾 assuming the hypotheses of our above results? We have to admit that, at least at the moment, we are not in a position to prove this statement. Another even more tantalizing problem is the natural question if it is possible to prove analogues of Theorems 1.1 and 1.2 for 𝑐 a primitive fourth or sixth root of unity, the other two cases, where ℚ(𝑐) is simultaneously a cyclotomic and an imaginary quadratic number field. On the difficulties with this problem we will make some comments at the end of Section 3 (see Remark 3.4).

To prove Theorems 1.1 and 1.2, we will essentially use our generalization [5] of Borwein's function theoretical method from [2]. In Section 2, the analytical tools are presented in a way suitable for both situations. Sections 3 and 4 contain the necessary arithmetic considerations to conclude the proofs of Theorems 1.1 and 1.2, respectively.

2. The Analytic Construction

The following extensive lemma contains all analytic information we need for the proofs of our main results.

Lemma 2.1. With 𝑞,𝑐₁,𝑐₂,𝑄₁,𝑄₂∈ℂ satisfying |𝑞|>max(1,|𝑐₁|,|𝑐₂|) define the meromorphic function 𝑉(𝑧)∶=^∞_𝑘=1(𝑄₁𝑞^𝑘−𝑐₁𝑧+𝑄₂𝑞^𝑘−𝑐₂𝑧) (2.1) and consider, for large parameters 𝐿₁,𝐿₂,𝑀,𝑁∈ℕ, the (positively oriented) integral 1𝐽∶=2𝜋𝑖_|𝑧|=1∏²_𝑡=1∏^𝐿𝑡_ℓ=1(𝑞^ℓ−𝑐_𝑡𝑧)𝑧^𝑀∏^𝑁_𝑛=1(1−𝑞^𝑛𝑧)𝑉(𝑧)𝑑𝑧. (2.2)
(a) Then the following explicit formula holds:𝐽=^𝑁_𝑛=1𝑅_𝑛(𝑉(1)−^𝑛2_𝑘=1𝑡=1𝑄_𝑡𝑞^𝑘−𝑐_𝑡)+_{𝜅+𝜆+𝜈=𝑀−1}𝑃_𝜅,𝜆⋅𝑄₁𝑐^𝜈₁+𝑄₂𝑐^𝜈₂𝑞^𝜈+1,−1 (2.3) where, for 𝑛=1,…,𝑁, 𝑅_𝑛∶=(−1)^{𝑁+1−𝑛}𝑞^{(𝑀−𝐿1−𝐿2)𝑛+𝑛(𝑛−1)/2}∏²_𝑡=1∏_{𝑛<ℓ≤𝑛+𝐿𝑡}(𝑞^ℓ−𝑐_𝑡)∏^𝑛−1_𝜈=1(𝑞^𝜈−1)⋅∏^𝑁−𝑛_𝜈=1(𝑞^𝜈−1) (2.4) and all 𝑃_𝜅,𝜆 in the above triple sum over all (𝜅,𝜆,𝜈) of nonnegative integers with 𝜅+𝜆+𝜈=𝑀−1 are in ℤ[𝑞,𝑐₁,𝑐₂].
(b) Supposing additionally that |𝑐₁|=|𝑐₂|=1,|𝐿₁−𝐿₂|=1 and defining 𝐿∶=max(𝐿₁,𝐿₂), one has the evaluation|^𝑁_𝑛=1𝑅_𝑛|=|𝑞|^{𝐿2+𝑀𝑁+𝑂(1)} (2.5) for 𝑀 large enough and, the 𝑂-constant depending on |𝑞| at most.
(c) Supposing, moreover, that 𝑐₁/𝑐₂∉𝑞^ℤ,𝐿₂=𝐿₁−1 if |𝑄₁|≤|𝑄₂| but 𝐿₂=𝐿₁+1 if |𝑄₁|>|𝑄₂|, then the asymptotic formula|𝐽|=max(|𝑄₁|,|𝑄₂|)⋅|𝑞|^{−(𝑀+𝑁−2𝐿)𝐿−𝑁2/2+𝑂(𝑁)} (2.6) holds as soon as 𝐿,𝑀,𝑁 have the same order of magnitude and 𝑀+𝑁−2𝐿 is large enough. Here the 𝑂-constant depends on |𝑞| and |𝑐₁−𝑐₂| at most.

Proof. (a) We apply the residue theorem to the integral 𝐽 defined in (2.2) and use the poles of the integrand in |𝑧|<1 noting that 𝑉(𝑧) is holomorphic in |𝑧|≤1, by (2.1) and the hypothesis |𝑐₁|,|𝑐₂|<|𝑞|. Thus, we obtain 𝐽=−^𝑁_𝑛=1𝑞^{(𝑀−𝐿1−𝐿2)𝑛}∏²_𝑡=1∏^𝐿𝑡_ℓ=1(𝑞^ℓ+𝑛−𝑐_𝑡)∏^𝑛−1_𝜈=1(1−𝑞^−𝜈)⋅∏^𝑁−𝑛_𝜈=1(1−𝑞^𝜈)⋅𝑉(𝑞^−𝑛)𝑞^𝑛+_{𝜅+𝜆+𝜈=𝑀−1}1(𝑑𝜅!)𝑑𝑧^𝜅{^2𝐿_𝑡=1^𝑡_ℓ=1(𝑞^ℓ−𝑐_𝑡𝑧)}|_𝑧=0⋅1(𝑑𝜆!)𝑑𝑧^𝜆𝑁_𝑛=1(1−𝑞^𝑛𝑧)⁻¹|_𝑧=0⋅𝑉^(𝜈)(0),𝜈! (2.7) the first sum coming from the simple poles at 𝑞^−𝑛(𝑛=1,…,𝑁) and the second one from the 𝑀-fold pole at the origin. The above first sum leads immediately to the 𝑛-sum in (2.3) using the definition of 𝑅_𝑛 in (2.4) and the fact 𝑉(𝑞^−𝑛)=𝑞^𝑛∞2_𝑘=1𝑡=1𝑄_𝑡𝑞^𝑘+𝑛−𝑐_𝑡=𝑞^𝑛2_{𝑘>𝑛𝑡=1}𝑄_𝑡𝑞^𝑘−𝑐_𝑡=𝑞^𝑛(𝑉(1)−^𝑛2_𝑘=1𝑡=1𝑄_𝑡𝑞^𝑘−𝑐_𝑡) (2.8) for any 𝑛∈ℕ. In the above triple sum, the factor in front of 𝑉^(𝜈)(0)/𝜈! is just what we denoted in (2.3) by 𝑃_𝜅,𝜆, hence 𝑃_𝜅,𝜆∈ℤ[𝑞,𝑐₁,𝑐₂] is evident. From (𝑑/𝑑𝑧)^𝜈(𝑞^𝑘−𝑐_𝑡𝑧)⁻¹=𝜈!𝑐^𝜈_𝑡(𝑞^𝑘−𝑐_𝑡𝑧)^−𝜈−1 for any 𝜈∈ℕ₀∶=ℕ∪{0} and from (2.1) we simply deduce that 1𝑉𝜈!^(𝜈)(0)=^∞_𝑘=1(𝑄₁𝑐^𝜈₁+𝑄₂𝑐^𝜈₂)𝑞^{−𝑘(𝜈+1)}, (2.9) whence, the triple sum in (2.3).
(b) If 𝑐∈ℂ satisfies |𝑐|=1, then we have for any ℓ∈ℕ1−|𝑞|^−ℓ≤|1−𝑐𝑞^−ℓ|≤1+|𝑞|^−ℓ<(1−|𝑞|^−ℓ)⁻¹, (2.10) whence, by (2.4), |𝑅_𝑛|<𝛾⁴₀|𝑞|^{(𝑀−𝐿1−𝐿2)𝑛+∑2𝑡=1∑𝑡𝑛+𝐿ℓ=𝑛+1ℓ−(1/2)(𝑁−𝑛)(𝑁+1−𝑛)} (2.11) with 𝛾₀∶=∏^∞_ℓ=1(1−|𝑞|^−ℓ)⁻¹. Notice that |𝑅_𝑛| can be bounded below by 𝛾₀⁻⁴ times the same |𝑞|-power as in (2.11). Our additional hypothesis on the 𝐿_𝑡's means that either 𝐿₁=𝐿,𝐿₂=𝐿−1 or 𝐿₁=𝐿−1,𝐿₂=𝐿; hence the exponent of |𝑞| in (2.11) equals 1(𝑀+1−2𝐿)𝑛+2²_𝑡=1𝐿_𝑡(𝐿_𝑡2+2𝑛+1)−𝑁+1−𝑛=𝑀𝑛+𝐿²−2.𝑁+1−𝑛 (2.12) Thus, we have for 𝑛=2,…,𝑁|𝑅_𝑛−1𝑅_𝑛|<𝛾⁸₀|𝑞|^{−𝑀−𝑁−1+𝑛}≤𝛾⁸₀|𝑞|^−𝑀−1 (2.13) and the right-hand side is ≤1/3 for 𝑀 large enough (in terms of |𝑞| only). Under this condition we find |𝑅_𝑁−1𝑅_𝑁+𝑅_𝑁−1𝑅_𝑁⋅𝑅_𝑁−2𝑅_𝑁−1𝑅+⋯+^𝑁−1𝑅_𝑁𝑅⋅⋯⋅¹𝑅₂1|<2, (2.14) whence, (1/2)|𝑅_𝑁|<|𝑅_𝑁+𝑅_𝑁−1+⋯+𝑅₁|<(3/2)|𝑅_𝑁| and the inequalities 𝛾₀⁻⁴|𝑞|^{𝐿2+𝑀𝑁}<|𝑅_𝑁|<𝛾⁴₀|𝑞|^{𝐿2+𝑀𝑁} (compare (2.11) and (2.12)) establish (2.5).
(c) We may assume that (𝑄₁,𝑄₂)≠(0,0) since otherwise (2.6) is trivial. In contrast to the situation in (a), to evaluate |𝐽| asymptotically from (2.2), we use the poles of the integrand outside the unit circle. As a matter of fact, one can easily show that −𝐽 is just the sum of the residues at all these (simple) poles appearing precisely at the points 𝑞^𝑘𝑡/𝑐_𝑡 with 𝑘_𝑡>𝐿_𝑡(𝑡=1,2). To justify this equality, the estimate |𝑉(𝑧)|=|𝒬|⋅𝑂(𝐻|𝑞|^−𝐻) on |𝑧|=|𝑞|^𝐻+(1/2) for large 𝐻∈ℕ is useful; the O-constant depends only on |𝑞|. The distinctness of the before-mentioned poles is guaranteed by our hypothesis 𝑐₁/𝑐₂∉𝑞^ℤ. Thus, we are led to an expression of 𝐽 as sum𝑄₁𝑐^{𝑀+𝑁−1−𝐿2}_1𝑘>𝐿1∏^𝐿1_ℓ=1(𝑞^ℓ−𝑞^𝑘)⋅∏^𝐿2_ℓ=1(𝑐₁𝑞^ℓ−𝑐₂𝑞^𝑘)𝑞^𝑁𝑀𝑘∏_𝑛=1(𝑐₁−𝑞^𝑛+𝑘) (2.15) plus the same sum, where the subscripts 1 and 2 are interchanged.
Denoting the 𝑘th summand in (2.15) by 𝑆_𝑘, similar considerations as for (2.11) show the existence of a constant 𝛾₁>1 depending only on |𝑞| and |𝑐₁−𝑐₂| such that𝛾₁⁻¹|𝑞|^{(2𝐿−1−𝑀−𝑁)𝑘−𝑁(𝑁+1)/2}≤|𝑆_𝑘|≤𝛾₁|𝑞|^{(2𝐿−1−𝑀−𝑁)𝑘−𝑁(𝑁+1)/2} (2.16) holds for every 𝑘>𝐿₁. This implies that |𝑆_𝑘+1𝑆_𝑘|≤𝛾²₁|𝑞|^{2𝐿−1−𝑀−𝑁} (2.17) for the same 𝑘, and here the right-hand side is bounded by 1/3, say, since 𝑀+𝑁−2𝐿 is supposed to be large enough. As in (b), this leads to 12|𝑆_𝐿1+1|≤|_𝑘>𝐿1𝑆_𝑘3|≤2|𝑆_𝐿1+1| (2.18) for the sum in (2.15). Thus, the absolute value of term (2.15) is bounded above by (3𝛾₁2)|𝑄₁||𝑞|^{−(2𝐿−1−𝑀−𝑁)(𝐿1+1)−𝑁(𝑁+1)/2} (2.19) and below by the same expression with 3𝛾₁/2 replaced by 1/(2𝛾₁) (compare (2.16)).
If 𝑄₁𝑄₂=0, we may assume that 𝑄₁≠0,𝑄₂=0 without loss of generality. Then we have 𝐿₁=𝐿−1,𝐿₂=𝐿, by one of our additional hypotheses in (c), and (2.15) and (2.19) together with the remark after (2.19) lead to (2.6) noting that all quotients of any two of 𝐿,𝑀,𝑁 are bounded above and below by certain absolute constants.
Suppose finally that 𝑄₁𝑄₂≠0. Denoting term (2.15) by 𝐽₁ and term (2.15) with subscripts 1 and 2 interchanged by 𝐽₂, we know that 𝐽₁𝐽₂≠0 (see (2.19) and the remark thereafter). Then, in the case |𝑄₁|≤|𝑄₂|, we have |𝐽|=|𝐽₂‖1+(𝐽₁/𝐽₂)| and|𝐽₁𝐽₂|≤(3𝛾₁/2)|𝑄₁|(1/(2𝛾₂))|𝑄₂||𝑞|^{(2𝐿−1−𝑀−𝑁)(𝐿1−𝐿2)}≤3𝛾₁𝛾₂|𝑞|^{2𝐿−1−𝑀−𝑁}, (2.20) by (2.19) and the corresponding lower bound for |𝐽₂|, where the constant 𝛾₁ from (2.19) is replaced by 𝛾₂. Hence |𝐽₁/𝐽₂|≤1/2 as soon as 𝑀+𝑁−2𝐿 is large enough, and this leads to (1/2)|𝐽₂|≤|𝐽|≤(3/2)|𝐽₂| giving (2.6) for |𝑄₁|≤|𝑄₂| if we use for |𝐽₂|evaluation (2.19) and thereafter with subscript 1 replaced by 2. The case |𝑄₁|>|𝑄₂| is treated analogously.

3. Proof of Theorem 1.1

For 𝑐 as in Theorem 1.1, we have (𝑋−𝑐)(𝑋−𝑐⁻¹)=𝑋²+𝑋+1, which is the cyclotomic polynomial Φ₃(𝑋). The main arithmetical tool for the proof of this theorem concerns certain divisibility properties of the polynomial Φ₃(𝑋^𝑘) and is contained in Lemma 3.2, the proof of which will be prepared in the following auxiliary result.

Lemma 3.1. For any 𝑘∈ℕ, one has Φ₃(𝑋^𝑘)=𝑋^2𝑘+𝑋^𝑘+1=_{𝑑|𝑘,3𝑑∤𝑘}Φ_3𝑑(𝑋). (3.1)

Proof. From the well-known formula 𝑋^𝑘−1=∏_𝑑|𝑘Φ_𝑑(𝑋) we find 𝑋^3𝑘−1=_{𝑑|3𝑘,3|𝑑}Φ_𝑑(𝑋)⋅_{𝑑|3𝑘,3∤𝑑}Φ_𝑑(𝑋)=_𝛿|𝑘Φ_3𝛿(𝑋)⋅_{𝑑|𝑘,3∤𝑑}Φ_𝑑(𝑋) (3.2) whence, Φ₃(𝑋^𝑘𝑋)=^3𝑘−1𝑋^𝑘=∏−1^𝑑|𝑘Φ_3𝑑(𝑋)⋅∏_{𝑑|𝑘,3∤𝑑}Φ_𝑑(𝑋)∏_{𝑑|𝑘,3∤𝑑}Φ_𝑑(𝑋)⋅∏_{𝑑|𝑘,3|𝑑}Φ_𝑑,(𝑋) (3.3) and after cancellation we obtain the desired result.

Lemma 3.2. For any 𝑛∈ℕ, all polynomials Φ₃(𝑋^𝑘)(𝑘=1,…,𝑛) divide (in ℤ[𝑋]) the product _{𝑛<ℓ≤2𝑛}Φ₃(𝑋^ℓ). (3.4)

Proof. According to Lemma 3.1, product (3.4) equals _{𝑛<ℓ≤2𝑛𝛿|ℓ,3𝛿∤ℓ}Φ_3𝛿(𝑋). (3.5) Since the factors on the right-hand side of (3.1) are pairwise coprime in ℤ[𝑋], it suffices to show that each Φ_3𝑑(𝑋) with 𝑑|𝑘,3𝑑∤𝑘 appears in product (3.5).
Clearly, Φ_3𝑘(𝑋) divides Φ₃(𝑋^𝑘). If 𝑘>𝑛/2, then we put ℓ^∗∶=2𝑘 and have 𝑛<ℓ^∗≤2𝑛, and we consider the contribution_{𝛿|ℓ∗,3𝛿∤ℓ∗}Φ_3𝛿(𝑋) (3.6) to product (3.5), in which Φ_3𝑘(𝑋) obviously occurs. We now suppose that 𝑑∣𝑘,𝑑≠𝑘, or 𝑑=𝑘 but 𝑘≤𝑛/2. In the first case, we have 𝑑𝑑^′=𝑘 with some integer 𝑑^′≥2, hence 𝑑≤𝑘/2. Thus, in both remaining situations, we have 𝑑≤𝑛/2. Then at least two successive multiples of 𝑑 are in the ℓ-set {𝑛+1,…,2𝑛} appearing in (3.4). Take two successive multiples of those multiples, 𝜌𝑑 and (𝜌+1)𝑑. Then 𝜌,𝜌+1 are not both divisible by 3, and with such a number 𝜌^∗ we put ℓ^∗∶=𝜌^∗𝑑. Clearly, Φ_3𝑑(𝑋) appears in product (3.6), whence it appears also in (3.5).

We are now in a position to prove Theorem 1.1. With 𝑐,𝑞,𝑄₁,𝑄₂ as there, we apply Lemma 2.1 to 𝑐₁=𝑐,𝑐₂=𝑐⁻¹, and obtain from  (2.1)𝑉(1)=𝑄₁𝐿_𝑞(𝑐)+𝑄₂𝐿_𝑞(𝑐⁻¹), (3.7) the “interesting” part of the linear form to be bounded below in Theorem 1.1. On the parameters 𝐿₁,𝐿₂ (hence on 𝐿), 𝑀,𝑁 we assume that all conditions mentioned in Lemma 2.1 (a), (b), (c). Rewriting (2.3) as𝐽=𝑄^∗𝑉(1)+𝑃^∗with𝑄^∗∶=^𝑁_𝑛=1𝑅_𝑛 (3.8) and obvious definition of 𝑃^∗, it is clear that 𝑃^∗ and 𝑄^∗ are both in 𝐾. Furthermore, |𝑄^∗| and |𝐽| are asymptotically evaluated in (2.5) and (2.6), respectively. Our next aim is to determine a 𝐷∈𝑂_𝐾⧵{0} such that 𝑄∶=𝐷⋅𝑄^∗ and 𝑃∶=𝐷⋅𝑃^∗ are both in 𝑂_𝐾.

To this purpose, we first remark that the double sum over 𝑘 and 𝑡 appearing in (2.3) equals^𝑛_𝑘=1𝑄₁(𝑞^𝑘−𝑐⁻¹)+𝑄₂(𝑞^𝑘−𝑐)Φ₃(𝑞^𝑘) (3.9) and 𝑅_𝑛 from (2.4) equals±𝑞^𝐸(𝑛)𝑁−1𝑛−1_𝑞(𝑞^𝑛+𝐿−𝑐^∗)∏_{𝑛<𝑙≤𝑛+𝐿−1}Φ₃(𝑞^ℓ)∏^𝑁−1_𝜈=1(𝑞^𝜈−1) (3.10) with 𝐸(𝑛)∶=(𝑀+1−2𝐿)𝑛+𝑛(𝑛−1)/2, with a suitable 𝑐^∗∈{𝑐,𝑐⁻¹} and the 𝑞-binomial coefficient [^𝑁−1_𝑛−1]_𝑞 being in ℤ[𝑞]. Assuming furthermore that 𝐿>𝑁, we see that, for every 𝑛∈{1,…,𝑁}, all numbers Φ₃(𝑞^𝑘)(𝑘≤𝑛) appearing in the denominators of (3.9) divide the product ∏_{𝑛<ℓ≤2𝑛}Φ₃(𝑞^ℓ) (in ℤ[𝑞], thus in 𝑂_𝐾), by Lemma 3.2, whence they divide the product appearing in the numerators of (3.10).

Secondly, considering the triple sum in (2.3), it is clear that 𝐷 must contain the factorlcm{𝑞−1,𝑞²−1,…,𝑞^𝑀−1}=_𝑑≤𝑀Φ_𝑑(𝑞)(uptoaunitin𝑂_𝐾); (3.11) see, for example, [6, Lemma 4(i)]. According to [6, Lemma 3(i)], we have for the denominator appearing in  (3.10)_𝜈<𝑁(𝑞^𝜈−1)=_𝑑<𝑁Φ_𝑑(𝑞)^{[(𝑁−1)/𝑑]}=𝑃_𝑁(𝑞)⋅_𝑑<𝑁Φ_𝑑(𝑞), (3.12) a formula by which 𝑃_𝑁(𝑞)∈𝑂_𝐾 is defined. Note that, therefore,|𝑃_𝑁(𝑞)|=|𝑞|^{(1/2−3/𝜋2)𝑁2+𝑂(𝑁log𝑁)} (3.13) holds with an O-constant depending only on |𝑞|.

To sum up, our above considerations and a comparison of (3.11) and (3.12) show that we may take𝐷∶=𝑞^𝐸∗⋅𝑃_𝑁(𝑞)⋅_𝑑≤𝑀Φ_𝑑(𝑞) (3.14) assuming additionally that 𝑀≥𝑁−1, where 𝐸^∗∈ℕ₀ has to satisfy the inequalities 𝐸(𝑛)+𝐸^∗≥0(𝑛=1,…,𝑁); see after (3.10). Clearly, 𝐸(𝑛)≥0 holds for all 𝑛∈ℕ if 𝑀≥2𝐿−1. But if 𝑀≤2𝐿−2, then min_𝑛≥1𝐸(𝑛)=−(1/2)(2𝐿−𝑀)(2𝐿−𝑀−1) and this minimum is attained at one of the successive positive integers 2𝐿−𝑀−1 or 2𝐿−𝑀, which, by 𝑀+𝑁>2𝐿−1⇔𝑁≥2𝐿−𝑀, are both in {1,…,𝑁}. Therefore we may choose in  (3.14)𝐸^∗0∶=if1𝑀≥2𝐿−1,2(2𝐿−𝑀)(2𝐿−𝑀−1)if𝑀≤2𝐿−2. (3.15)

Collecting all inequalities on 𝐿,𝑀,𝑁 we met so far, we now choose 𝐿=𝑁+1,𝑀=[𝛼𝑁] with suitable 𝛼>1 to be fixed later. In particular, if 1<𝛼≤2 holds, then 𝐸^∗=(1/2)(2−𝛼)²𝑁²+𝑂(𝑁), by the second alternative in (3.15). Thus, we may write (3.15) as𝐸^∗=𝜀^∗𝑁²+𝑂(𝑁)with𝜀^∗0∶=if1𝛼≥2,2(2−𝛼)²if1<𝛼≤2. (3.16) This, (3.13), and the well-known asymptotic formula for the product in (3.14) lead to|𝐷|=|𝑞|^{𝜏𝑁2+𝑂(𝑁log𝑁)}with𝜏∶=𝜀^∗+12+3𝜋²(𝛼²−1). (3.17) Further, we deduce from (3.8) and (2.5) that|𝑄^∗|=|𝑞|^{𝜎𝑁2+𝑂(𝑁)}with𝜎∶=𝛼+1 (3.18) and from  (2.6)|𝐽|=|𝒬||𝑞|^{−𝛽𝑁2+𝑂(𝑁)}with1𝛽∶=𝛼−2 (3.19) with all O-constants depending at most on |𝑞|.

With 𝑄,𝑃∈𝑂_𝐾 as defined after (3.8), (3.8) is equivalent to𝐷⋅𝐽=𝑄𝑉(1)+𝑃. (3.20) In the sequel, we have to be sure that this 𝑂_𝐾-linear form 𝑄𝑉(1)+𝑃 is “very small”, that is, 𝐷⋅𝐽 is “very small”. From (3.17) and (3.19) we have the inequalities|𝒬||𝑞|^{−(𝛽−𝜏)𝑁2−𝛾3𝑁log𝑁}≤|𝐷⋅𝐽|≤|𝒬||𝑞|^{−(𝛽−𝜏)𝑁2+𝛾3𝑁log𝑁}, (3.21) where 𝛾₃ (and all subsequent 𝛾’s) is a positive constant depending only on |𝑞|. To guarantee the “smallness” of 𝐷⋅𝐽, we have to suppose𝛽>𝜏. (3.22)

We now take our linear form 𝑄₀+𝑄₁𝐿_𝑞(𝑐)+𝑄₂𝐿_𝑞(𝑐⁻¹)=∶ℒ with large |𝒬|=max(|𝑄₁|,|𝑄₂|) and define 𝑁∈ℕ uniquely by|𝑞|^{(𝛽−𝜏)(𝑁−1)2−𝛾3(𝑁−1)log(𝑁−1)}<2|𝒬|≤|𝑞|^{(𝛽−𝜏)𝑁2−𝛾3𝑁log𝑁}. (3.23) Clearly 𝑁 becomes large exactly if |𝒬| does. Combination of the right-hand sides of (3.21) and (3.23) yields the right half of|𝑞|^{−𝛾4𝑁log𝑁}1≤|𝐷⋅𝐽|≤2, (3.24) whereas the left half comes from the left-hand inequalities in (3.21) and (3.23).

From our above definition of ℒ and from (3.7), we see that ℒ=𝑄₀+𝑉(1), whence𝑄ℒ=𝑄𝑄₀−𝑃+𝐷𝐽, (3.25) by (3.20). Thus, if 𝑃≠𝑄𝑄₀, then |𝑄ℒ|≥1/2 using the right-hand side of (3.24), whereas |𝑄ℒ|≥|𝑞|^{−𝛾4𝑁log𝑁} if 𝑃=𝑄𝑄₀. This remark and the asymptotic evaluation of |𝑄|=|𝐷𝑄^∗| from (3.17) and (3.18) give us|ℒ|≥|𝑞|^{−(𝜎+𝜏)𝑁2−𝛾5𝑁log𝑁}. (3.26) Since (3.23) implies that log|𝒬|=(𝛽−𝜏)𝑁²log|𝑞|+𝑂(𝑁log𝑁), we can eliminate 𝑁 from the right-hand side of (3.26) with the final result|ℒ|≥|𝒬|^{−𝜂−𝛾(loglog|𝒬|)/(log|𝒬|)1/2}with𝜂∶=𝜎+𝜏.𝛽−𝜏 (3.27)

For the numerical evaluation of 𝜂, we have to ensure (3.22), which, by (3.17) and (3.19), is equivalent to𝛼−1>𝜀^∗+3𝜋²(𝛼²−1). (3.28) If 𝛼≥2 this is equivalent to 𝛼<(𝜋²/3)−1=2.28986⋯=∶𝛼₂; see the definition of 𝜀^∗ in (3.16). If 1<𝛼≤2, then (3.28) reads as1𝛼−1>2(2−𝛼)²+3𝜋²(𝛼²−1), (3.29) which, after some calculation, yields 𝛼>1.50852⋯=∶𝛼₁.

Finally, we minimize 𝜂 in terms of 𝛼∈]𝛼₁,𝛼₂[=∶𝐼 or, more conveniently,𝜂+1=𝛽+𝜎,𝛽−𝜏 (3.30) where 𝜏 occurs only once. As a function of 𝛼, this is positive continuously differentiable in 𝐼 and tends to +∞ as 𝛼↑𝛼₂ and 𝛼↓𝛼₁. Furthermore, its derivative vanishes in 𝐼 exactly at 𝛼₀∶=(1/4)((121𝜋²−90)/(𝜋²+6)−1)=1.8353799…, whence (3.30) reaches its minimal value in 𝐼 exactly at 𝛼₀. Evaluation of (3.30) at 𝛼₀ yields the value 39.9475… for 𝜂 given in Theorem 1.1.

Remark 3.3. It should be pointed out that evaluation of (3.30) simply at 𝛼=2 would lead to the much weaker result (7𝜋²+18)/(2𝜋²−18)=50.0729… for 𝜂.

Remark 3.4. Here we will try to explain to some extent the difficulties, may be unexpected at first glance, of proving an analogue of Theorem 1.1 for primitive fourth or sixth roots of unity 𝑐. Let us restrict ourselves to the first case, where 𝐾 is the Gaussian field ℚ(𝑖) and (𝑋−𝑐)(𝑋−𝑐⁻¹)=𝑋²+1=Φ₄(𝑋). In this situation, we have the following analogue of Lemma 3.2: All polynomials Φ₄(𝑋^𝑘)(𝑘=1,…,𝑛) divide the product _{𝑛<ℓ≤3𝑛}Φ₄(𝑋^ℓ); (3.31) moreover, the upper bound 3𝑛 under the product sign cannot be replaced by something smaller. (Remember that in (3.4) this upper bound was 2𝑛.) Thus, we have to assume 𝐿>2𝑁 (instead of 𝐿>𝑁 before) to guarantee that, for every 𝑛∈{1,…,𝑁}, all Φ₄(𝑞^𝑘)(𝑘≤𝑛) divide ∏_{𝑛<ℓ≤3𝑛}Φ₄(𝑞^ℓ). Then the denominator 𝐷 from (3.14) remains unchanged, and we try it with the parameter choice 𝐿=2𝑁+1,𝑀=[𝛼𝑁], (3.32) where we need 𝛼>3 since 𝑀+𝑁−2𝐿 has to be large (see (2.6)). The crucial question is, of course, if we can ensure inequality (3.22) by a suitable choice of 𝛼. Clearly, comparing (2.6) and (3.19), we now have to work with 𝛽=2𝛼−(11/2) and with 𝜏 from (3.17) but with 𝜀^∗=0 if 𝛼≥4 and 𝜀^∗=(1/2)(4−𝛼)² if 4≥𝛼>3 (see (3.15) and (3.16)). It is easily checked that, unfortunately, there is no 𝛼>3, for which (3.22) holds.

It should be added that the situation becomes even worse if 𝑐 is a primitive sixth root of unity. Again the analogue of Lemma 3.2 is the main obstacle.

4. Proof of Theorem 1.2

With 𝑐,𝑞,𝑄₁,𝑄₂ as there, we now apply Lemma 2.1 to 𝑐₁=1,𝑐₂=𝑐 and obtain 𝑉(1)=𝑄₁𝐿_𝑞(1)+𝑄₂𝐿_𝑞(𝑐) from (2.1), and this is again the main part of the 𝑂_𝐾-linear form to be estimated from below in Theorem 1.2. Plainly, (3.8) remains valid but with new 𝑃^∗,𝑄^∗∈𝐾, namely,𝑄^∗=^𝑁_𝑛=1𝑅_𝑛=^𝑁_𝑛=1(−1)^{𝑁+1−𝑛}𝑞^𝐸(𝑛)∏^𝑛+𝐿1_𝜈=𝑛+1(𝑞^𝜈−1)∏^𝑛−1_𝜈=1(𝑞^𝜈−1)⋅∏^𝑁−𝑛_𝜈=1(𝑞^𝜈⋅−1)^𝑛+𝐿2_ℓ=𝑛+1(𝑞^ℓ−𝑐), (4.1) with 𝐸(𝑛) after (3.10), and𝑃^∗=−^𝑁_𝑛=1𝑅_𝑛^𝑛_𝑘=1(𝑄₁𝑞^𝑘+𝑄−1𝑞^𝑘−²𝑐)+_{𝜅+𝜆+𝜈=𝑀−1}𝑃_𝜅,𝜆⋅𝑄₁+𝑄₂𝑐^𝜈𝑞^𝜈+1,−1 (4.2) with all 𝑃_𝜅,𝜆∈ℤ[𝑞,𝑐].

Next, we determine a “denominator” 𝐷 for these 𝑃^∗,𝑄^∗. Assuming 𝐿≥𝑁 it is clear, by 𝐿₁≥𝐿−1≥𝑁−1, that the quotient appearing on the right-hand side of (4.1) is in ℤ[𝑞], whence 𝑞^𝐸∗ is a denominator for any 𝑅_𝑛 and so for 𝑄^∗, if 𝐸^∗ is defined as in (3.15). To multiply away the denominators from the triple sum in (4.2), 𝐷 must contain also factor (3.11), that is,_𝑑≤𝑀Φ_𝑑(𝑞). (4.3) Assuming that 𝑀≥𝑁, this factor takes also care of the 𝑞^𝑘−1(𝑘≤𝑛) appearing in the denominators of the first sum in (4.2). What about the additional 𝑞^𝑘−𝑐(𝑘≤𝑛) appearing there? Since 𝑞^𝑘−𝑐 divides 𝑞^3𝑘−1 in 𝑂_𝐾 for any 𝑘∈ℕ, it is clear: Because all 𝑞^3𝑘−1 with 𝑘≤𝑀/3 divide the above product (3.11), it is enough to take𝐷=𝑞^𝐸∗⋅_𝑑≤𝑀Φ_𝑑(𝑞)⋅_{𝑀/3<𝑘≤𝑁}(𝑞^𝑘−𝑐), (4.4) where, as always, empty products have to be interpreted as 1.

Choosing 𝐿,𝑀 exactly as in Section 3 (after (3.15)), we have to suppose again 𝛼>1. If 𝛼≥3, then the second product in (4.4) is empty and we obtain |𝐷|=|𝑞|^{(3𝛼2/𝜋2)𝑁2+𝑂(𝑁log𝑁)} hence (3.17) with 𝜏=3𝛼²/𝜋². Since the value of 𝛽 is as in (3.19), inequality (3.22) would mean 𝛼−1/2>3𝛼²/𝜋², which is never satisfied if 𝛼≥3. Hence we have to check the interval 1<𝛼<3, where we obtain formula (3.17) with 𝜏=𝜀^∗+3𝜋²𝛼²+12𝛼(1−²9), (4.5)𝜀^∗ being as in (3.16). Omitting the calculations, it is easily seen that the decisive inequality (3.22) holds exactly in the subinterval ]1.9119132…,2.1734402…[ of ]1,3[. So we have to minimize the function (3.30) in this subinterval. Calculation shows that this minimum appears exactly at 𝛼₀∶=(1/4)((54+359𝜋²)/(54−𝜋²)−1)=2.0071097… yielding the numerical value 704.2655… for 𝜂, much worse than our claim in Theorem 1.2.

Of course, our procedure to include in (4.4) the second product to take care of the denominators 𝑞^𝑘−𝑐 with “large” 𝑘’s in (4.2) is too trivial. To decrease the last value of 𝜂 considerably, we consider the even 𝑘 with (𝑀/3)<𝑘≤𝑁. Writing 𝑘=2𝑗,(𝑀/6)<𝑗≤(𝑁/2) and taking 𝑐=𝑐⁴ into account, we find𝑞^𝑘−𝑐=(𝑞^𝑗−𝑐²)(𝑞^𝑗+𝑐²). (4.6) If (𝑁/2)≤(𝑀/3) (or equivalently 𝛼≥3/2), then we know that, for every even 𝑘 with (𝑀/3)<𝑘≤𝑁, the corresponding factor 𝑞^𝑗−𝑐² in (4.6) is already considered in the first product of (4.4) since it divides 𝑞^3𝑗−1 in 𝑂_𝐾. Hence we may replace the second product in (4.4) by_{(𝑀/3)<𝑘≤𝑁,𝑘odd}(𝑞^𝑘−𝑐)⋅_{(𝑀/6)<𝑗≤(𝑁/2)}(𝑞^𝑗+𝑐²), (4.7) with the absolute value of which being asymptotically|𝑞|^{(3/8)(1−𝛼2/9)𝑁2+𝑂(𝑁)}. (4.8) Thus, for our refinement of formula (4.4), we obtain again (3.17) but 𝜏 from (4.5) is now replaced by𝜏=𝜀^∗+3𝜋²𝛼²+38𝛼(1−²9). (4.9) After a bit of calculation, we see that (3.22) is satisfied here exactly in the larger subinterval ]1.6507301…,2.4519699…[ of [3/2,3[. The minimum of the function (3.30) in this subinterval occurs exactly at 𝛼₀=(1/4)((72+1403𝜋²)/(72+11𝜋²)−1)=1.9449636… and the corresponding 𝜂-value is just the one given in Theorem 1.2.

Remark 4.1. Step by step we can further refine this procedure. One possibility is to extract from the first product in (4.7) certain factors, namely, from those with 𝑘 divisible by 5. For 𝑘=5(2𝑗−1) we note, by 𝑐=𝑐¹⁰, 𝑞^𝑘−𝑐=𝑞^5(2𝑗−1)−𝑐^5⋅2=(𝑞^2𝑗−1−𝑐²)(𝑞^4(2𝑗−1)+𝑞^3(2𝑗−1)𝑐²+⋯). (4.10) Since the factor 𝑞^2𝑗−1−𝑐² divides 𝑞^3(2𝑗−1)−1 in 𝑂_𝐾, we know that this factor of 𝑞^𝑘−𝑐, for odd 𝑘 divisible by 5, appears already in the product (3.11). Thus, we may refine (4.7) to _{(𝑀/3)<𝑘≤𝑁(𝑘,10)=1}(𝑞^𝑘−𝑐)⋅_{𝑘(𝑀/3)<𝑘≤𝑁odd,5∣𝑘}(𝑞^4𝑘/5+𝑞^3𝑘/5𝑐²+⋯)⋅_{(𝑀/6)<𝑗≤(𝑁/2)}(𝑞^𝑗+𝑐²). (4.11) The numerical calculations show that this step reduces the 𝜂-value from 56.6026… to 52.5205….

Remark 4.2. The “limit case” of reducing the 𝑘-product in (4.4) still further, possibly to a factor Ψ∈𝑂_𝐾⧵{0} with asymptotic |Ψ|=|𝑞|^𝑜(𝑁2), would yield the following. Let 𝛼>1. Then (3.22) holds if and only if 𝛼∈]1.2562872…,2.6749215…[ and the optimal choice is 𝛼₀=(1/4)((6+105𝜋²)/(6+𝜋²)−1)=1.776073… leading to 𝜂=12.8692….