For fixed complex with , the -logarithm is the meromorphic continuation of the series , into the whole complex plane. If is an algebraic number field, one may ask if are linearly independent over for satisfying . In 2004, Tachiya showed that this is true in the Subcase , , , and the present authors extended this result to arbitrary integer from an imaginary quadratic number field , 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 is the Eisenstein number field , an integer from , and a primitive third root of unity. Under these conditions, the linear independence holds also for , and both results are quantitative.

1. Introduction and Results

For fixed complex of absolute value greater than 1, the -logarithm is defined by the power series which converges in and has the meromorphic continuation into the whole complex plane. In the early 1990s, the irrationality investigations on this -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 and if , then both numbers and are irrational, with the second result appearing only in [2].

The next important step was made by Tachiya [3], who succeeded in proving, for , the linear independence of over using Borwein's function theoretic method from [2]. Shortly later, quantitative refinements of this result and also of the linear independence of were obtained independently by Zudilin [4] and by the present authors [5]; here the dash indicates differentiation with respect to . Somehow related to Tachiya's above-mentioned theorem is the linear independence over of for squares , which was established in [6]. Another result, proved in [7], is the linear independence of for any . It should be noted that all these linear independence statements remain true if one replaces by an arbitrary imaginary quadratic number field and if one supposes 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 (, 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 . The parameter 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 at both primitive third roots of unity is to be answered quantitatively as follows.

Theorem 1.1. Let be a primitive third root of unity, and let denote the ring of integers of . Then, for any with , the numbers are linearly independent over . Moreover, there exists a constant depending at most on such that, for any with large enough, the inequality holds with .

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 series for and rational (with some necessary exceptions). Namely, it is easily seen that holds for 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 are linearly independent over . Moreover, there exists a constant depending at most on such that, for any with large enough, the inequality holds with .

It should be noted that here the value of 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 are linearly independent over 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 a primitive fourth or sixth root of unity, the other two cases, where 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 satisfying define the meromorphic function and consider, for large parameters , the (positively oriented) integral
(a) Then the following explicit formula holds: where, for , and all in the above triple sum over all of nonnegative integers with are in .
(b) Supposing additionally that and defining , one has the evaluation for large enough and, the -constant depending on at most.
(c) Supposing, moreover, that if but if , then the asymptotic formula holds as soon as have the same order of magnitude and is large enough. Here the -constant depends on and at most.

Proof. (a) We apply the residue theorem to the integral defined in (2.2) and use the poles of the integrand in noting that is holomorphic in , by (2.1) and the hypothesis . Thus, we obtain the first sum coming from the simple poles at and the second one from the -fold pole at the origin. The above first sum leads immediately to the -sum in (2.3) using the definition of in (2.4) and the fact for any . In the above triple sum, the factor in front of is just what we denoted in (2.3) by , hence is evident. From for any and from (2.1) we simply deduce that whence, the triple sum in (2.3).
(b) If satisfies , then we have for any whence, by (2.4), with . Notice that can be bounded below by times the same -power as in (2.11). Our additional hypothesis on the 's means that either or ; hence the exponent of in (2.11) equals Thus, we have for and the right-hand side is ≀1/3 for large enough (in terms of only). Under this condition we find whence, and the inequalities (compare (2.11) and (2.12)) establish (2.5).
(c) We may assume that since otherwise (2.6) is trivial. In contrast to the situation in (a), to evaluate 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 is just the sum of the residues at all these (simple) poles appearing precisely at the points with . To justify this equality, the estimate on for large is useful; the O-constant depends only on . The distinctness of the before-mentioned poles is guaranteed by our hypothesis . Thus, we are led to an expression of as sum plus the same sum, where the subscripts 1 and 2 are interchanged.
Denoting the th summand in (2.15) by , similar considerations as for (2.11) show the existence of a constant depending only on and such that holds for every . This implies that for the same , and here the right-hand side is bounded by , say, since is supposed to be large enough. As in (b), this leads to for the sum in (2.15). Thus, the absolute value of term (2.15) is bounded above by and below by the same expression with replaced by (compare (2.16)).
If , we may assume that without loss of generality. Then we have , 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 are bounded above and below by certain absolute constants.
Suppose finally that . Denoting term (2.15) by and term (2.15) with subscripts 1 and 2 interchanged by , we know that (see (2.19) and the remark thereafter). Then, in the case , we have and by (2.19) and the corresponding lower bound for , where the constant from (2.19) is replaced by . Hence as soon as is large enough, and this leads to giving (2.6) for if we use for evaluation (2.19) and thereafter with subscript 1 replaced by 2. The case is treated analogously.

3. Proof of Theorem 1.1

For as in Theorem 1.1, we have , which is the cyclotomic polynomial . The main arithmetical tool for the proof of this theorem concerns certain divisibility properties of the polynomial and is contained in Lemma 3.2, the proof of which will be prepared in the following auxiliary result.

Lemma 3.1. For any , one has

Proof. From the well-known formula we find whence, and after cancellation we obtain the desired result.

Lemma 3.2. For any , all polynomials divide (in ) the product

Proof. According to Lemma 3.1, product (3.4) equals Since the factors on the right-hand side of (3.1) are pairwise coprime in , it suffices to show that each with appears in product (3.5).
Clearly, divides . If , then we put and have , and we consider the contribution to product (3.5), in which obviously occurs. We now suppose that , or but . In the first case, we have with some integer , hence . Thus, in both remaining situations, we have . Then at least two successive multiples of are in the -set appearing in (3.4). Take two successive multiples of those multiples, and . Then are not both divisible by 3, and with such a number we put . Clearly, appears in product (3.6), whence it appears also in (3.5).

We are now in a position to prove Theorem 1.1. With as there, we apply Lemma 2.1 to , and obtain from  (2.1) the β€œinteresting” part of the linear form to be bounded below in Theorem 1.1. On the parameters (hence on ), we assume that all conditions mentioned in Lemma 2.1 (a), (b), (c). Rewriting (2.3) as and obvious definition of , it is clear that and are both in . Furthermore, and are asymptotically evaluated in (2.5) and (2.6), respectively. Our next aim is to determine a such that and are both in .

To this purpose, we first remark that the double sum over and appearing in (2.3) equals and from (2.4) equals with , with a suitable and the -binomial coefficient being in . Assuming furthermore that , we see that, for every , all numbers appearing in the denominators of (3.9) divide the product (in , thus in ), 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 must contain the factor see, for example, [6, Lemma (i)]. According to [6, Lemma (i)], we have for the denominator appearing in  (3.10) a formula by which is defined. Note that, therefore, holds with an O-constant depending only on .

To sum up, our above considerations and a comparison of (3.11) and (3.12) show that we may take assuming additionally that , where has to satisfy the inequalities ; see after (3.10). Clearly, holds for all if . But if , then and this minimum is attained at one of the successive positive integers or , which, by , are both in . Therefore we may choose in  (3.14)

Collecting all inequalities on we met so far, we now choose with suitable to be fixed later. In particular, if holds, then , by the second alternative in (3.15). Thus, we may write (3.15) as This, (3.13), and the well-known asymptotic formula for the product in (3.14) lead to Further, we deduce from (3.8) and (2.5) that and from  (2.6) with all O-constants depending at most on .

With as defined after (3.8), (3.8) is equivalent to In the sequel, we have to be sure that this -linear form is β€œvery small”, that is, is β€œvery small”. From (3.17) and (3.19) we have the inequalities where (and all subsequent ’s) is a positive constant depending only on . To guarantee the β€œsmallness” of , we have to suppose

We now take our linear form with large and define uniquely by Clearly becomes large exactly if does. Combination of the right-hand sides of (3.21) and (3.23) yields the right half of whereas the left half comes from the left-hand inequalities in (3.21) and (3.23).

From our above definition of and from (3.7), we see that , whence by (3.20). Thus, if , then using the right-hand side of (3.24), whereas if . This remark and the asymptotic evaluation of from (3.17) and (3.18) give us Since (3.23) implies that , we can eliminate from the right-hand side of (3.26) with the final result

For the numerical evaluation of , we have to ensure (3.22), which, by (3.17) and (3.19), is equivalent to If this is equivalent to ; see the definition of in (3.16). If , then (3.28) reads as which, after some calculation, yields .

Finally, we minimize in terms of or, more conveniently, where occurs only once. As a function of , this is positive continuously differentiable in and tends to as and . Furthermore, its derivative vanishes in exactly at , whence (3.30) reaches its minimal value in exactly at . Evaluation of (3.30) at yields the value for given in Theorem 1.1.

Remark 3.3. It should be pointed out that evaluation of (3.30) simply at would lead to the much weaker result for .

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 . Let us restrict ourselves to the first case, where is the Gaussian field and . In this situation, we have the following analogue of Lemma 3.2: All polynomials divide the product moreover, the upper bound under the product sign cannot be replaced by something smaller. (Remember that in (3.4) this upper bound was .) Thus, we have to assume (instead of before) to guarantee that, for every , all divide . Then the denominator from (3.14) remains unchanged, and we try it with the parameter choice where we need since 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 . Clearly, comparing (2.6) and (3.19), we now have to work with and with from (3.17) but with if and if (see (3.15) and (3.16)). It is easily checked that, unfortunately, there is no , for which (3.22) holds.

It should be added that the situation becomes even worse if 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 as there, we now apply Lemma 2.1 to and obtain from (2.1), and this is again the main part of the -linear form to be estimated from below in Theorem 1.2. Plainly, (3.8) remains valid but with new , namely, with after (3.10), and with all .

Next, we determine a β€œdenominator” for these . Assuming it is clear, by , that the quotient appearing on the right-hand side of (4.1) is in , whence is a denominator for any and so for , if is defined as in (3.15). To multiply away the denominators from the triple sum in (4.2), must contain also factor (3.11), that is, Assuming that , this factor takes also care of the appearing in the denominators of the first sum in (4.2). What about the additional appearing there? Since divides in for any , it is clear: Because all with divide the above product (3.11), it is enough to take where, as always, empty products have to be interpreted as 1.

Choosing exactly as in Section 3 (after (3.15)), we have to suppose again . If , then the second product in (4.4) is empty and we obtain hence (3.17) with . Since the value of is as in (3.19), inequality (3.22) would mean , which is never satisfied if . Hence we have to check the interval , where we obtain formula (3.17) with being as in (3.16). Omitting the calculations, it is easily seen that the decisive inequality (3.22) holds exactly in the subinterval of . So we have to minimize the function (3.30) in this subinterval. Calculation shows that this minimum appears exactly at yielding the numerical value for , 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 with β€œlarge” ’s in (4.2) is too trivial. To decrease the last value of considerably, we consider the even with . Writing and taking into account, we find If (or equivalently ), then we know that, for every even with , the corresponding factor in (4.6) is already considered in the first product of (4.4) since it divides in . Hence we may replace the second product in (4.4) by with the absolute value of which being asymptotically Thus, for our refinement of formula (4.4), we obtain again (3.17) but from (4.5) is now replaced by After a bit of calculation, we see that (3.22) is satisfied here exactly in the larger subinterval of . The minimum of the function (3.30) in this subinterval occurs exactly at and the corresponding -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 divisible by 5. For we note, by , Since the factor divides in , we know that this factor of , for odd divisible by 5, appears already in the product (3.11). Thus, we may refine (4.7) to The numerical calculations show that this step reduces the -value from to .

Remark 4.2. The β€œlimit case” of reducing the -product in (4.4) still further, possibly to a factor with asymptotic , would yield the following. Let . Then (3.22) holds if and only if and the optimal choice is leading to .