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Volume 2010, Article ID 839695,14pages doi:10.1155/2010/839695

Research Article

Linear Independence of

q-Logarithms over

the Eisenstein Integers

Peter Bundschuh

1

and Keijo V ¨a ¨an ¨anen

2

1Mathematisches Institut, Universit¨at zu K¨oln, Weyertal 86-90, 50931 K¨oln, Germany

2Department of Mathematics, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland

Correspondence should be addressed to Peter Bundschuh,pb@math.uni-koeln.de

Received 16 July 2009; Accepted 28 March 2010

Academic Editor: Kenneth Berenhaut

Copyrightq2010 P. Bundschuh and K. V¨a¨an¨anen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For fixed complexqwith|q|>1, theq-logarithmLqis the meromorphic continuation of the series

n>0zn/qn−1,|z|<|q|, into the whole complex plane. IfKis an algebraic number field, one may

ask if 1, Lq1, Lqcare linearly independent overKforq, cK×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 integerqfrom an imaginary quadratic number fieldK, 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 Kis the Eisenstein number fieldQ√−3,qan integer fromK, andca primitive third root of unity. Under these conditions, the linear independence holds also for 1, Lqc, Lqc−1, and both results

are quantitative.

1. Introduction and Results

For fixed complex qof absolute value greater than 1, theq-logarithm Lq is defined by the

power series

Lqz:

n1

zn

qn1, 1.1

which converges in|z|<|q|and has the meromorphic continuation

z

n1

1

(2)

into the whole complex plane. In the early 1990s, the irrationality investigations on thisq -logarithm got a fresh impetus by two papers of Borwein1,2, where he introduced new

analytic tools to demonstrate quantitative versions of the following. Ifq∈Z\ {0,±1}and if c∈Q×\qN, then both numbersLqcandLq2cLq2cqare irrational, with the second result appearing only in2.

The next important step was made by Tachiya 3, who succeeded in proving, for

q∈ Z\ {0,±1}, the linear independence of 1, Lq1, Lq−1overQusing Borwein’s function

theoretic method from 2. Shortly later, quantitative refinements of this result and also

of the linear independence of 1, Lq1, Lq1 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 overQ of 1, Lqq, Lq−√qfor squaresq∈Z\ {0,1}, which was established in6. Another result,

proved in7, is the linear independence of 1, Lq1, Lq1for anyq ∈ Z\ {0,1}. It should

be noted that all these linear independence statements remain true if one replacesQby an arbitrary imaginary quadratic number field and if one supposesqto 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 courseby a primitive third root of unity. As we will see inTheorem 1.2below, this is indeed true if we study linear independence over the particular quadratic number fieldQ√−3. The parameterqhas to be from its ring of integers, which is sometimes called ring of Eisenstein integers since Eisenstein1844was 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 Lqat both primitive third roots of unity is to be answered quantitatively as follows.

Theorem 1.1. Letc be a primitive third root of unity, and let OK denote the ring of integers of

K:Qc. Then, for anyqOKwith|q|>1, the numbers 1, Lqc, Lqc−1are linearly independent

overK. Moreover, there exists a constantγ ∈ R depending at most on|q|such that, for anyQ

Q0, Q1, Q2∈O3Kwith|Q|:max|Q1|,|Q2|large enough, the inequality

Q0 Q1Lqc Q2Lq

c−1≥ |Q|−ηγlog log|Q|/log|Q|1/2

1.3

holds withη:39.9475. . ..

This question arose when preparing our recent work8, where, as a very particular

application, we obtained a quantitative version of the irrationality of the series

Sqa:

n1

qn

q2n aqn 1 1.4

forq∈Z\ {0,±1}and rationalawith some necessary exceptions. Namely, it is easily seen that

Sq1

LqcLq

c−1

cc−1 1.5

(3)

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, Lq1, Lqc are linearly

independent overK. Moreover, there exists a constant γ ∈ R depending at most on|q|such that, for anyQ Q0, Q1, Q2∈O3Kwith|Q|:max|Q1|,|Q2|large enough, the inequality

Q0 Q1Lq1 Q2Lqc≥ |Q|ηγlog log|Q|/log|Q|1/2

1.6

holds withη: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

Remarks4.1and4.2.

Of course, Theorems 1.1and 1.2 together suggest the following problem. Is it true that the four numbers 1, Lq1, Lqc, Lqc−1are linearly independent overK 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 Theorems1.1and1.2forca primitive fourth or sixth root of unity, the other two cases, whereQcis simultaneously a cyclotomic and an imaginary quadratic number field. On the difficulties with this problem we will make some comments at the end ofSection 3seeRemark 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 Theorems1.1and1.2, 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. Withq, c1, c2, Q1, Q2 ∈ Csatisfying|q| > max1,|c1|,|c2|define the meromorphic function

Vz:

k1

Q1

qkc1z

Q2

qkc2z

2.1

and consider, for large parametersL1, L2, M, N∈N, the (positively oriented) integral

J: 1 2πi

|z|1

2

t1 Lt

1

qc tz

zMN n1

(4)

aThen the following explicit formula holds:

J

N

n1

Rn V1− n

k1 2

t1

Qt

qkct

κ λ νM−1

Pκ,λ·

Q11 Q2c2ν

11 , 2.3

where, forn1, . . . , N,

Rn: −1N 1−nqML1−L2n nn−1/2

2

t1

n<n Lt

qc t

n−1

ν1

1·Nn ν1

1 2.4

and allPκ,λin the above triple sum over allκ, λ, νof nonnegative integers withκ λ νM−1

are inZq, c1, c2.

bSupposing additionally that|c1||c2|1,|L1−L2|1 and definingL:maxL1, L2, one has the evaluation

N

n1

Rn

qL

2 MN O1

2.5

forMlarge enough and, theO-constant depending on|q|at most.

c Supposing, moreover, that c1/c2/qZ, L2 L1−1 if |Q1| ≤ |Q2| but L2 L1 1 if |Q1|>|Q2|, then the asymptotic formula

|J|max|Q1|,|Q2|·qM N−2LLN

2/2 ON

2.6

holds as soon asL, M, Nhave the same order of magnitude andM N−2Lis large enough. Here theO-constant depends on|q|and|c1−c2|at most.

Proof. aWe apply the residue theorem to the integralJdefined in2.2and use the poles of the integrand in|z|<1 noting thatVzis holomorphic in|z| ≤1, by2.1and the hypothesis

|c1|,|c2|<|q|. Thus, we obtain

J

N

n1

qML1−L2n2

t1 Lt

1

q nc t

n−1

ν1

1−qν·Nn ν1

1− ·

Vqn

qn

κ λ νM−1

1 κ!

d dz

κ2

t1

Lt

1

qctz z0

·1 λ! d dz λ N

n1

1−qnz−1

z0

·0

(5)

the first sum coming from the simple poles atqn n1, . . . , Nand the second one from the

M-fold pole at the origin. The above first sum leads immediately to then-sum in2.3using the definition ofRnin2.4and the fact

Vqnqn

k1 2

t1

Qt

qk nct q n

k>n

2

t1

Qt

qkct q

n V1n k1 2

t1

Qt

qkct

2.8

for anyn∈N. In the above triple sum, the factor in front of0! is just what we denoted in2.3byPκ,λ, hencePκ,λ∈Zq, c1, c2is evident. Fromd/dzνqkctz−1ν!ctνqkctzν−1

for anyν∈N0:N∪ {0}and from2.1we simply deduce that

1 ν!V

ν0

k1

Q11 Q22

q 1, 2.9

whence, the triple sum in2.3.

bIfc∈Csatisfies|c|1, then we have for any∈N

1−q≤1−cq≤1 q <1−q−1, 2.10

whence, by2.4,

|Rn|< γ04q

ML1−L2n 2t1 nLt

n1−1/2NnN 1−n 2.11

withγ0:∞11− |q|−−1. Notice that|Rn|can be bounded below byγ0−4times the same|q|

-power as in2.11. Our additional hypothesis on theLt’s means that eitherL1L, L2L−1

orL1L−1, L2L; hence the exponent of|q|in2.11equals

M 1−2Ln 1 2

2

t1

LtLt 2n 1−

N 1−n 2

Mn L2− N 1−n

2

. 2.12

Thus, we have forn2, . . . , N

Rn−1

Rn

< γ08qMN−1 nγ08qM−1 2.13

and the right-hand side is≤1/3 forMlarge enoughin terms of|q|only. Under this condition we find

RN−1

RN

RN−1

RN ·

RN−2

RN−1 · · ·

RN−1

RN · · · · ·

R1

R2 < 1

2, 2.14

whence,1/2|RN| < |RN RN−1 · · · R1|< 3/2|RN|and the inequalitiesγ0−4|q|L

2 MN <

|RN|< γ04|q|L

2 MN

(6)

cWe may assume thatQ1, Q2/ 0,0since otherwise2.6is trivial. In contrast to

the situation ina, to evaluate|J|asymptotically from2.2, we use the poles of the integrand

outside the unit circle. As a matter of fact, one can easily show that−Jis just the sum of the residues at all thesesimplepoles appearing precisely at the pointsqkt/c

twithkt> Lt t

1,2. To justify this equality, the estimate|Vz||Q| ·OH|q|−Hon|z||q|H 1/2for large H∈Nis useful; the O-constant depends only on|q|. The distinctness of the before-mentioned poles is guaranteed by our hypothesisc1/c2/qZ. Thus, we are led to an expression ofJ as

sum

Q1cM1 N−1−L2

k>L1

L1

1

qqk·L2

1

c1qc2qk

qMkN n1

c1−qn k

2.15

plus the same sum, where the subscripts 1 and 2 are interchanged.

Denoting thekth summand in2.15bySk, similar considerations as for2.11show

the existence of a constantγ1 >1 depending only on|q|and|c1−c2|such that

γ1−1q2L−1−MNkNN 1/2≤ |Sk| ≤γ1q2L−1−MNkNN 1/2 2.16

holds for everyk > L1. This implies that

Sk 1

Sk

γ12q2L−1−MN 2.17

for the same k, and here the right-hand side is bounded by 1/3, say, sinceM N −2L is supposed to be large enough. As inb, this leads to

1

2|SL1 1| ≤

k>L1 Sk

3

2|SL1 1| 2.18

for the sum in2.15. Thus, the absolute value of term2.15is bounded above by

3

γ1

2

|Q1|q−2L−1−MNL1 1−NN 1/2 2.19

and below by the same expression with 3γ1/2 replaced by 1/2γ1 compare2.16.

IfQ1Q2 0, we may assume thatQ1/0, Q2 0 without loss of generality. Then we

haveL1 L−1, L2 L, by one of our additional hypotheses inc, and2.15and 2.19

(7)

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

J1

J2

3γ1/2

|Q1|

1/2γ2

|Q2|

q2L−1−MNL1−L23

γ1γ2q2L−1−MN, 2.20

by2.19and the corresponding lower bound for |J2|, where the constantγ1 from2.19is

replaced byγ2. Hence|J1/J2| ≤1/2 as soon asM N−2Lis large enough, and this leads to

1/2|J2| ≤ |J| ≤ 3/2|J2|giving2.6for|Q1| ≤ |Q2|if we use for|J2|evaluation2.19and

thereafter with subscript 1 replaced by 2. The case|Q1|>|Q2|is treated analogously.

3. Proof of

Theorem 1.1

For cas in Theorem 1.1, we haveXcXc−1 X2 X 1, which is the cyclotomic

polynomialΦ3X. The main arithmetical tool for the proof of this theorem concerns certain

divisibility properties of the polynomialΦ3Xkand is contained inLemma 3.2, the proof of

which will be prepared in the following auxiliary result.

Lemma 3.1. For anyk∈N, one has

Φ3

XkX2k Xk 1 d|k,3dk

Φ3dX. 3.1

Proof. From the well-known formulaXk1

d|kΦdXwe find

X3k−1

d|3k,3|d

ΦdX·

d|3k,3d

ΦdX

δ|k

Φ3δX·

d|k,3d

ΦdX 3.2

whence,

Φ3

Xk X3k−1

Xk1

d|kΦ3dX·

d|k,3dΦdX

d|k,3dΦdX·d|k,3|dΦdX, 3.3

and after cancellation we obtain the desired result.

Lemma 3.2. For anyn∈N, all polynomialsΦ3Xk k1, . . . , ndivide (inZX) the product

n<≤2n

Φ3

X. 3.4

Proof. According toLemma 3.1, product3.4equals

n<≤2n

δ|,3δ

(8)

Since the factors on the right-hand side of3.1are pairwise coprime inZX, it suffices to show that eachΦ3dXwithd|k,3dkappears in product3.5.

Clearly,Φ3kXdividesΦ3Xk. Ifk > n/2, then we put∗:2kand haven < ∗≤2n,

and we consider the contribution

δ|,3δ

Φ3δX 3.6

to product3.5, in whichΦ3kXobviously occurs. We now suppose thatd|k, d /k, ordk

butkn/2. In the first case, we haveddkwith some integerd≥2, hencedk/2. Thus, in both remaining situations, we havedn/2. Then at least two successive multiples ofdare in the-set{n 1, . . . ,2n}appearing in3.4. Take two successive multiples of those multiples,

ρdandρ 1d. Thenρ, ρ 1 are not both divisible by 3, and with such a numberρ∗we put ∗:ρd. Clearly,Φ3dXappears in product3.6, whence it appears also in3.5.

We are now in a position to proveTheorem 1.1. Withc, q, Q1, Q2 as there, we apply

Lemma 2.1toc1c, c2c−1, and obtain from2.1

V1 Q1Lqc Q2Lq

c−1, 3.7

the “interesting” part of the linear form to be bounded below in Theorem 1.1. On the parametersL1, L2hence onL,M, Nwe assume that all conditions mentioned inLemma 2.1

a,b,c. Rewriting2.3as

JQV1 P∗ with Q∗:

N

n1

Rn 3.8

and obvious definition ofP∗, it is clear thatP∗andQ∗are both inK. Furthermore,|Q∗|and

|J|are asymptotically evaluated in2.5and2.6, respectively. Our next aim is to determine

aDOK\ {0}such thatQ:D·Q∗andP:D·P∗are both inOK.

To this purpose, we first remark that the double sum overkandtappearing in2.3

equals

n

k1

Q1

qkc−1 Q2

qkc Φ3

qk 3.9

andRnfrom2.4equals

±qEn

N−1 n−1

q

qn Lc

n<ln L−1Φ3

q

N−1

ν1

1 3.10

with En : M 1−2Ln nn−1/2, with a suitable c∗ ∈ {c, c−1} and theq-binomial coefficientN−1

n−1

q being inZq. Assuming furthermore thatL > N, we see that, for every

(9)

productn<≤2nΦ3q inZq, thus inOK, byLemma 3.2, whence they divide the product

appearing in the numerators of3.10.

Secondly, considering the triple sum in2.3, it is clear thatDmust contain the factor

lcmq−1, q2−1, . . . , qM−1

dM

Φd

q up to a unit inOK

; 3.11

see, for example,6, Lemma 4i. According to6, Lemma 3i, we have for the denominator

appearing in3.10

ν<N

−1

d<N

Φd

qN−1/dPN

q·

d<N

Φd

q, 3.12

a formula by whichPNqOKis defined. Note that, therefore,

PN

qq1/2−32N2 ONlogN 3.13

holds with an O-constant depending only on|q|.

To sum up, our above considerations and a comparison of3.11and3.12show that we may take

D:qE∗·PN

q·

dM

Φd

q 3.14

assuming additionally thatMN−1, whereE∗∈N0has to satisfy the inequalitiesEn E∗≥

0 n 1, . . . , N; see after3.10. Clearly,En≥ 0 holds for alln∈NifM≥ 2L−1. But if M≤2L−2, then minn≥1En −1/22LM2LM−1and this minimum is attained at

one of the successive positive integers 2LM−1 or 2LM, which, byM N >2L−1 ⇔ N≥2LM, are both in{1, . . . , N}. Therefore we may choose in3.14

E∗:

⎧ ⎪ ⎨ ⎪ ⎩

0 ifM≥2L−1,

1

22LM2LM−1 ifM≤2L−2.

3.15

Collecting all inequalities onL, M, Nwe met so far, we now chooseL N 1, M αNwith suitableα >1 to be fixed later. In particular, if 1< α≤2 holds, thenE∗ 1/22− α2N2 ON, by the second alternative in3.15. Thus, we may write3.15as

EεN2 ON withε∗:

⎧ ⎪ ⎨ ⎪ ⎩

0 ifα≥2,

1 22−α

2 if 1< α2. 3.16

This,3.13, and the well-known asymptotic formula for the product in3.14lead to

|D|qτN2 ONlogN with τ:ε∗ 1 2

3 π2

(10)

Further, we deduce from3.8and2.5that

|Q∗|qσN2 ON with σ:α 1 3.18

and from2.6

|J||Q|qβN2 ON withβ:α−1

2 3.19

with all O-constants depending at most on|q|.

WithQ, POKas defined after3.8,3.8is equivalent to

D·JQV1 P. 3.20

In the sequel, we have to be sure that thisOK-linear formQV1 Pis “very small”, that is,

D·Jis “very small”. From3.17and3.19we have the inequalities

|Q|qβτN2−γ3NlogN≤ |

D·J| ≤ |Q|qβτN2 γ3NlogN

, 3.21

whereγ3and all subsequentγ’sis a positive constant depending only on|q|. To guarantee

the “smallness” ofD·J, we have to suppose

β > τ. 3.22

We now take our linear form Q0 Q1Lqc Q2Lqc−1 : L with large |Q|

max|Q1|,|Q2|and defineN∈Nuniquely by

qβτN−12−γ3N−1logN−1

<2|Q| ≤τN2−γ3NlogN

. 3.23

ClearlyN becomes large exactly if|Q|does. Combination of the right-hand sides of 3.21

and3.23yields the right half of

qγ4NlogN≤ |

D·J| ≤ 1

2, 3.24

whereas the left half comes from the left-hand inequalities in3.21and3.23.

From our above definition ofLand from3.7, we see thatLQ0 V1, whence

QLQQ0−P DJ, 3.25

by3.20. Thus, if P /QQ0, then|QL| ≥ 1/2 using the right-hand side of3.24, whereas |QL| ≥ |q|−γ4NlogN ifP QQ

0. This remark and the asymptotic evaluation of|Q| |DQ∗|

from3.17and3.18give us

|L| ≥qσ τN2−γ5NlogN

(11)

Since3.23implies that log|Q| βτN2log|q| ONlogN, we can eliminateN from

the right-hand side of3.26with the final result

|L| ≥ |Q|−ηγlog log|Q|/log|Q|1/2

with η: σ τ

βτ. 3.27

For the numerical evaluation ofη, we have to ensure3.22, which, by3.17and3.19,

is equivalent to

α−1> ε∗ 3 π2

α2−1. 3.28

Ifα≥2 this is equivalent toα <π2/312.28986· · ·:α

2; see the definition ofε∗in3.16.

If 1< α≤2, then3.28reads as

α−1> 1 22−α

2 3

π2

α2−1, 3.29

which, after some calculation, yieldsα >1.50852· · ·:α1.

Finally, we minimizeηin terms ofαα1, α2:Ior, more conveniently,

η 1 β σ

βτ, 3.30

whereτ occurs only once. As a function ofα, this is positive continuously differentiable in I and tends to ∞asαα2 andαα1. Furthermore, its derivative vanishes inIexactly at

α0 : 1/4

121π290/π2 61 1.8353799. . ., whence3.30reaches its minimal

value inI exactly atα0. Evaluation of3.30atα0yields the value 39.9475. . .forηgiven in

Theorem 1.1.

Remark 3.3. It should be pointed out that evaluation of3.30simply atα2 would lead to the much weaker result7π2 18/2π218 50.0729. . .forη.

Remark 3.4. Here we will try to explain to some extent the difficulties, may be unexpected

at first glance, of proving an analogue ofTheorem 1.1for primitive fourth or sixth roots of unity c. Let us restrict ourselves to the first case, where K is the Gaussian field Qiand

XcXc−1 X2 1 Φ

4X. In this situation, we have the following analogue of

Lemma 3.2: All polynomialsΦ4Xk k1, . . . , ndivide the product

n<≤3n

Φ4

X; 3.31

(12)

dividen<≤3nΦ4q. Then the denominatorDfrom3.14remains unchanged, and we try

it with the parameter choice

L2N 1, M αN, 3.32

where we needα >3 sinceM N−2Lhas to be largesee2.6. The crucial question is, of

course, if we can ensure inequality3.22by a suitable choice ofα. Clearly, comparing2.6

and3.19, we now have to work withβ2α−11/2and withτfrom3.17but withε∗0 ifα≥ 4 andε∗ 1/24−α2 if 4 ≥ α > 3see3.15and3.16. It is easily checked that,

unfortunately, there is noα >3, for which3.22holds.

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

4. Proof of

Theorem 1.2

With c, q, Q1, Q2 as there, we now applyLemma 2.1 to c1 1, c2 c and obtain V1

Q1Lq1 Q2Lqcfrom 2.1, and this is again the main part of the OK-linear form to be

estimated from below inTheorem 1.2. Plainly,3.8remains valid but with newP, Q∗ ∈K, namely,

Q

N

n1

Rn N

n1

−1N 1−n

qEn

n L1

νn 1

−1

n−1

ν1

1·Nn ν1

1· nL2

n 1

qc, 4.1

withEnafter3.10, and

P∗−

N

n1

Rn n

k1

Q1

qk1

Q2

qkc

κ λ νM−1

Pκ,λ·Q1 Q2c ν

1−1 , 4.2

with allPκ,λ∈Zq, c.

Next, we determine a “denominator”Dfor theseP, Q∗. AssumingLNit is clear, byL1 ≥L−1≥N−1, that the quotient appearing on the right-hand side of4.1is inZq,

whenceqEis a denominator for anyR

nand so forQ∗, ifE∗is defined as in3.15. To multiply

away the denominators from the triple sum in4.2,Dmust contain also factor3.11, that is,

dM

Φd

q. 4.3

Assuming thatMN, this factor takes also care of theqk 1 k nappearing in the

denominators of the first sum in4.2. What about the additionalqkc k nappearing

there? Sinceqkcdividesq3k−1 in OKfor any k ∈ N, it is clear: Because allq3k−1 with

kM/3 divide the above product3.11, it is enough to take

DqE∗·

dM

Φd

q·

M/3<kN

qkc, 4.4

(13)

ChoosingL, Mexactly as inSection 3after3.15, we have to suppose againα >1. Ifα≥3, then the second product in4.4is empty and we obtain|D||q|3α22N2 ONlogN hence 3.17with τ 3α22. Since the value ofβ is as in3.19, inequality3.22 would

meanα−1/2>3α22, which is never satisfied ifα3. Hence we have to check the interval

1< α <3, where we obtain formula3.17with

τ ε∗ 3 π2α

2 1

2 1− α2

9

, 4.5

ε∗ being as in3.16. 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 359π2/54π21 2.0071097. . . yielding the

numerical value 704.2655. . .forη, much worse than our claim inTheorem 1.2.

Of course, our procedure to include in 4.4 the second product to take care of the denominators qk c with “large”k’s in4.2is too trivial. To decrease the last value ofη

considerably, we consider the evenkwithM/3< kN. Writingk2j,M/6< jN/2 and takingcc4into account, we find

qkcqjc2qj c2. 4.6

If N/2 ≤ M/3 or equivalently α ≥ 3/2, then we know that, for every even k with M/3 < kN, the corresponding factorqjc2 in4.6is already considered in the first

product of4.4since it dividesq3j1 inO

K. Hence we may replace the second product in

4.4by

M/3<kN, kodd

qkc· M/6<jN/2

qj c2, 4.7

with the absolute value of which being asymptotically

q3/81−α2/9N2 ON

. 4.8

Thus, for our refinement of formula4.4, we obtain again 3.17 butτ from 4.5is now replaced by

τ ε∗ 3 π2α

2 3

8 1− α2

9

. 4.9

After a bit of calculation, we see that 3.22 is satisfied here exactly in the larger subinterval1.6507301. . . ,2.4519699. . .of3/2,3. The minimum of the function3.30in this subinterval occurs exactly atα0 1/4

72 1403π2/72 11π21 1.9449636. . .

(14)

Remark 4.1. Step by step we can further refine this procedure. One possibility is to extract

from the first product in4.7certain factors, namely, from those withkdivisible by 5. For k52j−1we note, bycc10,

qkcq52j−1−c5·2q2j−1−c2q42j−1 q32j−1c2 · · ·. 4.10

Since the factorq2j−1c2dividesq32j−11 inO

K, we know that this factor ofqkc, for odd

kdivisible by 5, appears already in the product3.11. Thus, we may refine4.7to

M/3<kN

k,101

qkc· M/3<kN

kodd,5|k

q4k/5 q3k/5c2 · · ·· M/6<jN/2

qj 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 thek-product in4.4still further, possibly to a factor Ψ ∈ OK\ {0}with asymptotic |Ψ| |q|oN

2

, would yield the following. Let α > 1. Then 3.22 holds if and only ifα ∈1.2562872. . . ,2.6749215. . .and the optimal choice is α0

1/46 105π2/6 π21 1.776073. . .leading toη12.8692. . ..

References

1 P. B. Borwein, “On the irrationality of1/qn r,” Journal of Number Theory, vol. 37, no. 3, pp.

253–259, 1991.

2 P. B. Borwein, “On the irrationality of certain series,” Mathematical Proceedings of the Cambridge

Philosophical Society, vol. 112, no. 1, pp. 141–146, 1992.

3 Y. Tachiya, “Irrationality of certain Lambert series,” Tokyo Journal of Mathematics, vol. 27, no. 1, pp. 75–85, 2004.

4 W. Zudilin, “Approximations toq-logarithms andq-dilogarithms, with applications toq-zeta values,”

Zapiski Nauchnykh Seminarov Sankt-Peterburgskoe Otdelenie. Matematicheski˘ıInstitut im. V. A. Steklova (POMI), vol. 322, pp. 107–124, 2005.

5 P. Bundschuh and K. V¨a¨an¨anen, “Linear independence ofq-analogues of certain classical constants,”

Results in Mathematics, vol. 47, no. 1-2, pp. 33–44, 2005.

6 P. Bundschuh, “Linear independence of values of a certain Lambert series,” Results in Mathematics, vol. 51, no. 1-2, pp. 29–42, 2007.

7 P. Bundschuh, “Remarks on linear independence of q-harmonic series” Russian, to appear in

Fundamental’naya i Prikladnaya Matematika.

References

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