IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE

2nd Reading

October 3, 2013 10:26 WSPC/S1793-0421 203-IJNT 1350112

International Journal of Number Theory(2013)c© World Scientific Publishing CompanyDOI: 10.1142/S1793042113501121

IRRATIONALITY OF LAMBERT SERIES ASSOCIATEDWITH A PERIODIC SEQUENCE

FLORIAN LUCA

Fundacion Marcos Moshinsky, UNAMCircuito Exterior, C.U., Apdo. Postal 70-543

Mexico D.F. 04510, [email protected]

YOHEI TACHIYA

Graduate School of Science and TechnologyHirosaki University

Hirosaki 036-8561, [email protected]

Received 2 April 2013Accepted 2 September 2013Published 4 October 2013

Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of ratio-nal numbers, not identically zero from some point on. Then the number

P∞n=1 an/(qn−1)

is irrational. In particular, if the periodic sequences {a(i)n }n≥1 (i = 1, . . . , m) of rational

numbers are linearly independent over Q, then so are the following m + 1 numbers:

1,∞X

n=1

a(i)n

qn − 1, i = 1, . . . , m.

This generalizes a result of Erdos who treated the case of m = 1 and a(1)n = 1 (n ≥ 1).

The method of proof is based on the original approaches of Chowla and Erdos, togetherwith some results about primes in arithmetic progressions with large moduli of Ahlford,

Granville and Pomerance.

Keywords: Lambert series; irrationality; linear independence.

Mathematics Subject Classification 2010: 11J72

1. Introduction

Let {an}n≥1 be an eventually periodic sequence of rational numbers, not identicallyzero from some point on, namely, there exist positive integers � and N such thatan = an+� for every integer n ≥ N and an �= 0 for infinitely many n. Then the

1

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

http://dx.doi.org/10.1142/S1793042113501121

2nd Reading


2 F. Luca & Y. Tachiya

Lambert series L(z) associated with {an}n≥1 is defined by

L(z) :=∞∑

n=1

anzn

1 − zn=

∞∑n=1

∑

d|nad

zn, (1.1)

and converges for |z| < 1 because the sequence {an}n≥1 is bounded. In the simplestcase of an = 1 for all n ≥ 1, Erdos [10] proved the irrationality of the value

L(1/q) =∞∑

n=1

1qn − 1

=∞∑

n=1

d(n)qn

(1.2)

for any integer q with q ≥ 2, where d(n) denotes the number of divisors of n. Inparticular, he showed that the q-adic expansion of the number (1.2) contains anyarbitrary long string of zeros without being identically zero from some point on.This irrationality result was generalized in [14] to the case of a periodic sequence{an}n≥1 with a period length at most two. The method is based on suitable contour-integral constructions and is a variant of a method used previously by Borwein [2](see also [5–7]). Zudilin [16] obtained the same result by using Pade approximationsof the second kind. Both methods are completely different from those of Erdosand are powerful in the sense that they yield quantitative results concerning theirrationality measures for such numbers.

In this paper, we prove Theorem 1.1 below as a generalization of Erdos’s result.The method is based on the original approaches of Chowla [8] and Erdos [10],together with some results about primes in arithmetic progressions with large mod-uli of Ahlford, Granville, and Pomerance [1]. We mention that Erdos’ original argu-ment has also been revisited recently by Vandehey [15].

Theorem 1.1. Let q (|q| > 1) be an integer and {an}n≥1 be an eventually periodicsequence of rational numbers, which is not identically zero from some point on.Then the number

∞∑n=1

an

qn − 1(1.3)

is irrational.

The sequences {a(i)n }n≥1 (i = 1, 2, . . . , m) are called linearly dependent over Q,

if there exist rational numbers c1, . . . , cm, not all zero, such that the sequences

{c1a(1)n + c2a

(2)n + · · · + cma(m)

n }n≥1

are identically zero from some point on. If such rational numbers do not exist, thenthe sequences are said to be linearly independent over Q.

In what follows, let q be an integer with |q| > 1. As an application of Theo-rem 1.1, we obtain the following results.

Theorem 1.2. Let {a(i)n }n≥1 (i = 1, . . . , m) be eventually periodic sequences of

rational numbers. If the sequences {a(i)n }n≥1 (i = 1, . . . , m) are linearly independent

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading


Irrationality of Lambert Series Associated with Periodic Sequence 3

over Q, then so are the m + 1 numbers

1, βi :=∞∑

n=1

a(i)n

qn − 1, i = 1, 2, . . . , m.

Proof. Suppose on the contrary that there exist rational numbers ci with i =0, . . . , m, not all zero, such that

c0 + c1β1 + · · · + cmβm = 0.

Then we have cj �= 0 for some j ∈ {1, . . . , m} and

−c0 = c1α1 + c2β2 + · · · + cmβm =∞∑

n=1

θn

qn − 1, (1.4)

where θn := c1a(1)n + · · · + cma

(m)n (n ≥ 1). Since the periodic sequences {a(i)

n }n≥1

(i = 1, . . . , m) are linearly independent over Q, the sequence {θn}n≥1 is also even-tually periodic and not identically zero. Hence, by Theorem 1.1, the number (1.4)is irrational. This is a contradiction.

Corollary 1.3. For any integer � ≥ 1, the numbers

1,

∞∑n=1

1qn − 1

,

∞∑n=1

1q2n − 1

, . . . ,

∞∑n=1

1q�n − 1

.

are linearly independent over Q.

Proof. We put

∞∑n=1

1qin − 1

=∞∑

n=1

a(i)n

qn − 1, i = 1, 2, . . . , �,

where {a(i)n }n≥1 is a periodic sequence with a

(i)n = 1 if n ≡ 0 (mod i), and a

(i)n = 0

otherwise. Clearly the sequences {a(i)n }n≥1 are linearly independent over Q, and

hence the corollary follows immediately from Theorem 1.2.

Corollary 1.4. Let {an}n≥1 be a purely periodic sequence of rational numbers witha period length � ≥ 1. If

a1 + a2ζ + a3ζ2 + · · · + a�ζ

�−1 �= 0

for any �th root of unity ζ, then the � + 1 numbers

1,

∞∑n=1

an+i

qn − 1, i = 1, . . . , �,


Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Proof. Let ζ� be a primitive �th root of unity. Then the circulant determinant∣∣∣∣∣∣∣∣∣

a1 a2 · · · a�

a� a1 · · · a�−1

.... . .

...a2 a3 · · · a1

∣∣∣∣∣∣∣∣∣=

�−1∏k=0

(a1 + a2ζk� + a3ζ

2k� + · · · + a�ζ

(�−1)k� ),

is nonzero by the assumption. Hence, the sequences {an+i}n≥1 (i = 1, . . . , �) arelinearly independent over Q and the corollary follows from Theorem 1.2.

Example 1.5. Let � ≥ 1 be an integer and

{an}n≥1 = {1, 2, . . . , �, 1, 2, . . . , �, . . .} := {1, 2, . . . , �}.Then the � + 1 numbers

1,

∞∑n=1

an+i

qn − 1, i = 1, 2, . . . , �,


Proof. This follows immediately from Corollary 1.4 and the fact that

0 �= 1 + 2ζ + 3ζ2 + · · · + �ζ�−1 =

�(� + 1)2

if ζ = 1,

− �

1 − ζotherwise,

for any �th root of unity ζ.

Corollary 1.6. For any integer s ≥ 1, the 2s + 1 numbers

1,∞∑

n=1n≡i (mod s)

1qn − 1

,∞∑

n=1n≡j (mod s)

(−1)n

qn − 1, i, j = 1, 2, . . . , s,


Proof. Let ε = 1 or −1. Then, putting∞∑

n=1n≡i (mod s)

εn

qn − 1=

∞∑n=1

a(i)ε,n

qn − 1, i = 1, 2, . . . , s,

we have

a(i)ε,n =

1 if n ≡ i (mod 2s),

ε if n ≡ s + i (mod 2s),

0 otherwise.

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Hence, the 2s periodic sequences {a(i)ε,n}n≥1 (ε = ±1, i = 1, 2, . . . , s) are linearly

independent over Q and we derive the conclusion from Theorem 1.2.

The following Examples 1.7 and 1.8 follow immediately from Corollary 1.6 withs = 1 and s = 2, respectively.

Example 1.7 (cf. [14, 16]). The three numbers

1,∞∑

n=1

1qn − 1

,∞∑

n=1

(−1)n

qn − 1(1.5)


We note that quantitative linear independence results for the numbers (1.5) havebeen obtained in [5, 12, 16].

Example 1.8. The numbers

1,

∞∑n=1

1q2n − 1

,

∞∑n=1

(−1)n

q2n − 1,

∞∑n=1

1q2n−1 − 1

,

∞∑n=1

(−1)n

q2n−1 − 1


Example 1.8 gives the linear independence over Q for the following sets of num-bers:

1, α :=∞∑

n=1

1q2n−1 + 1

, β :=∞∑

n=1

1q2n−1 − 1

; (1.6)

1,

∞∑n=1

1qn − 1

,

∞∑n=1

1(−q)n − 1

. (1.7)

This was first proved by Bundschuh (cf. [3, 4]) together with their quantitativelinear independence results. Indeed, for the numbers (1.6), we get

α =∞∑

n=1

1qn + 1

−∞∑

n=1

1q2n + 1

= −∞∑

n=1

(−1)n

qn − 1+

∞∑n=1

(−1)n

q2n − 1

= β −∞∑

n=1

1q2n − 1

+∞∑

n=1

(−1)n

q2n − 1.

Putting c0 + c1α + c2β = 0 (cj ∈ Q for j = 0, 1, 2), we obtain

c0 + (c1 + c2)β − c1

∞∑n=1

1q2n − 1

+ c1

∞∑n=1

(−1)n

q2n − 1= 0,

and hence, by Example 1.8, we obtain c0 = c1 = c2 = 0. The same holds for thenumbers (1.7) from the equalities

∞∑n=1

1qn − 1

= β +∞∑

n=1

1q2n − 1

,

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



and∞∑

n=1

1(−q)n − 1

= −α +∞∑

n=1

1q2n − 1

= −β + 2∞∑

n=1

1q2n − 1

−∞∑

n=1

(−1)n

q2n − 1.

2. Lemmas

In this section, we collect some lemmas needed for the proof of Theorem 1.1. Foreach j = 1, 2, . . . , �, we define

dj(n) :=∑d|n

d≡j (mod �)

1. (2.1)

Lemma 2.1. Suppose that the positive integer n has the form n = m∏s

i=1 pei

i ,

where pi are distinct prime numbers with pi ≡ 1 (mod �) and coprime with m.Then dj(n) is expressed as

dj(n) = dj(m)s∏

i=1

(ei + 1), j = 1, . . . , �.

Proof. Let m =∏t

i=s+1 pei

i . Then, we have for each j = 1, . . . , �

dj(n) =∑d|n

d≡j (mod �)

1

= �

{(u1, . . . , ut)

∣∣∣∣∣ pu11 · · · pus

s

t∏i=s+1

pui

i ≡ j (mod �), 0 ≤ ui ≤ ei

}

=s∏

i=1

(ei + 1) × �

{(us+1, . . . , ut)

∣∣∣∣∣t∏

i=s+1

pui

i ≡ j (mod �), 0 ≤ ui ≤ ei

}

= dj(m)s∏

i=1

(ei + 1),

where �S denotes the number of elements in the set S.

In what follows, we write π(x) for the number of primes p ≤ x. For coprimeintegers a and b, we write π(x; b, a) for the number of primes p ≤ x in the arithmeticprogression p ≡ a (mod b). The following result is due to Ahlford, Granville andPomerance [1] and played an important role in the proof of the existence of infinitelymany Carmichael numbers. It was used in our previous work [11] to deduce the linearindependence over Q of the collection of numbers

1,

∞∑n=1

1qni − 1

(i = 1, . . . , m)

for any integer m ≥ 1.

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Lemma 2.2 (cf. [1]). There exist numbers x0 and D such that the inequality

π(y; d, a) ≥ π(y)2φ(d)

holds whenever d and a are coprime integers, x > x0, d ≤ min{x2/3, y/x1/3}, andd is not divisible by any number from D(x), a set of at most D elements each ofwhich exceeds log x.

Lemma 2.3. Let A and � be coprime. There exists a number n0 depending on �

but not on A such that the inequality

π(An/(3�); A�, a) ≥ n

9�2 log n

holds for all n ≥ max{n0, (A�)2}, assuming the integers A� and a are coprime, andthat A� is not divisible by any number from D(n3/2).

Proof. Let x0 be as in Lemma 2.2. Let d = A� and n0 := max{�x0�, 51�4}. For aninteger n with n ≥ max{n0, (A�)2}, we put x := n3/2 and y := An/(3�). Then wehave

d = A� ≤ A√

n/(3�) = min{x2/3, y/x1/3}.Hence, by Lemma 2.2, we obtain

π(An/(3�); A�, a) ≥ π(An/(3�))2φ(A�)

≥ An/(3�)2φ(A)φ(�) log(An/(3�))

≥ n

9�2 log n.

For the second inequality above, we used the fact that π(x) > x/log x for all x ≥ 17(see [13, Corollary 1]), which for us holds because An/(3�) ≥ n0/(3�) ≥ 17. Thelast one follows from m ≥ φ(m) and

log(An/(3�)) ≤ log(A�n) ≤ (3/2) logn.

Let B be a fixed positive integer such that A and B are coprime and 1 ≤ B < 2A.Define

SA,B(n) := {Ai + B | i = 0, 1, . . . , n}.Lemma 2.4. Let A and �! be coprime and let n0 be the integer as in Lemma 2.3.Assume that n is an integer with n ≥ max{n0, (A�)2} and that the integer A� isnot divisible by any number from D(n3/2). Then for each integer s = 1, . . . , �, theset SA,B(n) contains at least T := �n/(9�2 log n)� numbers mi,s (i = 1, . . . , T ) suchthat

dj(mi,s) = 2dj(s), i = 1, . . . , T, j = 1, . . . , �.

Proof. We fix an integer s (s = 1, . . . , �). Since A and �! are coprime, there existsan integer bs such that

s(bs� + 1) ≡ B (mod A), 1 ≤ bs ≤ A. (2.2)

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Note that the integers A� and bs� + 1 are coprime because so are A and B. Let

ms,u := s(A�u + bs� + 1), u = 0, 1, . . . ,⌊ n

2�2

⌋.

Note that ms,u ∈ SA,B(n) for every u. Indeed, for n ≥ n0 ≥ 4�2,

0 < ms,u = A�su + s(bs� + 1) ≤ A�2⌊ n

2�2

⌋+ �(A� + 1) ≤ An + B

and ms,u ≡ B (mod A) by (2.2).Consider the primes of the form

A�u + bs� + 1, u = 0, 1, . . . ,⌊ n

2�2

⌋.

Since n ≥ n0 ≥ 6�2, we have

A�⌊ n

2�2

⌋+ bs� + 1 > A�

⌊ n

2�2

⌋≥ An

3�.

Hence, by Lemma 2.3, the number of such primes is at least

π(An/(3�); A�, bs� + 1) ≥ n

9�2 log n.

Let u1, . . . , uT be indices corresponding to T of such primes. Then A�ui + bs� + 1and s are coprime, since A�ui + bs� + 1 > � ≥ s. Furthermore, noting that A�ui +bs� + 1 ≡ 1 (mod �), for i = 1, . . . , T , we see, by Lemma 2.1, that

dj(ms,ui) = dj(s(A�ui + bs� + 1)) = 2dj(s), j = 1, 2, . . . , �.

Taking mi,s := ms,ui for i = 1, . . . , T , the proof of Lemma 2.4 is completed.

Lemma 2.5. Suppose that N ≥ 2A. Then we haveN∑

i=1

dj(Ai + B) ≤ 8N log N

for any j = 1, . . . , �.

Proof. For each j, we have

dj(m) =∑d|m

d≡j (mod �)

1 ≤∑d|m

1 ≤ 2∑

d≤√m

d|m

1,

so thatN∑

i=1

dj(Ai + B) ≤ 2

√AN+B∑d=1

∑1≤i≤Nd|Ai+B

1 ≤ 2

√AN+B∑d=1

(1 +

N

d

)

≤ 4N

√AN+B∑d=1

1d≤ 4N(1 + log N) ≤ 8N log N.

where we used the inequalities d ≤ √AN + B ≤ √

2AN ≤ N and 2 ≤ 2B < N .

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



The following result was obtained by Duverney [9].

Lemma 2.6. Let q be an integer with |q| ≥ 2 and {θ(n)}n≥1 be a sequence ofintegers. Assume that there exists a sequence of nonnegative integers {nk}k≥1 withnk ≥ 2k such that, for every k sufficiently large,{

q|θ(nk − k + 1), q2|θ(nk − k + 2), . . . , qk−1|θ(nk − 1),

qk+1|θ(nk + 1), qk+2|θ(nk + 2), . . . , q2k|θ(nk + k),

and satisfying

limk→+∞

1|q|k

∞∑n=0

|θ(n + nk + k + 1)||q|n = 0.

Assume further that∑∞

n=0θ(n)qn is convergent and is a rational number. Then

qk|θ(nk) for every large k.

3. Proof of Theorem 1.1

Let {an}n≥1 be an eventually periodic sequence with a period length � ≥ 1. Thenwe can choose an integer N ≥ 1 such that

{an}n≥1 = {a1, a2, . . . , a�N , b1, . . . , b�, b1, . . . , b�, . . .},so that

∞∑n=1

an

qn − 1=

�N∑n=1

an

qn − 1+

∞∑n=1

bn

qn+�N − 1

=�N∑n=1

an

qn − 1−

�N∑n=1

bn

qn − 1+

∞∑n=1

bn

qn − 1.

Hence, to obtain the irrationality of the number shown at (1.3), it is sufficient toshow that of

∞∑n=1

bn

qn − 1.

Therefore, in Theorem 1.1, we may assume that the sequence {an}n≥1 is a purelyperiodic sequence of rational numbers, not identically zero:

{an}n≥1 = {a1, . . . , a�, a1, . . . , a�, . . .}. (3.1)

Let dj(n) (j = 1, 2, . . . , �) be given as in (2.1); then we have by (3.1)

∑d|n

ad =�∑

j=1

ajdj(n).

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Thus, for any integer q (|q| > 1), we obtain by (1.1)

L(1/q) =∞∑

n=1

an

qn − 1=

�∑j=1

aj

( ∞∑n=1

dj(n)qn

). (3.2)

Suppose on the contrary that the number (3.2) is a rational number. Then, mul-tiplying both sides of (3.2) by the greatest common denominator of a1, . . . , a�, wefind the rational integers b1, . . . , b�, not all zero, such that

�∑j=1

bj

( ∞∑n=1

dj(n)qn

)=

∞∑n=1

θ(n)qn

∈ Q,

where θ(n) := b1d1(n) + · · · + b�d�(n). In what follows, we denote by c1, c2, . . .

the positive constants dependent on q, �, and bj , but not on k. Let p1, p2, . . . beincreasing sequence of prime numbers in arithmetic progression pi ≡ 1 (mod �).Then we recall that the following upper bound of the nth prime pn

pn ≤ c1n logn (3.3)

is valid for large n. Here, we can take c1 = 2� (or even c1 = � if � > 1). Let k > 0be a sufficiently large integer and we put Nk := 2k3

and x := N3/2k . Let D = D(x)

be a set of at most D elements each of which exceeds log x = (log 23/2)k3 ≥ k3 ≥ 2.For each d ∈ D, if there exist prime factors of d which do not divide �, let pd be oneof them. Let

P = {pd : d ∈ D} ∪ {p : p | �}.

Clearly, P has at most ω(�) + D ≤ � + D elements, where ω(�) denotes the numberof distinct prime factors of �. Define

tk =k(k + 1)

2, rk = tk + 1.

Let q1, q2, . . . , qt2kbe first t2k odd prime numbers in the arithmetic progression

qi ≡ 1 (mod �) all greater than 4k3 and which do not belong to P . By the ChineseRemainder Theorem, we can find a natural number βk satisfying all the congruences

βk − k + 1 ≡ q|q|−11 (mod q

|q|1 )

βk − k + 2 ≡ (q2q3)|q|−1 (mod (q2q3)|q|)...

...βk − 1 ≡ (qrk−2 · · · qtk−1)

|q|−1 (mod (qrk−2 · · · qtk−1)|q|)

βk + 1 ≡ (qrk· · · qtk+1)

|q|−1 (mod (qrk· · · qtk+1)

|q|)...

...βk + k ≡ (qr2k−1 · · · qt2k

)|q|−1 (mod (qr2k−1 · · · qt2k)|q|).

(3.4)

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



In particular, putting

Ak :=t2k∏i=1

i�=rk−1,...,tk

q|q|i ,

we obtain an integer βk which is unique subject to the condition

1 ≤ βk ≤ Ak.

Since qi > 4k3, we have, by (3.3),

βk ≤ Ak ≤t2k∏i=1

p|q|i+�+D+4k3 ≤ ec2k2 log k, (3.5)

where the last inequality holds for all sufficiently large k. Let

S(k) := SAk,βk(Nk)\{βk} = {uk,i := Aki + βk | i = 1, . . . , Nk}.

In what follows, we fix s (s = 1, . . . , �). Define the subsets Ts(k) of S(k) by

Ts(k) := {u ∈ S(k) | dj(u) = 2dj(s), j = 1, . . . , �}.We show that the hypotheses of Lemma 2.4 are satisfied for A = Ak, B = βk,

and n = Nk. Note that Ak and βk are coprime. Otherwise, there exists some prime qi

which divides βk. Hence, by (3.4), qi divides a nonzero integer t with −k+1 ≤ t ≤ k.This is impossible since 0 < |t| ≤ k < qi. Further, it is clear that Nk ≥ (Ak�)2 forlarge k by inequality (3.5). Let d ∈ D(N3/2

k ). If there exists a prime factor of d notdividing �, then we see d � Ak� because we have chosen our primes q1, . . . , q2k notto belong to P . If not, all prime factors of d consist of those of �. Hence, at leastone of the exponents of the prime factors of d is greater than the correspondingexponent of that prime factor of �, because d > log x ≥ k3 is large. In any case, weobtain that d � Ak�. Thus, by Lemma 2.4, we see that the set S(k) contains at least� Nk

9�2 log Nk� − 1 numbers in Ts(k). Next, we estimate an upper bound for

αk,s :=Nk∑i=1

uk,i∈Ts(k)

3k3−1∑

m=0

|θ(m + uk,i + k + 1)|. (3.6)

We see that Ak and m + βk + k + 1 are coprime by an argument similar to the onewhich proved the coprimality of Ak and βk. Hence, by Lemma 2.5 with A = Ak,B = m + βk + k + 1, and n = Nk, and for all sufficiently large k and uniformly inm ∈ {0, 1, . . . , 3k3 − 1}, we have

αk,s ≤3k3−1∑m=0

Nk∑i=1

�∑

j=1

|bj |dj(Aki + m + βk + k + 1)

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



≤3k3−1∑m=0

�∑j=1

|bj |(

Nk∑i=1

dj(Aki + m + βk + k + 1)

)

≤ 24k3(|b1| + · · · + |b�|)Nk log Nk

= c3k62k3

.(3.7)

Define

mk,s := mini=1,2,...,Nk

uk,i∈Ts(k)

3k3−1∑

m=0

|θ(m + uk,i + k + 1)|.

Since the number of elements in Ts(k) is at least⌊Nk

9�2 log Nk

⌋− 1 =

⌊2k3

9�2k3 log 2

⌋− 1 ≥ 2k3

9�2k3,

by (3.6) and (3.7), we have

2k3

9�2k3mk,s ≤ αk,s ≤ c3k

62k3.

Thus, mk,s ≤ 9c3�2k9 ≤ k10 for large k. Hence, for large k, there exists ik,s ∈

{1, 2, . . . , Nk} such that uk,ik,s∈ Ts(k) and

3k3−1∑m=0

|θ(m + uk,ik,s+ k + 1)| ≤ k10. (3.8)

Define nk,s := uk,ik,s∈ Ts(k). Then

nk,s = uk,ik,s= Akik,s + βk ≥ Ak ≥ q1 ≥ 4k3 > 2k. (3.9)

On the other hand, by (3.4), we have

nk,s + µ = Akik,s + βk + µ = Nk,s,µ

tµ+k∏i=rµ+k−1

q|q|−1i ,

for each nonzero integer µ = −k+1, . . . , k, where Nk,s,µ is a positive integer coprimewith primes qi for i ∈ {rµ+k−1, . . . , tµ+k}. Hence, by Lemma 2.1, we obtain

dj(nk,s + µ) = |q|tµ+k−rµ+k−1+1dj(Nk,s,µ) = |q|k+µdj(Nk,s,µ)

for j = 1, 2, . . . , �. Since

θ(nk,s + µ) =�∑

j=1

bjdj(nk,s + µ) = |q|k+µ�∑

j=1

bjdj(Nk,s,µ),

we obtain {q|θ(nk,s − k + 1), q2|θ(nk,s − k + 2), . . . , qk−1|θ(nk,s − 1),

qk+1|θ(nk,s + 1), qk+2|θ(nk,s + 2), . . . , q2k|θ(nk,s + k).(3.10)

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Furthermore, since

3k3 + nk,s + k ≤ 3k3 + Akik,s + βk + k

≤ 3k3 + 2k3ec1k2 log k + ec1k2 log k + k

≤ 22k3, (3.11)

for all sufficiently large k, we obtain, from (3.8) and (3.11),∞∑

m=0

|θ(m + nk,s + k + 1)||q|m ≤ k10 +

∞∑m=3k3

|θ(m + nk,s + k + 1)||q|m

≤ k10 + c4

∞∑m=3k3

m + nk,s + k + 1|q|m

≤ k10 +c4

|q|3k3

∞∑m=0

m + 3k3 + nk,s + k + 1|q|m

≤ k10 +c4

|q|3k3

∞∑m=0

m + 22k3+ 1

|q|m

≤ k10 + c4

(22

|q|3)k3 ∞∑

m=0

m + 1|q|m ≤ k11. (3.12)

Thus, by (3.9), (3.10), and (3.12), the assumptions in Lemma 2.6 are satisfied for thesequence {θ(n)}n≥1 and for each of the � subsequences nk,s ∈ Ts(k) (s = 1, . . . , �).Hence, by Lemma 2.6, the numbers

θ(nk,s) =�∑

j=1

bjdj(nk,s) = 2�∑

j=1

bjdj(s)

=

2b1, (s = 1)

2b1 + 2b2, (s = 2)

2b1 + 2b3, (s = 3)...2b1 + · · · + 2b�, (s = �)

are multiples of qk for every large k. This implies that b1 = · · · = b� = 0, which isa contradiction. The proof of Theorem 1.1 is completed.

Acknowledgments

We thank the referee for comments which improved the quality of this paper. Workby F. L. started in Winter of 2012 when this author visited Hirosaki University asa JSPS fellow. This author thanks Professor Takao Komatsu for useful advice, the

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

2nd Reading



Mathematics Department of the Hirosaki University for its hospitality and JSPSfor support. In addition, work of F. L. was supported in part by Projects PAPIITIN 104512, CONACyT 193539, CONACyT 163787 and a Marcos Moshinsky Fel-lowship.

References

[1] W. R. Ahlford, A. Granville and C. Pomerance, There are infinitely many Carmichaelnumbers, Ann. Math. 140 (1994) 703–722.

[2] P. B. Borwein, On the irrationality of certain series, Math. Proc. Cambridge Philos.Soc. 112 (1992) 141–146.

[3] P. Bundschuh, Linear independence of values of a certain Lambert series, ResultsMath. 51 (2007) 29–42.

[4] , Remarks on the linear independence of q-harmonic series, Fundam. Prikl.Mat. 16 (2010) 31–39.

[5] P. Bundschuh and K. Vaananen, Linear independence of q-analogues of certain clas-sical constants, Results Math. 47 (2005) 33–44.

[6] , Linear independence of certain Lambert series and allied series, Acta Arith.120 (2005) 197–209.

[7] P. Bundschuh and W. Zudilin, Irrationality measures for certain q-mathematical con-stants, Math. Scand. 101 (2007) 104–122.

[8] S. Chowla, On series of the Lambert type which assume irrational values for rationalvalues of the argument, Proc. Nat. Inst. Sci. India 13 (1947) 171–173.

[9] D. Duverney, Arithmetical functions and irrationality of Lambert series, AIP Conf.Proc. 1385 (2011) 5–16.

[10] P. Erdos, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.)12 (1948) 63–66.

[11] F. Luca and Y. Tachiya, Linear independence of certain Lambert series, to appear inProc. Amer. Math. Soc.

[12] V. Merila, On arithmetical properties of certain q-series, Results Math. 53 (2009)129–151.

[13] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of primenumbers, Illinois J. Math. 6 (1962) 64–94.

[14] Y. Tachiya, Irrationality of certain Lambert series, Tokyo J. Math. 27 (2004) 75–85.[15] J. Vandehey, On an incomplete argument of Erdos on the irrationality of Lambert

series, preprint (2012); http://arxiv.org/abs/1205.0090.[16] W. Zudilin, Approximations to q-logarithms and q-dilogarithms with applications to

q-zeta values, Zap Nauchn Sem S-Peterburg Otdel Math Inst Steklov (POMI ) 322(2005) 107–124.

Int.

J. N

umbe

r T

heor

y D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by "

UN

IV O

F M

INN

ESO

TA

, TW

IN C

ITIE

S" o

n 10

/19/

13. F

or p

erso

nal u

se o

nly.

Documents

IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE