A criterion to estimate the least common multiple of sequences and asymptotic formulas for ζ(3) arising from

recurrence relation of an elliptic function ^{∗}

Shigeki Akiyama

§0. Introduction

In [2], the author studied the asymptotic behavior of the least common multiple of a sequences {a_{n}}_{n=1}^{∞} provided that it satisfies certain axioms (A1) and (A2) (see page 4). Sequences defined by binary linear recurrence, for example, were handled there. A typical result was

\frac{log |a_{1}a_{2} · · · · · · a_{n}|}{log[a_{1}, a_{2}, …, a_{n}]} = ζ(2) + O(\frac{log n}{n}),  (1)

where [a_{1}, a_{2}, …, a_{n}] is the least common multiple of the terms a_{1}, a_{2}, …, a_{n} and ζ(·) is the Riemann zeta function. On the origin of these problems and related works, see [7] [5] [1] [2] [10]. To prove (1) in [2], the fundamental tool employed was to rewrite the least common multiple by ”an inclusion exclusion principle”. This was done in [2] with the essential use of the axioms (A1) and (A2). In this article, we employ a more sophisticated axiom

(S) (a_{n}, a_{m}) = |a_{(n,m)}|.

We say that the non zero sequence {a_{n}}_{n=1}^{∞} satisfying (S) to be a strongly divisible sequence. This ”strong divisibility” was studied by several authors in [4] [3] [9]. And we prove that the assertion of [2] still holds (See Theorem 1). It is easily seen that (S) is weaker than (A1) and (A2). Furthermore, it is shown in Theorem 1 that the relation (2), which is the ”inclusion exclusion principle of the least common multiple”, is equivalent to the axiom (S).

Our next problem is to find the good example of a strongly divisible sequence which has appropriate asymptotic behavior. Let σ(u) be the Weierstrass sigma function associated with some lattice. Put ψ_{n}(u) = σ(nu)/(σ(u)^{n^{2}}). Then ψ_{n}(u) is an elliptic function with respect to u and satisfies the recurrence relation:

∗Only for the private copy. See Japanese Journal of Math., Vol.22, no.1 (1996), 129 – 146, for the exact published version.

ψ_{m+n}(u)ψ_{m−n}(u) = ψ_{m+1}(u)ψ_{m−1}(u)ψ_{n}(u)^{2} − ψ_{n+1}(u)ψ_{n−1}(u)ψ_{m}(u)^{2},

and ψ_{0}(u) = 0, ψ_{1}(u) = 1. This relation was classically known and used to calculate the algebraic relation between 𝔭(nu) and 𝔭(u), where 𝔭(u) is the Weierstrass p-function. The sequence {ψ_{n}(u)}_{n=1}^{∞} is determined completely by the initial values ψ_{2}(u), ψ_{3}(u) and ψ_{4}(u) when ψ_{2}(u)ψ_{3}(u) ≠ 0. Now let ψ_{i}(u)(i = 2, 3, 4) be integers. The sequences of this type are systematically studied by Ward. He showed in [12] and [13] that, when

ψ_{2}(u)|ψ_{4}(u) and (ψ_{3}(u), ψ_{4}(u)) = 1,

each ψ_{n}(u) is an integer and the sequence {ψ_{n}(u)} satisfies (S). In section 3, we prove the average asymptotic formula for log |ψ_{n}(u)|. So, under certain conditions, we can derive the asymptotic formula:

\frac{log |ψ_{1}(u)ψ_{2}(u) · · · · · · ψ_{n}(u)|}{log[ψ_{1}(u), ψ_{2}(u), …, ψ_{n}(u)]} = ζ(3) + O(\frac{1}{n}),

(Theorem 4), by using the method developed in [2]. As a by-product, in some special cases, we can calculate log |σ(u)|, where u is a division point of some period of the elliptic function (see section 4).

§1. The Fundamental Theorem.

Let {a_{n}}_{n=1}^{∞} be a non zero integer sequence. We say that {a_{n}} is divisible when it has the property:

(D) n|m implies a_{n}|a_{m}.

It is easily seen that the axiom (S) implies (D).

Theorem 1. Let {a_{n}}_{n=1}^{∞} be a strongly divisible sequence. Then we have

[a_{1}, a_{2}, …, a_{n}] = ∏_{i=1}^{n} |M(i)|  (2)

where M(i) = ∏_{d|i}(a_{i/d})^{μ(d)} and μ(·) is the Möbius function. Conversely, if a sequence of non zero integers {a_{n}}_{n=1}^{∞} satisfies (2) then {a_{n}}_{n=1}^{∞} is strongly divisible.

Proof. We first prove the sufficiency. The case n = 1 is obvious. Assume the equality (2) for n − 1. First we see that

[a_{1}, a_{2}, …, a_{n}]/[a_{1}, a_{2}, …, a_{n−1}] = G.C.D._{i=1,2,... ,n−1} (\frac{a_{n}}{(a_{n}, a_{i})})
= |a_{n}|/ ( L.C.M._{i=1,2,... ,n−1} a_{(n,i)}) = |a_{n}|/ (L.C.M._{d|n} a_{n/d}) .

Since {a_{n}} is divisible, we may restrict to the prime divisor p of n:

[a_{1}, a_{2}, …, a_{n}]/[a_{1}, a_{2}, …, a_{n−1}] = |a_{n}|/ (L.C.M._{p|n} a_{n/p}) .

Using the inclusion exclusion principle, we have

= |a_{n}|\frac{∏_{p_{1},p_{2}}(a_{n/p_{1}}, a_{n/p_{2}}) ∏_{p_{1},p_{2},p_{3},p_{4}}(a_{n/p_{1}}, a_{n/p_{2}}, a_{n/p_{3}}, a_{n/p_{4}}) ⋯}{∏_{p_{1}} |a_{n/p_{1}}| ∏_{p_{1},p_{2},p_{3}}(a_{n/p_{1}}, a_{n/p_{2}}, a_{n/p_{3}}) ⋯} (3)
= |a_{n}|\frac{∏_{p_{1},p_{2}} |a_{n/p_{1}p_{2}}| ∏_{p_{1},p_{2},p_{3},p_{4}} |a_{n/p_{1}p_{2}p_{3}p_{4}}| ⋯}{∏_{p_{1}} |a_{n/p_{1}}| ∏_{p_{1},p_{2},p_{3}} |a_{n/p_{1}p_{2}p_{3}}| ⋯}
= |M(n)|.

Thus the relation (2) is proved by induction.

Now we prove the necessity. So we assume the relation (2). Then every M(n) is an integer because

|M(n)| = [a_{1}, a_{2}, …, a_{n}]/[a_{1}, a_{2}, …, a_{n−1}].  (4)

The axiom (S) is equivalent to the following statement:

if d_{1} |̸d_{2} and d_{2} |̸d_{1} then (M(d_{1}), M(d_{2})) = 1.

This can easily be shown by the inverse relation a(n) = ∏_{d|n} M(d). In fact, let us assume that there exists a pair d_{1} and d_{2} which satisfies

d_{1} < d_{2}, d_{1} |̸d_{2}, d_{2} |̸d_{1}, p|M(d_{1}) and p|M(d_{2}),

where p is a prime. We also assume that d_{1} is chosen to be minimum under these conditions. Denote by ord_{p}(x) the multiplicities of p in the prime factorization of an integer x. Then if d|d_{1}, we have d|d_{2} or p |̸M(d) by the minimality of d_{1}. Thus

ord_{p} a_{d_{1}} = ∑_{d|d_{1}} ord_{p} M(d) = ∑_{d|(d_{1},d_{2})} ord_{p} M(d) = ord_{p} a_{(d_{1},d_{2})}.

But from (4), this means ord_{p} M(d_{1}) = 0. This is a contradiction.

The equality (3) can be found in Ward [14] Lemma 9.1. He also noticed in the same paper that the axiom (S) is equivalent to:

(R) There exists a function f from N to N such that

M|a_{n} is equivalent to f(M)|n.

We can easily show this equivalence itself. It seems that the main interest of Ward [14] is to characterize this axiom (R) by using M(n). Now let us recall the axioms (A1) and (A2) in [2]:

(A1) For each prime p, we denote by S_{p} the set of positive integer n’s so that a_{n} is divisible by p. If S_{p} ≠ ϕ, there exists an integer r(p) such that S_{p} coincides with the set of all positive r(p) multiples,

(A2) For each prime p, there exists a weakly increasing function f_{p} from N∪{0} to itself satisfying the property ord_{p}(a_{n}) = f_{p}(ord_{p}(n)), for n ∈ S_{p}.

We see that the axioms (A1) + (A2) are stronger than (R). So Theorem 1 of [2] is a consequence of the above Theorem 1. But the author does not have a good, not too artificial, example of strongly divisible sequence which does not satisfy both (A1) and (A2).

Proposition 1. Let {a_{n}} and {b_{n}} be strongly divisible sequences. Then the greatest common divisor sequence {(a_{n}, b_{n})}_{n=1}^{∞} is strongly divisible. Moreover, if we assume that each b_{n} is positive, then the composition sequence {a_{b_{n}}}_{n=1}^{∞} is also strongly divisible.

Proof. These are easily verified by the relations,

((a_{n}, b_{n}), (a_{m}, b_{m})) = ((a_{n}, a_{m}), (b_{n}, b_{m})) = (a_{(n,m)}, b_{(n,m)})

and

(a_{b_{n}}, a_{b_{m}}) = |a_{(b_{n},b_{m})}| = |a_{b_{(n,m)}}|.

The similar assertion holds for the axioms (A1) and (A2) (see Proposition 1 and 2 in [2]). Once we establish Theorem 1, the next two theorems follow immediately by the same proof as for Proposition 3 of [2].

Theorem 2. Let {a_{n}} be a strongly divisible sequence which has the following asymptotic behavior:

log |a_{n}| = An^{l} + O(n^{l−1}ω(n)),

with positive constant A and

ω(n) = \{ \begin{matrix}1 & if l > 1 \\ log n & if l = 1. \end{matrix}

Then we have

log[a_{1}, a_{2}, …, a_{n}] = \frac{n^{l+1}}{(l + 1)ζ(l + 1)} + O(n^{l}ω(n))  (5)

and

\frac{log |a_{1}a_{2} · · · · · · a_{n}|}{log[a_{1}, a_{2}, …, a_{n}]} = ζ(l + 1) + O(\frac{ω(n)}{n}).  (6)

Theorem 3. Let {a_{n}} be a strongly divisible sequence. If we have

log |a_{1}a_{2} · · · · · · a_{n}| = An^{l+1} + O(n^{l}),

with l ≥ 1. Then we have (5) and (6).

In the case l > 1, the assumption of Theorem 3 is weaker than that of Theorem 2. Consider the case l = 1. If we know more precise average asymptotic behavior, we can proceed further. Assume that there exist constants ε, A and B such that

\frac{1}{n} log |a_{1}a_{2} · · · · · · a_{n}| = An + B + O(n^{−ε}),  (7)

where A and ε are positive. Then we easily see, for the strongly divisible sequence {a_{n}} it holds that

\frac{log |a_{1}a_{2} · · · · · · a_{n}|}{log[a_{1}, a_{2}, …, a_{n}]} = ζ(2) + O(\frac{E(n)}{n^{2}}),  (8)

where

E(n) = ∑_{k=1}^{n} φ(k) − \frac{3}{π^{2}} n^{2},

and φ(·) is Euler’s totient function. The error term of (8) is better than those of Theorem 2 and 3 (see [11], [8]) and is best possible. The estimate of type (7) is established in the case of Lucas sequence in [6] by using the estimation of discrepancy of a sequence {nθ}_{n=1}^{∞} where θ is a certain irrational number. See also [10].

§2. Sequences arising from an Elliptic Function.

In this section, we treat an example of strongly divisible sequences. Let L = 2ω_{1}Z+2ω_{3}Z be a lattice in C. We choose ω_{i}(i = 1, 3) so that τ = ω_{3}/ω_{1} is in the upper half plane H= {z ∈C | Im(z) > 0}. Denote by σ_{i}(u) = σ_{i}(u, L) the Weierstrass sigma function:

σ_{i}(u, L) = u ∏_{w∈L′}(1 − u/w) exp(u/w + (1/2)(u/w)^{2}),

where L′ = L − {0}. Put ψ_{n}(u) = σ(nu)/(σ(u)^{n^{2}}). Then ψ_{n}(u) is an elliptic function and satisfies the following recurrence relation for m ≥ n ≥ 1:

ψ_{m+n}(u)ψ_{m−n}(u) = ψ_{m+1}(u)ψ_{m−1}(u)ψ_{n}(u)^{2} − ψ_{n+1}(u)ψ_{n−1}(u)ψ_{m}(u)^{2}.

This relation is crucial in calculating the n-th multiple value of Weierstrass p-function in the classical elliptic function theory. Note that when u ∉ L then ψ_{0}(u) = 0 and ψ_{1}(u) = 1. Let {h_{n}} be the sequence defined by the recurrence

h_{m+n}h_{m−n} = h_{m−1}h_{m+1}h_{n}^{2} − h_{n+1}h_{n−1}h_{m}^{2}  (9)

and h_{0} = 0, h_{1} = 1. The systematical study of this sequence in the rational number field was done by Ward in [12] and [13]. We quote some of his results in this section.

• Let h_{i}(i = 2, 3, 4) be integers, h_{2}h_{3} ≠ 0 and h_{2}|h_{4}. Then {h_{n}} is well defined and every h_{n} is an integer.

Hereafter, we assume h_{i}(i = 2, 3, 4) to be integers. We call this type of sequence {h_{n}} as an elliptic sequence.

• If h_{2}h_{3}h_{4}h_{5} ≠ 0 then every h_{n} ≠ 0 for n ≥ 1.

• The sequence {h_{n}} is divisible. Moreover if (h_{3}, h_{4}) = 1, then {h_{n}} is strongly divisible.

Now let Δ be the discriminant of the elliptic curve corresponding to the lattice L. Under the assumption h_{2}h_{3} ≠ 0, we can determine the values g_{2}(u, L), g_{3}(u, L) and 𝔭(u, L) formally in terms of h_{2}, h_{3} and h_{4} by solving simultaneous equations:

ψ_{i}(u) = ψ_{i}(u, L) = h_{i},

for i = 2, 3 and 4. In fact, g_{2}(L), g_{3}(L) and 𝔭(u, L) are written in a form of rather complicated rational function of h_{2}, h_{3} and h_{4} in page 50 of [12]. Thus Δ is given by

Δ = g_{2}(L)^{3} − 27g_{3}(L)^{2}
= (−h_{2}^{15}h_{4} + h_{2}^{12}h_{3}^{3} − 3h_{2}^{10}h_{4}^{2} + 20h_{2}^{7}h_{3}^{3}h_{4} − 3h_{2}^{5}h_{4}^{3} − 16h_{2}^{4}h_{3}^{6} − 8h_{2}^{2}h_{3}^{3}h_{4}^{2} − h_{4}^{4})/(h_{2}^{8}h_{3}^{3}).

Note that the corresponding formula (19.3) of [12] is not valid. When Δ ≠ 0, then the sequence {h_{n}} does correspond to some elliptic curve. In other words, there exist a lattice L with j-invariant g_{2}(L)^{3}/Δ and the value g_{2}(L) and g_{3}(L) fit together with the above calculated ones. Thus, when h_{2}h_{3} ≠ 0 and Δ ≠ 0, we can express h_{n} = ψ_{n}(u) for every n.

Let {a_{n}}, {b_{n}} be any complex valued sequences. We say that {a_{n}} is equivalent to {b_{n}} when there exists a non-zero constant C such that

a_{n} = C^{n^{2}−1}b_{n}.

We have

• Let {h_{n}} be an elliptic sequence, h_{2}h_{3} ≠ 0 and Δ = 0. Then {h_{n}} is equivalent either to the sequence 0, 1, 2, …, n, … of non negative integers or to a sequence {U_{n}} where U_{n} = (α^{n} − β^{n})/(α − β), αβ = 1 and α + β is contained in a quadratic extension of Q.

Notice that log[1, 2, …, n] is the Chebyshev’s psi function which is widely studied, and the least common multiple of Lucas type sequence was treated in [2]. As we are interested in the estimation of the least common multiple of a sequence, there will be no problem in the case Δ = 0. Taking into account of Theorem 2, we can state our problem as follows:

Problem. Let {h_{n}} be an elliptic sequence with h_{2}h_{3} ≠ 0 and Δ ≠ 0. Study the asymptotic behavior of log |h_{n}|.

§3. Asymptotic Behavior of Elliptic Sequences.

Let z = exp(π\sqrt{−1}v), v = u/(2w_{1}) and θ_{1}(v) be the elliptic theta function:

θ_{1}(v) = \sqrt{−1} ∑_{n∈Z}(−1)^{n}q^{(n−1/2)^{2}}z^{2n−1},

where q = exp(π\sqrt{−1}τ). Denote by θ_{1}^{′0} the value of θ1^{′}(v) at 0, that is,

θ_{1}^{′0} = 2π ∑_{n≥1}(−1)^{n−1}(2n − 1)q^{(n−1/2)^{2}}.

Then we have

Lemma 1. The function F(v) = π(Im(v))^{2}/ Im(τ) − log |θ_{1}(v)| is invariant under parallel translations v → v + 1 and v → v + τ. In other words, the value F(u/(2w_{1})) is determined by u mod L.

Proof. This can be shown by the well known formulas:

θ_{1}(v + 1) = −θ_{1}(v)

and

θ_{1}(v + τ) = − exp(−π\sqrt{−1}(2v + τ))θ_{1}(v).

Lemma 2. Let

A(u) = F(v) + log |θ_{1}^{′0}| − log |2w_{1}|.  (10)

Then we have

log |ψ_{n}(u)| = A(u)n^{2} − A(nu)

Proof. The function σ(u) is written in the form:

σ(u) = 2w_{1} exp(2η_{1}w_{1}v^{2})θ_{1}(v)/θ_{1}^{′0},

with a constant η_{1} which satisfies

σ(u + 2w_{1}) = −σ(u) exp(2η_{1}(u + w_{1})).

Then we have

A(u) = Re(η_{1}u^{2}/(2w_{1})) + π(Im(v))^{2}/ Im(τ) − log |σ(u)|.  (11)

This formula implies

n^{2} (A(u) + log |σ(u)|) = A(nu) + log |σ(nu)|.

This proves the lemma.

Lemma 3. Let |z| > 1 and q^{2m}z^{2n} ≠ 1 for all m ∈Z. Denote by [x] the maximal integer which does not exceed x. Then we have

log | ∏_{m≥1}(1 − q^{2m}z^{2n})| = log |1 − q^{2m_{0}}z^{2n}| − (n log |z|)^{2}/ log |q| + O(n),

where m_{0} = [ −n log |z|/ log |q| + 1/2 ].

Proof. We can show that |q^{2m}z^{2n}| < |q| for m > m_{0} and |q^{−2m}z^{−2n}| ≤ |q| for m < m_{0}. Thus we have

log | ∏_{m≥1}(1 − q^{2m}z^{2n})| = log |1 − q^{2m_{0}}z^{2n}| + ∑_{m<m_{0}} log |q^{2m}z^{2n}|

+ ∑_{m<m_{0}} log |1 − q^{−2m}z^{−2n}| + ∑_{m>m_{0}} log |1 − q^{2m}z^{2n}|

= log |1 − q^{2m_{0}}z^{2n}| + 2m_{0}n log |z| + m_{0}^{2} log |q| + O(n)
= log |1 − q^{2m_{0}}z^{2n}| − (n log |z|)^{2}/ log |q| + O(n).

Lemma 4. For all positive integer n, let nu ∉ L . Then A(nu) is bounded from below as a function of n and we have

A(nu) = − min_{m∈Z} log |1 − e(mτ + nv)| + O(n),

where e(z) = exp(2π\sqrt{−1}z).

Proof. By (10) and Lemma 1, A(u) is determined by u mod L. Since θ_{1}(v) is entire, log |θ_{1}(v)| is bounded from above in the fundamental parallelotope { v = ξ_{1} + ξ_{3}τ ∈C | ξ_{1}, ξ_{3} ∈ [0,1] }. This shows that A(nu) is bounded from below as a function of n. Changing the sign of v if necessary, we may assume that |z| > 1. Then by using Lemma 3 and the infinite product representation of θ_{1}(v):

θ_{1}(v) = −\sqrt{−1}q^{1/4}z ∏_{m≥1}(1 − q^{n})(1 − q^{2m}z^{2})(1 − q^{2m−2}z^{−2}),

we have

log |θ_{1}(nv)| = log |1−q^{2m_{0}}z^{2n}|−(n log |z|)^{2}/ log |q|+∑_{m≥1} log |1−q^{2m−2}z^{−2n}|+O(n).

Since |z| > 1, the third term of the right hand side is O(1). Thus we have

log |θ_{1}(nv)| = log |1 − q^{2m_{0}}z^{2n}| + π(Im(nv))^{2}/ Im(τ) + O(n).

By (10), we proved the assertion.

Lemma 5. Let Θ be a positive number smaller than 1/2 and z be a complex number with |1 − z| < Θ. Then the inequality

k|1 − z|/2 < |1 − z^{k}| < 2k|1 − z|

holds for any positive integer k ≤ (2Θ)^{−1}.

Proof. We easily see that |(1−z^{k})/(1−z)| ≤ k max{1, |z^{k−1}|}. Let r = z −1 then

k log |1 + r| ≤ k log(1 + |r|) ≤ k|r| ≤ 1/2.

So |(1 + r)^{k−1}| ≤ \sqrt{e} · (1 + |r|)^{−1} ≤ \sqrt{e} < 2. This shows the right inequality. So we have

|(1 − z^{k})/(1 − z) − k| ≤ ∑_{i=1}^{k−1} |z^{i} − 1| ≤ 2|z − 1| ∑_{i=1}^{k−1} i
= k(k − 1)|z − 1| < (k − 1)/2 < k/2 (12)

This implies that |(1−z^{k})/(1−z)| > k−k/2, which is the left inequality.

We are now able to prove A(nu) = O(n) for almost all n.

Lemma 6. Assume that there exists no integer n such that nu ∈ L. Consider a set C = {n ∈N | A(nu) > Ln } for sufficiently large fixed L. Assume that C contains infinitely many elements. We put C = {n_{i}}_{i=1}^{∞} with

n_{1} < n_{2} < n_{3} < · · · · · · .

Then there exists a positive constant T ≥ 2 and a sequence of positive integers {ν_{k}}_{k=1,2,...} which satisfies three conditions:

(a) ν_{k+1} > exp(T ν_{k} ),

(b) if ν_{k} ≤ n_{i} < ν_{k+1}, then n_{i} is a multiple of ν_{k},

(c) Let ξ(n_{i}) = − min_{m∈Z} log |1 − e(mτ + n_{i}v)|, then ξ is a decreasing function of n_{i} ∈ C in each interval [ν_{k}, ν_{k+1}).

Proof. By Lemma 4, there exists L_{0} such that

A(nu) + min_{m∈Z} log |1 − e(mτ + nv)| ≤ L_{0}n

for any n. Take a positive constant L greater than L_{0}. Then we have

log |1 − e(m_{i}τ + n_{i}v)| < −(L − L_{0})n_{i},

for a certain integer m_{i}. This shows that

|1 − e(m_{i}τ + n_{i}v)| < exp(−Ln_{i}).

for a positive constant L(= L − L_{0}). Define a new set

C′ = {n ∈ N | ^{∃}m ∈ Z, |1 − e(mτ + nv)| < exp(−Ln)}.

Then the set C is a subset of C′. Thus our aim is to verify the above three conditions for the set C′ for a sufficiently large L. More precisely, we shall show that there exist two sequences of positive integers {ν_{k}}_{k=1}^{∞} and {r_{k}}_{k=1}^{∞} so that C′ is the set consisting of elements of the form

ν_{1} < 2ν_{1} < ⋯ < r_{1}ν_{1} < ν_{2} < 2ν_{2} < ⋯ < r_{2}ν_{2} < ν_{3} < · · · · · · .

We proceed by induction. Let ν_{1} = n_{1}, μ_{1} = m_{1} and r_{1} be the biggest integer so that

|1 − e(k(μ_{1}τ + ν_{1}v))| < exp(−Lkν_{1})  (13)

holds for any integer k ≤ r_{1}. This inequality implies that

|1 − e(μ_{1}τ + ν_{1}v)| < exp(−Lr_{1}ν_{1}).

In fact, by using Lemma 5 and induction, we can show

|1 − e(μ_{1}τ + ν_{1}v)| < exp(−Lkν_{1}).

for 1 ≤ k ≤ r_{1}. The existence of r_{1} easily follows from this. By (13), we see that ν_{1}, 2ν_{1}, …, r_{1}ν_{1} ∈ C′. It is shown that the minimum of |1 − e(mτ + kν_{1}v)| is attained by m = kμ_{1}, if we take sufficiently large L. We can also show that ξ(kν_{1}) is a decreasing function. To see this, we note that the inequality (12) implies that

|\frac{1 − e(k(μ_{1}τ + ν_{1}v))}{1 − e(μ_{1}τ + ν_{1}v)} − k| < \frac{k}{2}

and

\frac{1 − e((k + 1)(μ_{1}τ + ν_{1}v))}{1 − e(μ_{1}τ + ν_{1}v)} − \frac{1 − e(k(μ_{1}τ + ν_{1}v))}{1 − e(μ_{1}τ + ν_{1}v)} = e(k(μ_{1}τ + ν_{1}v)).

The value e(k(μ_{1}τ + ν_{1}v)) is close enough to 1, which show that ξ(kν_{1}) ≥ ξ((k + 1)ν_{1}) for k ≤ r_{1} − 1.

Now we assume that

ν_{1} < 2ν_{1} < ⋯ < r_{1}ν_{1} < ν_{2} < 2ν_{2} < ⋯ < r_{2}ν_{2} · · · · · · < ν_{i} < 2ν_{i} < ⋯ < r_{i}ν_{i}

are the elements of C′ satisfying the following:

(a’) There exists a constant T ≥ 2 such that ν_{j+1} > exp(T r_{j}ν_{j}) for j = 1, 2, …, i − 1,

(b’) ξ(lν_{j}) is a decreasing function of l for l = 1, 2, …, r_{j},

(c’) There exist no other elements of C′ smaller than ν_{i}.

(d’) r_{j} is defined to be the biggest integer k so that |1 − e(k(μ_{j}τ + ν_{j}v))| < exp(−Lkν_{j}) holds for j = 1, 2, …, i.

Take the smallest element ν_{i+1} of C′ − {lν_{j}}_{j,l} where j = 1, 2, …, i and l = 1, 2, …, r_{j}. Let μ_{i+1} be the corresponding m. We see ν_{i+1} > ν_{i} by (c’). At first, we shall show ν_{i+1} > r_{i}ν_{i}. If ν_{i+1} ≤ r_{i}ν_{i} then we have

|1 − e((μ_{i+1} − μ_{i})τ + (ν_{i+1} − ν_{i})v)|
≤ |1 − e(μ_{i+1}τ + ν_{i+1}v)| + |1 − e(μ_{i}τ + ν_{i}v)|
≤ exp(−Lν_{i+1}) + exp(−Lr_{i}ν_{i}) ≤ exp(−L(ν_{i+1} − ν_{i})),

which contradicts the definition of ν_{i+1}. Let E and F be integers such that ν_{i+1} = Er_{i}ν_{i} +F and |F | ≤ r_{i}ν_{i}/2. Let G be a certain positive integer which will be chosen suitably later. Then we have

r_{i}ν_{i} − G · F = (G · E + 1)r_{i}ν_{i} − G · ν_{i+1}.

Thus we have

|1 − e (((G · E + 1)r_{i}μ_{i} − G · μ_{i+1})τ + (r_{i}ν_{i} − G · F)v) |
≤ |1 − e(G · (μ_{i+1}τ + ν_{i+1}v))| + |1 − e((G · E + 1)r_{i}(μ_{i}τ + ν_{i}v))|
≤ 2G · exp(−Lν_{i+1}) + 2(G · E + 1)r_{i} exp(−Lr_{i}ν_{i}).

Here we used Lemma 5. (The conditions of G in Lemma 5 cause no problem in showing the assertion.) Hence if we have the inequality

2G · exp(−Lν_{i+1}) + 2(G · E + 1)r_{i} exp(−Lr_{i}ν_{i}) < exp(−L(ν_{i}r_{i} − G · F))

then ν_{i}r_{i} − G · F ∈ C′. Firstly, we consider the case F > 0. Then we have

2G · exp(−Lν_{i+1}) + 2(G · E + 1)r_{i} exp(−Lr_{i}ν_{i}) < 2r_{i}G · (E + 2) exp(−Lr_{i}ν_{i}).

Thus if 2r_{i}G · (E + 2) < exp(LG · F) then ν_{i}r_{i} − G · F ∈ C′. Put G = 1 when r_{i}ν_{i}/4 ≤ |F |. If r_{i}ν_{i}/4 > |F | then take G such that r_{i}ν_{i}/4 ≤ |G·F | ≤ 3r_{i}ν_{i}/4. In this case, we remark that there are at least two ways to choose G. If E ≤ exp(Lr_{i}ν_{i}/5) then we have

2r_{i}G · (E + 2) ≤ 3/2 · r_{i}^{2}ν_{i} · (E + 2) ≤ exp(L · r_{i}ν_{i}/4) ≤ exp(LG · F)

for L ≥ 60. This shows ν_{i}|(F · G). Moreover, we see ν_{i}|F. In fact, when G ≠ 1, we can choose G with ν_{i}|F among the several possible candidates.

Secondly, consider the case F < 0. Note that |1 − exp(z)| = |1 − exp(−z)|. So, in the same way, we can prove ν_{i}|F, substituting ν_{i}r_{i} − G · F with G · F − ν_{i}r_{i}. Now, we consider the case ν_{i}|F, which includes the case F = 0. Then we have ν_{i+1} = Hν_{i} for a positive integer H. When μ_{i+1} = Hμ_{i}, by using Lemma 5, if H ≤ exp(Lr_{i}ν_{i})/2 then

|1 − e(μ_{i+1}τ + ν_{i+1}v)| = |1 − e(H · (μ_{i}τ + ν_{i}v))| ≥ H/2 · |1 − e(μ_{i}τ + ν_{i}v)|

≥ \frac{H}{4(r_{i} + 1)} exp(−L(r_{i} + 1)ν_{i})

The last inequality follows from the definition of r_{i} and Lemma 5. On the other hand

|1 − e(μ_{i+1}τ + ν_{i+1}v)| ≤ exp(−L · Hν_{i}),

and H ≥ r_{i} + 2, which gives a contradiction. This shows that H > exp(Lr_{i}ν_{i})/2 and ν_{i+1} > exp(T r_{i}ν_{i}). If μ_{i+1} ≠ Hμ_{i} , then there exists a positive constant c which depends only on τ such that c < |1−e((μ_{i+1}−Hμ_{i})τ)|. But we have

|1 − e((μ_{i+1} − Hμ_{i})τ)| ≤ |1 − e(μ_{i+1}τ + ν_{i+1}v)| + |1 − e(H · (μ_{i}τ + ν_{i}v))|.
≤ exp(−Lν_{i+1}) + |1 − e(H · (μ_{i}τ + ν_{i}v))|

Choose L sufficiently large so that exp(−L) < c/2. Applying Lemma 5, we obtain H ≫ exp(Lr_{i}ν_{i}) and ν_{i+1} > exp(T r_{i}ν_{i}). Let r_{i+1} be the integer defined by the property (d’) for j = i + 1. Then, in a similar way as above, we can show that (b’) for j = i + 1 is valid. This completes the proof.

Now we prove the average asymptotic behavior of elliptic sequence.

Lemma 7. Let {h_{n}} be an elliptic sequence satisfying h_{2}|h_{4}, (h_{3}, h_{4}) = 1, Δ ≠ 0 and h_{2}h_{3}h_{4}h_{5} ≠ 0. Then we have

\frac{1}{n} log |h_{1}h_{2} · · · · · · h_{n}| = \frac{A(u)}{3}n^{2} + O(n),

where u is a complex number determined by h_{n} = ψ_{n}(u), and A(·) is defined by (10).

Proof. Let C be the set defined in Lemma 6. If C is finite, then by Lemma 4, we see that A(nu) = O(n). Thus we have, by Lemma 2,

log |h_{1}h_{2} · · · · · · h_{n}| = ∑_{i=1}^{n} (A(u)i^{2} + O(i))

= \frac{A(u)}{3}n^{3} + O(n^{2}).

This shows the assertion. Now we assume that C is an infinite set.

log |h_{1}h_{2} · · · · · · h_{n}| = ∑_{i=1}^{n} A(u)i^{2} − ∑_{i=1}^{n} A(iu)

= \frac{A(u)}{3}n^{3} + O(n^{2}) − ∑_{i∉C(n)} A(iu) − ∑_{i∈C(n)} A(iu),  (14)

where C(n) = {i ∈N | i ≤ n, n ∈ C}. By the definition of C and Lemma 4, we have

∑_{i∉C(n)} A(iu) = O (∑_{i=1}^{n} i) = O(n^{2}).  (15)

Without loss of generality, we may assume that n is an integer in the interval lν_{k} ≤ n < (l + 1)ν_{k}, l ≤ r_{k}, where k, l are positive integers. Then we have

∑_{i∈C(n)} A(iu) = ∑_{i=1}^{k−1} ∑_{j=1}^{r_{i}} ξ(jν_{i}) + ∑_{j=1}^{l} ξ(jν_{k}) + O(∑_{i=1}^{n} i).

By Lemma 6, we have

∑_{i=1}^{k−1} ∑_{j=1}^{r_{i}} ξ(jν_{i}) ≤ ∑_{i=1}^{k−1} r_{i}ξ(ν_{i}),

and

∑_{j=1}^{l} ξ(jν_{k}) ≤ lξ(ν_{k}).

Since each h_{n}(n ≥ 1) is a non zero integer, we see A(nu) ≤ A(u)n^{2}. So, by Lemma 4, we have A(nu) = O(n^{2}). This implies that ξ(x) = O(x^{2}). Thus

∑_{i=1}^{k−1} ∑_{j=1}^{r_{i}} ξ(jν_{i}) = O (∑_{i=1}^{k−1} r_{i}ν_{i}^{2}) = O (kr_{k−1}ν_{k−1}^{2}) ,

and

∑_{j=1}^{l} ξ(jν_{i}) = O(lν_{k}^{2}) = O(n^{2}).

Using the inequality exp(T r_{k−1}ν_{k−1}) < ν_{k} appearing in the proof of Lemma 6, we have

k − 1 ≤ ν_{k−1} ≤ r_{k−1}ν_{k−1} ≤ log(ν_{k})/T = log(n)/T.

This shows that

∑_{i=1}^{k−1} ∑_{j=1}^{r_{i}} ξ(jν_{i}) = O(log^{3} n) = O(n^{2}).

Summing up, we have shown

∑_{i∈C(n)} A(iu) = O(n^{2}).

By (14) and (15), we see the assertion.

Our last task is to show that A(u) > 0 for the elliptic sequence of Lemma 7.

Lemma 8. Let {h_{n}} be an elliptic sequence satisfying h_{2}|h_{4}, (h_{3}, h_{4}) = 1. Then, for any prime p, there exists n ≤ 2p + 1 such that p|h_{n}.

Proof. This is the Theorem 5.1 of Ward [12]. Consider h_{n−1}h_{n+1}/h_{n}^{2} (mod p) for n = 2, 3, …, p+1 and use Dirichlet’s box principle with (9).

Lemma 9. Let {h_{n}} be an elliptic sequence satisfying h_{2}|h_{4}, Δ ≠ 0 and h_{2}h_{3}h_{4}h_{5} ≠ 0. Let u be the complex number determined by h_{n} = ψ_{n}(u). Then we have A(u) > 0.

Proof. By Lemma 2, we have log |h_{n}| = A(u)n^{2} − A(nu). If A(u) < 0, by Lemma 4, log |h_{n}| < 0 for a certain n, which contradicts the fact |h_{n}| ≥ 1. This shows that A(u) ≥ 0. Assume that A(u) = 0. Then log |h_{n}| = −A(nu) holds for all n. By Lemma 4, there exists a positive constant K such that |h_{n}| ≤ K. Let p be a prime greater than K. Then Lemma 8 implies that p|h_{n} for a certain n. This shows h_{n} = 0. This is a contradiction.

Theorem 4. Let {h_{n}} be an elliptic sequence satisfying h_{2}|h_{4}, (h_{3}, h_{4}) = 1, Δ ≠ 0 and h_{2}h_{3}h_{4}h_{5} ≠ 0. Then we have

\frac{log |h_{1}h_{2} · · · · · · h_{n}|}{log[h_{1}, h_{2}, …, h_{n}]} = ζ(3) + O(\frac{1}{n}).

Proof. Combine Theorem 3, Lemma 7 and Lemma 9.

§4. Concluding Remarks.

In section 1, we proposed axiom (S) in order to treat the least common multiple of sequences. How can we generalize these situations? The asymptotic behaviour of the least common multiples of sequences admits a rather big error term such as

log[a_{1}, a_{2}, …, a_{n}] = (main term) + O(n^{l}),

to obtain our type of results. It seems that the axiom (S) is too strict. We want a weaker axiom to obtain this asymptotic formula without using (2).

It seems natural to expect A(nu) = O(n). In other words, the estimation log |ψ_{n}(u)| = A(u)n^{2} + O(n) for every n is expected. (However, it would not cause any improvement of our asymptotic formulas.) In the case of sequences of Lucas type, this individual estimation is established in the light of Baker’s estimation of the summation of logarithm of algebraic numbers. The analogue of this for our case is needed. Up to now, the author does not know any result of this type. However if it exists, the above argument has an independent merit. To derive the average asymptotic behavior of this sequence, we do not use the fact that the corresponding elliptic curve is defined over rational number field. Thus our argument, elementary but a little complicated, might be used in other problems.

There exist examples of elliptic sequences with A(u) = 0, which stimulated some interest to the author. Let (h_{2}, h_{3}, h_{4}) = (1, 1, 1) or (1,1,0). In this case we see Δ ≠ 0 and h_{κ} = 0 for a certain κ. Take the smallest such κ(≥ 4). Choose u such that h_{n} = ψ_{n}(u). Then u is a κ-division point of the corresponding elliptic curve. This shows that h_{nκ} = 0 for every n ∈N, h_{n} ≠ 0 for κ |̸n and h_{n} = h_{n+κ} for every n. The formula log |ψ_{n}(u)| = A(u)n^{2} − A(nu) still holds for κ |̸n. Using the argument of Lemma 9, we have A(u) ≥ 0. If A(u) > 0 then, by Lemma 6, ψ_{n}(u) = h_{n} is not bounded, which contradicts the periodicity of h_{n}. This shows that A(u) = 0, which gives a relation:

log |σ(u)| = Re(η_{1}u^{2}/(2w_{1})) + π(Im(u/(2w_{1})))^{2}/ Im(τ),

by the use of (11).

Some numerical examples are found in the following Table 1.

Acknowledgements.

The author would like to express his gratitude to the referee for careful reading of the manuscript.

Table 1.

(h_{2}, h_{3}, h_{4}) = (1, 1, −1)
𝔭 = 1/3
g_{2} = 4/3
g_{3} = −35/27
Δ = −43
J = −64/1161
τ = −1/2 + 1.002926948197305394966\sqrt{−1}
w_{1} = 0 + 1.363182418170433596392\sqrt{−1}
u = 1.531551051899213180325
A(u) = 0.031408253543743824633
(h_{2}, h_{3}, h_{4}) = (1, 2, 1)
𝔭 = 3/4
g_{2} = 19/4
g_{3} = −23/8
Δ = −116
J = −6859/7424
τ = −1/2 + 1.228990832129246397562\sqrt{−1}
w_{1} = 0 + 1.111804814975135231767\sqrt{−1}
u = 1.301268905063793834665
A(u) = 0.084840615679859699533
(h_{2}, h_{3}, h_{4}) = (1, 3, 1)
𝔭 = 28/27
g_{2} = 2812/243
g_{3} = −168083/19683
Δ = −11321/27
J = −22235451328/6016443561
τ = −1/2 + 1.411429696528812658305\sqrt{−1}
w_{1} = 0 + 0.907350665488871173676\sqrt{−1}
u = 1.252539774071157056878
A(u) = 0.136695663405057918491
(h_{2}, h_{3}, h_{4}) = (1, −1, 1)
𝔭 = 0
g_{2} = 4
g_{3} = −1
Δ = 37
J = 64/37
τ = 0 + 1.221127360764627252496\sqrt{−1}
w_{1} = 0 + 1.225694690993395030427\sqrt{−1}
u = −0.92959271528539567440519
+1.22569469099339503042711\sqrt{−1}
A(u) = 0.025555704119984420117

\begin{aligned}(h_{2}, h_{3}, h_{4}) & = (2, 1, 4) \\ 𝔭 & = 41/6 \\ g_{2} & = 1573/3 \\ g_{3} & = −62387/27 \\ Δ & = −676 \\ J & = −23030293/108 \\ τ & = −1/2 + 3.139317204766964341216\sqrt{−1} \\ w_{1} & = 0 + 0.352738304847112422947\sqrt{−1} \\ u & = 0.660057981718722555872 \\ A(u) & = 0.126242626431163986909 \\ (h_{2}, h_{3}, h_{4}) & = (3, 5, 6) \\ 𝔭 & = 821/300 \\ g_{2} & = 425041/7500 \\ g_{3} & = −277119161/3375000 \\ Δ & = −2129/125 \\ J & = −76787844018343921/7185375000000 \\ τ & = −1/2 + 2.662903182751486527437\sqrt{−1} \\ w_{1} & = 0 + 0.615192114179878726930\sqrt{−1} \\ u & = 0.759101080162039522511 \\ A(u) & = 0.162312085677860900909 \\ (h_{2}, h_{3}, h_{4}) & = (1, 1, 0) \\ 𝔭 & = 5/12 \\ g_{2} & = 1/12 \\ g_{3} & = −161/216 \\ Δ & = −15 \\ J & = −1/25920 \\ τ & = −1/2 + 0.877437661348222505688\sqrt{−1} \\ w_{1} & = 0 + 1.596242222131783510148\sqrt{−1} \\ u & = 1.400603042332602023180 \\ A(u) & = 0.000000000000000000000 \\ (h_{2}, h_{3}, h_{4}) & = (1, 1, 1) \\ 𝔭 & = 2/3 \\ g_{2} & = 4/3 \\ g_{3} & = −19/27 \\ Δ & = −11 \\ J & = −64/297 \\ τ & = −1/2 + 1.087533286862971250700\sqrt{−1} \\ w_{1} & = 0 + 1.458816616938495229330\sqrt{−1} \\ u & = 1.269209304279553421688 \\ A(u) & = 0.000000000000000000000 \end{aligned}

20

References

[1] S. Akiyama, Lehmer numbers and an asymptotic formula for π, J. Number Theory 36 (1990) 328-331.

[2] S. Akiyama, A new type of inclusion exclusion principle for sequences and asymptotic formulas for ζ(n), J. Number Theory 45 (1993) 200-214.

[3] P. Horak and L. Skula, A characterization of the second order strong divisibility sequences, Fibonacci Quart. 23 (1985) no. 2, 126–132.

[4] C. Kimberling, Strong divisibility sequences and some conjectures, Fibonacci Quart. 17 (1979) no. 1, 13 - 17.

[5] P. Kiss and F. Mátyás, An asymptotic formula for π, J. Number Theory 31 (1989) 255-259.

[6] P. Kiss and B. Tropak, Average order of logarithms of terms in binary recurrences, Discussion Math. 10 (1990) 29-39.

[7] Y.V. Matiyasevich and R.K. Guy, A new formula for π, Amer. Math. Monthly 93 (1986) 631-635.

[8] A.I. Saltykov, On Euler’s function, Vestnik M.G.U., 15 (1960), 34-50

[9] A. Schinzel, Second order strong divisibility sequences in an algebraic number field, Arch. Math. (Brno) 23 (1987) no. 3, 181–186.

[10] B. Tropak, Some asymptotic properties of Lucas numbers, Algebra and number theory, Pedagogical Univ. (1990) 49-55, ed. A Grytczuk.

[11] A. Walfisz, Weylsche Exponentialsummen in der neuren Zahlen Theorie, Mathematische Forschungsberichte 15, Deutsher Verlag Wissen. , Berlin 1963.

[12] M. Ward, Memoir on elliptic divisibility sequence, Amer. J. Math. 70 (1948) 31-74.

[13] M. Ward, The law of repetition of primes in an elliptic divisibility sequence, Duke Math. J. 15 (1948) 941-946.

[14] M. Ward, The mapping of the positive integers into themselves which preserve division, Pacific J. Math. 5 (1955) 1013-1023.

FACULTY OF SCIENCE, NIIGATA UNIVERSITY NIIGATA, 950-21, JAPAN