Asymptotic relations for the products of elements of some positive sequences

1 Introduction

Let p i be the ith prime number. Let us denote ∏ p k ≔ ∏ i = 1 k p i . Consider the sequence b k ≔ k 2 k ∏ p k . Alzer and Sándor [1] proved the following inequality:

for every k ≥ 5 , where constants c 0 and c 1 are equal to

respectively. In this study, another estimation of the sequence { b k } connected with the limit lim k → ∞ f ( k ) = 2 will be shown, where

and p runs over all prime numbers less than or equal to x. The above limit was proven by Alzer and Sándor [1]. Nevertheless, we decided to obtain a new proof of this limit as the original proof was obtained in an over complicated way (in a certain sense). In consequence, we also found a new result (Theorem 1), which gives a simple and universal tool for generating asymptotic relations of many known sequences of real numbers, especially for the products of elements of certain sequences.

2 Main result

The discussion is based on the following well-known fact, which is connected to d′ Alembert’s ratio test.

This result will be used for finding the estimation of sequence { b k } k = 1 ∞ in the following way. If we can find sequences { m k } k = 1 ∞ , { M k } k = 1 ∞ of positive real numbers such that

then lim k → ∞ b k m k = ∞ , lim k → ∞ b k M k = 0 , thereby m k ≤ b k ≤ M k for sufficiently large k.

Using Stirling’s approximation we can write

From the prime number theorem, we get p k ∼ k log k . Therefore, we have

If we denote c k ( t ) = e k t ( log k ) − k , then we get c k ( t ) c k + 1 ( t ) ∼ e − t log ( k + 1 ) , and thereby

where e 2 − t > 1 if t < 2 and e 2 − t < 1 if t > 2 . Then, for every t 1 < 2 < t 2 and for sufficiently large k ∈ ℕ the following inequalities hold

hence

In a similar way, we may find the estimation of sequence ∏ p k k = 1 ∞ . We have

Let us choose the sequence d k ≔ ( e t k log k ) k . Then, d k + 1 ( t ) d k ( t ) ∼ e t + 1 ( k + 1 ) log ( k + 1 ) and

where e − t − 1 > 1 if t < − 1 and e − t − 1 < 1 if t > − 1 . Therefore, we obtain

for any t 3 < − 1 < t 4 and for sufficiently large k. Moreover, we also have

i.e.,

The last result reminds another known limit

which is not an incident and comes from the general relationship presented in Theorem 1.

From now on we will use the symbol ∏ x k to denote product ∏ i = 1 k x i , where { x i } i = 1 ∞ is any real sequence. Now, we will prove our main result.

3 Applications

Some applications of Theorem 1 are given as follows.

(3.1) Let f k = k . Then, lim k → ∞ k ! k k = 1 e .

(3.2) Let f k = 1 ∏ i = 1 k 1 − 1 p i . Then, f k ∼ e γ log k (see [2,3]) and

where γ denotes Euler’s constant. Let α > 0 . If we replace f k by

for k ≥ i 0 , where i 0 ≔ min { i ∈ ℕ : p i > α } , then f k ( α ) ∼ e − log ( c ( α ) ) log α k for some c ( α ) > 0 (see [2,4,5]), and consequently

For example, c ( 2 ) ≈ 0.832429065662 .

(3.3) Let α ∈ ℝ , 0 ≤ α < 1 . Then (see [6])

which implies the relation

(3.4) Let us set f n = ∑ i = 1 n i n . Then (see [6,14,15]), we have f n ∼ e e − 1 n n , which implies (see also final remark 2)

i.e.,

(3.5) Next, we set f n = ∏ k = 1 n k k n 2 2 . By the definition of the Glaischer-Kinkelin constant (see [6,7,8])

we also have

(see [7,9,10,11]) we obtain f n ∼ e − 1 2 n and

which by Theorem 1 gives us the relation

(3.6) At last, if we set

then f n ∼ A n n ( n + 1 ) 2 + 1 12 , where A denotes the Glaisher-Kinkelin constant. Therefore, we get

Now we present an application of Corollary 2.

(3.7) In [7, Problem 1.5], it was proved that

Hence, by Theorem 1, we get

and by Corollary 2

We note that from (4) we obtain the solution of Problem 51, p. 45 from Pólya and Szegő [12]:

Note also that, applying Corollary 2, monotonicity of sequences { f k } k = 1 ∞ and ∏ f k k k = 1 ∞ is not important at all, because the following lemma holds.

4 Final remarks

1. Using the other asymptotic expansions, especially the ones given in the study of Kellner [6] (e.g., for product of Bernoulli numbers, for products of the special values of gamma function, etc.), we can generate many new relations that are omitted here. Other relations were published recently in [13] as well.

2. In the history of the following asymptotic relation (see [14,15,16])

one thread – Dutch connection is missing. The above relationship was found by the last author as one of the problems in the problem section of the known Dutch journal Nieuw Archief voor Wiskunde but in the equivalent form:

It is interesting that in both proofs of these equalities (from Nieuw Archief voor Wiskunde and by Lampret – see [15]) Tannery’s theorem was used.

1. Another estimation for a number of primes is discussed by Meštrović in [17].

5 Problems

The natural question about asymptoticity of the following expressions:

and

arises from (1) and (2). For now, this question remains unanswered. The following formula (see [18,19])

is an inspiration for solving this problem.