On Sándor’s Inequality for the Riemann Zeta Function

Horst Alzer Morsbacher Straße 10 51545 Waldbröl Germany

                        h.alzer@gmx.de

Man Kam Kwong Department of Applied Mathematics The Hong Kong Polytechnic University Hunghom, Hong Kong

                   mankwong@connect.polyu.hk

Abstract Let ω(n) denote the number of distinct prime factors of n. We prove an analogue of a recently published inequality of Sándor that relates a series involving ω(n) to a quotient of zeta functions.

1 Introduction and statement of the main results

The classical Riemann zeta function is defined for real numbers s > 1 by

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime} \frac{1}{1 − p^{−s}} .

There are several Dirichlet series whose coefficients are arithmetical functions which can be expressed in terms of the zeta function. As an example we have the elegant identity

\frac{ζ^{2}(s)}{ζ(2s)} = ∑_{n=1}^{∞} \frac{2^{ω(n)}}{n^{s}} , s > 1,  (1)

where ω(n) denotes the number of distinct prime factors of n; see Apostol [1, p. 247], Hardy and Wright [2, p. 265]. The additive analogues of (1),

ζ^{2}(s) + ζ(2s) = 2 ∑_{n=1}^{∞} \frac{ζ_{n}(s)}{n^{s}}, s > 1,

and

ζ^{2}(s) − ζ(2s) = 2 ∑_{n=2}^{∞} \frac{ζ_{n−1}(s)}{n^{s}}, s > 1,

where ζ_{m}(s) = ∑_{k=1}^{m} 1/k^{s}, are due to Hassani and Rahimpour [3].

In 2018, Sándor [5] provided an interesting counterpart of (1),

∑_{n=1}^{∞} \frac{λ^{ω(n)}}{n^{s}} < \frac{ζ^{λ}(s)}{ζ(λs)}, λ > 2, s > 1.  (2)

Two questions arise naturally. Is there a similar inequality which offers an upper bound for ζ^{λ}(s)/ζ(λs)? Do there exist related results for real parameters λ with 1 < λ < 2? The aim of this note is to give affirmative answers to both questions.

Our first theorem presents a complement of (2).

Theorem 1. For all real numbers λ > 2 and s > 1, we have

\frac{ζ^{λ}(s)}{ζ(λs)} < ∑_{n=1}^{∞} \frac{(2^{λ} − 2)^{ω(n)}}{n^{s}} .  (3)

Next, we provide upper and lower bounds for ζ^{λ}(s)/ζ(λs) which are valid for all real numbers λ with 1 < λ < 2.

Theorem 2. Let 1 < λ < 2 and s > 1. Then

∑_{n=1}^{∞} \frac{(2^{λ} − 2)^{ω(n)}}{n^{s}} < \frac{ζ^{λ}(s)}{ζ(λs)} < ∑_{n=1}^{∞} \frac{λ^{ω(n)}}{n^{s}} .  (4)

In the next section, we collect three lemmas. The proofs of the two theorems are given in Section 3.

2 Lemmas

A proof of the first lemma can be found in Hardy and Wright [2, p. 249].

Lemma 3. Let f be a multiplicative arithmetical function with f(1) = 1 such that ∑_{n=1}^{∞} f(n)/n^{s} is absolutely convergent. Then

∑_{n=1}^{∞} \frac{f(n)}{n^{s}} = ∏_{p prime}(1 + \frac{f(p)}{p^{s}} + \frac{f(p^{2})}{p^{2s}} + ⋯).

The next lemma might be known. Since we cannot give a reference, we include a proof. Lemma 4. Let a > 0 and s > 1 be real numbers. The series

∑_{n=1}^{∞} \frac{a^{ω(n)}}{n^{s}}  (5)

is convergent. Proof. Let s > 1. From (1) we conclude that if 0 < a ≤ 2, then the series is convergent. Next, let a > 2. A result of Robin [4] states that there exists a number b ≈ 1.384 such that

ω(n) ≤ b\frac{log n}{log log n}, n ≥ 3.

This leads to

\frac{a^{ω(n)}}{n^{s}} ≤ \frac{a^{b log n/ log log n}}{n^{s}} , n ≥ 3.  (6)

Moreover, there exists a positive integer m such that

\frac{a^{b log n/ log log n}}{n^{s}} ≤ \frac{1}{n^{(s+1)/2}} , n ≥ m.  (7)

From (6) and (7) we conclude that the series in (5) is convergent. Lemma 5. Let c and x be real numbers.

(i) If 1 < c < 2 and 0 < x < 1/2, then

\frac{1 + (2^{c} − 3)x}{1 − x} < \frac{1 − x^{c}}{(1 − x)^{c}} < \frac{1 + (c − 1)x}{1 − x}.  (8)

(ii) If c > 2 and 0 < x < 1/2, then (8) holds with “>” instead of “<”. Proof. Let c > 0 and 0 < x < 1. We define

u_{c}(x) = \frac{1 − x^{c}}{(1 − x)^{c−1}} ,
v_{c}(x) = \frac{u_{c}(x) − 1}{x},
w_{c}(x) = xu′_{c}(x) − u_{c}(x) + 1.

Then

x^{2}v′_{c}(x) = w_{c}(x),
w′_{c}(x) = xu′′_{c}(x) = \frac{(c − 1)cx(1 − x^{c−2})}{(1 − x)^{c+1}}.

(i) Let 1 < c < 2 and 0 < x < 1/2. Then w′_{c}(x) < 0. Thus,

w_{c}(x) < w_{c}(0) = 0.

It follows that

2^{c} − 3 = v_{c}(1/2) < v_{c}(x) < v_{c}(0) = u′_{c}(0) = c − 1.

This implies (8). (ii) Let c > 2 and 0 < x < 1/2. Then w′_{c}(x) > 0 and w_{c}(x) > w_{c}(0) = 0. This gives

v_{c}(0) < v_{c}(x) < v_{c}(1/2)

which is equivalent to (8) with “>” instead of “<”.

3 Proof of the theorems

Proof. We follow the method of proof given in [5]. Let a > 0 and s > 1. For positive integers n, we define

F_{a}(n) = a^{ω(n)}.

Since

ω(1) = 0 and ω(mn) = ω(m) + ω(n), gcd(m, n) = 1,

we obtain

F_{a}(1) = 1 and F_{a}(mn) = F_{a}(m)F_{a}(n), gcd(m, n) = 1.

From Lemma 3 and Lemma 4 we get

∑_{n=1}^{∞} \frac{F_{a}(n)}{n^{s}} = ∏_{p prime}(1 + \frac{F_{a}(p)}{p^{s}} + \frac{F_{a}(p^{2})}{p^{2s}} + ⋯).  (9)

We have

∑_{n=1}^{∞} \frac{F_{a}(p^{n})}{p^{ns}} = ∑_{n=1}^{∞} \frac{a^{ω(p^{n})}}{p^{ns}} = ∑_{n=1}^{∞} \frac{a}{p^{ns}} = a\frac{p^{−s}}{1 − p^{−s}} .  (10)

(i) Let 1 < λ < 2. Using (9), (10) and Lemma 5, part (i) with x = p^{−s} yields

∑_{n=1}^{∞} \frac{(2^{λ} − 2)^{ω(n)}}{n^{s}} = ∏_{p prime} \frac{1 + (2^{λ} − 3)p^{−s}}{1 − p^{−s}} < ∏_{p prime} \frac{1 − p^{−sλ}}{(1 − p^{−s})^{λ}}  (11)

and

∑_{n=1}^{∞} \frac{λ^{ω(n)}}{n^{s}} = ∏_{p prime} \frac{1 + (λ − 1)p^{−s}}{(1 − p^{−s})^{λ}} > ∏_{p prime} \frac{1 − p^{−sλ}}{(1 − p^{−s})^{λ}} .  (12)

From (11), (12) and

ζ^{λ}(s) = ∏_{p prime} \frac{1}{(1 − p^{−s})^{λ}} , ζ(λs) = ∏_{p prime} \frac{1}{1 − p^{−λs}}  (13)

we conclude that (4) is valid.

(ii) Let λ > 2. We apply Lemma 5, part (ii) with x = p^{−s}, then we obtain (11) with “>” instead of “<”. Using this result and (13) leads to (3).

References

[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.

[2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford University Press, 1959.

[3] M. Hassani and S. Rahimpour, L-Summing method, RGMIA Research Report Collection 7(4) (2004), article 10.

[4] G. Robin, Estimation de la fonction de Tchebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, Acta Arith. 42 (1983), 367–389.

[5] J. Sándor, An inequality involving a ratio of zeta functions, Notes Number Th. Disc. Math. 24 (2018), 92–94.

2020 Mathematics Subject Classification: Primary 11M06; Secondary 11B83, 26D15. Keywords: Riemann zeta function, prime omega function, inequality.

(Concerned with sequence \underline{A129251}.)

Received January 8 2023; revised versions received March 3 2023; March 5 2023. Published in Journal of Integer Sequences, March 16 2023.

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