On Landau’s Inequality for the Prime Counting 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 We prove that the inequality 2π(n) − π(2n) ≥ 2ω(n) is valid for all n ≥ 71. Here, π(n) denotes the prime counting function and ω(n) denotes the number of distinct prime factors of n. Our inequality refines a recently published result by Zhang.

1 Introduction and statement of the main result

In 1909, Landau [4] conjectured that the inequality

π(2n) ≤ 2π(n)  (1)

is valid for all integers n ≥ 2. Here, π(n) denotes the number of primes which are less than or equal to n. The first proof of (1) was given by Rosser and Schoenfeld [11] in 1966. Moreover, they showed that the sign of equality holds in (1) if and only if n ∈ {2, 4, 10}.

Landau’s inequality attracted the attention of several mathematicians, who presented various extensions and counterparts of (1). For more information on this subject we refer to Ehrhart [3], Mitrinović, Sándor, Crstici [6, Chapter VII], Panaitopol [7, 8, 9], and Vlamos [12].

Our work was inspired by an interesting paper published by Zhang [13] in 2020. He obtained a positive lower bound for the difference 2π(n) − π(2n). More precisely, he proved that for n ≥ 59,

2π(n) − π(2n) > ω(2n),  (2)

where ω(n) denotes the number of distinct prime factors of n. Here, we offer the following improvement of (2) for n ≥ 71.

Theorem 1. Let n ≥ 71 be an integer. Then

2π(n) − π(2n) ≥ 2ω(n),  (3)

with equality if and only if n ∈ {78, 100, 102, 126}.

We note that n = 70 is the largest integer such that (3) is not true. In the next section, we present a proof of Theorem 1 and we show that (3) refines (2).

2 Proof of Theorem 1

Proof. Let

F (n) = 2π(n) − π(2n) − 2ω(n).  (4)

We consider two cases.

Case 1. 71 ≤ n ≤ 30091.

We used MAPLE 16 and the following computer program to verify (3).

     with(NumberTheory):
     w := n -> nops(PrimeFactors(n)):
     F := n -> 2*pi(n)-pi(2*n)-2*w(n):
     for k from 71 to 30091 do
        if F(n) <= 0 then print(k,F(n)) end if
     end do;

For all n we obtain that F (n) is positive with exactly four exceptions:

F (78) = F (100) = F (102) = F (126) = 0.

Case 2. n ≥ 30092.

We apply the estimates

\frac{n}{log(n) − 1} ≤ π(n) (n ≥ 5393),  (5)

π(n) ≤ \frac{n}{log(n) − 1.1} (n ≥ 60184),  (6)
ω(n) ≤ c\frac{log(n)}{log(log(n))} (c = 1.3841; n ≥ 3).  (7)

The inequalities (5) and (6) are due to Dusart [2], whereas (7) was proved by Robin [10]. Better bounds for π(n) were given by Berkane and Dusart [1].

Let F (n) be the function defined in (4). Using (5), (6) and (7) gives for n ≥ 30092,

F (n) ≥ \frac{2n}{log(n) − 1} − \frac{2n}{log(2n) − 1.1} − \frac{2c log(n)}{log(log(n))} = G(n), say.  (8)

We set n = e^{x} with x ≥ log(30092) ≈ 10.31 and a = 1.1 − log(2) ≈ 0.40. Then,

\frac{1}{2}G(n) = \frac{1}{2}G(e^{x}) = (1 − a)\frac{e^{x}}{(x − 1)(x − a)} − c\frac{x}{log(x)}.

Using

\frac{x − y}{log(x) − log(y)} < \frac{x + y}{2} (0 < y < x),

(see Mitrinović [5, p. 273]) with y = 1 gives

\frac{1}{2(1 − a)}(x − 1)(x − a)G(e^{x}) = e^{x} − \frac{c}{1 − a}x(x − a)\frac{x − 1}{log(x)}

> e^{x} − \frac{c}{1 − a}x(x − a)\frac{x + 1}{2}

> 1 + ∑_{k=1}^{5} \frac{x^{k}}{k!} − \frac{6}{5}x(x + 1)(x − \frac{2}{5})

= 1 + \frac{x}{600}P (x)

with

P (x) = 5x^{4} + 25x^{3} − 620x^{2} − 132x + 888.

Since P is positive on [10, ∞), we obtain G(e^{x}) > 0 for x ≥ log(30092). From (8) we conclude that F (n) > 0 for n ≥ 30092.

Finally, we show that (3) improves Zhang’s inequality (2).

Lemma 2. For all integers n ≥ 2, we have

2ω(n) ≥ ω(2n).  (9)

Equality holds in (9) if and only if n = p^{k}, where p is an odd prime number and k is a positive integer.

Proof. Let

n = ∏_{j=1}^{r} p_{j}^{k_{j}},

where p_{1}, ⋯, p_{r} are prime numbers with p_{1} < ⋯ < p_{r} and k_{1}, ⋯, k_{r} are positive integers. If p_{1} = 2, then

2ω(n) − ω(2n) = 2r − r = r > 0,

and if p_{1} > 2, then

2ω(n) − ω(2n) = 2r − (r + 1) = r − 1 ≥ 0,

with equality if and only if n = p_{1}^{k_{1}}.

Let n ≥ 71. If n = p^{k}, where p ≥ 3 is a prime number and k ≥ 1 is an integer, then n ∉ {78, 100, 102, 126}. From Theorem 1 and Lemma 2 we conclude that

2π(n) − π(2n) > 2ω(n) = ω(2n).

And, if n ≠ p^{k}, then

2π(n) − π(2n) ≥ 2ω(n) > ω(2n).

References

[1] D. Berkane and P. Dusart, On a constant relating to the prime counting function, Mediterr. J. Math. 13 (2016), 929–938.

[2] P. Dusart, Estimates of some functions over primes without R.H., arxiv preprint arXiv:1002.0442 [math.NT], 2010. Available at https://arxiv.org/abs/1002.0442.

[3] E. Ehrhart, On prime numbers, Fibonacci Quart. 26 (1988), 271–274.

[4] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, 1909.

[5] D. S. Mitrinović, Analytic Inequalities, Springer, 1970.

[6] D. S. Mitrinović, J. Sándor, and B. Crstici, Handbook of Number Theory, Mathematics and its Applications 351, Kluwer, 1996.

[7] L. Panaitopol, On the inequality p_{a}p_{b} > p_{ab}, Bull. Math. Soc. Sc. Math. Roumanie (N.S.) 41 (1998), 135–139.

[8] L. Panaitopol, On a theorem of Edmund Landau, An. Univ. Bucureşti Mat. 49 (2000), 67–72.

[9] L. Panaitopol, Some generalizations for a theorem by Landau, Math. Inequal. Appl. 4 (2001), 327–330.

[10] 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.

[11] J. B. Rosser and L. Schoenfeld, Abstracts of scientific communications, Intern. Congr. Math. Moscow, 1966, Section 3: Theory of Numbers.

[12] P. Vlamos, Properties of the function f(x) = x/π(x), Int. J. Math. Math. Sci. 28 (2001), 307–311.

[13] S. Zhang, An improved inequality of Rosser and Schoenfeld and its application, Integers 20 (2020), #A103.

2020 Mathematics Subject Classification: Primary 11A41. Keywords: Landau’s inequality, prime counting function, estimate, prime omega function.

(Concerned with sequences \underline{A000720}, \underline{A001221}, and \underline{A060208}.)

Received June 28 2022; revised versions received September 8 2022; September 10 2022. Published in Journal of Integer Sequences, September 11 2022.

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