Abstract

Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that where This is an improvement to the estimate found in the literature.

2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

1. Introduction

Let us set , where counts the number of primes no more than a fixed real number. The irregularity of the primes makes it difficult to obtain an exact pratical formula for the prime counting function. It is one of the well studied functions in number theory and the whole of mathematics given its profound connection to the grand Riemann hypothesis, but not much is known concerning their distribution. In the following sequel, our mission is to improve a known estimate for the prime counting function involving the chebyshev theta function, given by

2. Notations

Through out this paper a prime number will either be denoted by or the subscripts of . Any other letter will be clarified. The function counts the number of prime factors of with multiplicity. The inequality for sufficiently large values of will be compactly written as or .

3. Preliminary Results

Lemma 3.1. There exist some constant c > 0 such that

Proof. For a proof, see for instance ¹.

Lemma 3.2. For all x ≥ 2

Proof. For a proof see for instance ².

Lemma 3.3. For all x ≥ 2

Proof. For a proof see for instance ².

4. Main Result

Theorem 4.1. For every positive integer x

The plan of attack is to examine the distribution of odd natural numbers and even numbers. We break the proof of this result into two cases; The case is odd and the case is even. For the case is odd, we argue as follows: We first observe that there are as many even numbers as odd numbers less than any given odd number . That is, for , there are such possibilities. On the other hand, consider the sequence of even numbers less than , given as such that clearly there are such number of terms in the sequence. Again consider those of the form such that ; Clearly there are such terms in this sequence. We terminate the process by considering those of the form such that and ; Clearly there are such number of terms in this sequence. The upshot is that . We now turn to the case is even. For the case is even, we argue as follows: First we observe that there are even numbers less than or equal to On the other hand, there are even numbers less than or equal to This culminates into the assertion that By combining both cases, the result follows immediately.

Remark 4.2. Understanding the fractional parts of real numbers is an important problem in number theory and gains much paramountcy. The next result gives a little insight into the problem.

Corollary 4.1.

Proof. Stirling’s formula ³ gives

(4.1)

Also from Theorem 4.1, we obtain

(4.2)

Comparing equation (4.1) and (4.2), we have

thereby establishing the estimate.

The prime counting function is one of the most important functions in number theory, given it’s connection with the famous Riemann hypothesis. The prime counting function has been studied by many authors in the past decade. For instance in 2010, Dusart ⁴ showed that

holds for every x ≥ 2953652287 and that

for every . We now obtain, as an immediate consequence of Theorem 4.1, an explicit formula for the prime counting function It relates the prime counting function to the Chebychev theta function

Theorem 4.3. For all positive integers x

where

denotes the fractional part of any real number.

Proof. We use Theorem 4.1 to establish an explicit formula for the prime counting function. Theorem 4.1 gives which can then be recast as

where p runs over the primes. It follows by futher simplification that

It follows that

where

and the result follows immediately.

Remark 4.4. Using Theorem 4.3, we obtain an estimate for the prime counting and it’s allied functions, with very small error term in the following sequel.

Theorem 4.5. For all positive integers x ≥ 2

where

Proof. By Theorem 4.3, we can write

Where

Now, we estimate the term Clearly we can write

It follows that

where we have used Corollary 4.1 and the proof is complete.

Corollary 4.2. There exist a contant c > 0 such that

where

Proof. The result follows by plugging the estimate in Lemma 3.1 into Theorem 4.5.

It is known that

¹, where . The above estimate derived indicates that we can do better than this with

where is some explicit function.

Remark 4.6. It needs to be said that Theorem 4.5 relates the prime counting function to the Chebyshev theta function. Thus, very good and sharp estimates for will eventually yield a sharp estimate for the prime counting function. For the purpose of applications, we stick to the current estimate involving the Chebyshev theta function.

Theorem 4.7. For all positive integers x ≥ 2, we have

where

Proof. It is known that

for all x ≥ 2 ². Comparing this result with Theorem 4.5, the result follows immediately.

Theorem 4.8. For all positive integers x ≥ 2, we have

where

Proof. Recall the well-known identity ²

By comparing this identity with Theorem 4.5, the result follows immediately.

Remark 4.9. Theorem 4.7 measures the discrepancies of the prime counting function and _(x)/ log x.

5. Final Remarks

In this paper we have obtained a new estimate for the prime counting function. We have shown that indeed the estimate

where

is better than

found in the literature.

References