Derivation of an Exact Prime Counting Function Between Intervals and a Possible Parameterization for the Prime Numbers

DERIVATION OF AN EXACT PRIME COUNTING FUNCTION BETWEEN INTERVALS AND A POSSIBLE PARAMETERIZATION FOR THE PRIME NUMBERS.

JASON R. SOUTH JUNE 2, 2020

Abstract. The following analysis will present the necessary and suﬃcient conditions needed to establish when the sum of two, non-zero, natural numbers produce a prime number. In so doing, it will also be shown that the prime interval counting function given by π(a, a + b) gives the exact number of primes between some natural number a and a + b for appropriate conditions on b.

1. Introduction Finding an explicit formula for counting the prime numbers has profound consequences for many areas of mathematics, including the Riemann Hypothesis. In the following section it will be demonstrated that an exact count can be made of the prime numbers between any interval with appropriate bounds.

2. The Exact Prime Interval Counting Function. In this section, the set of non-zero natural numbers will be given by N, the set of primes by P. The primorial, a#, is deﬁned as the product of all prime numbers less than or equal to a. Finally, the greatest common divisor function is given by GCD.

Proposition 2.1. For every a ∈ N where a > 1, there will exist some b ∈ N where b < a and a + b is an odd, prime number. Proof. Chebyshev’s Theorem, see page 108 in [2], shows that there will exist a prime number between a and 2a for all a ∈ N. Since two is the only even prime number, it follows that for every natural number, a, where a > 1, there will exist some natural number b < a where a + b is an odd, prime number. Lemma 2.2. Given a, b ∈ N where a ≥ b, the sum a + b ∈ P only when GCD(a + b, a#) = 1. Proof. For the forward conditional where a, b ∈ N and a ≥ b is given by a + b ∈ P ⇒ GCD(a + b, a#) = 1 If a + b is a prime number where a > b, it will have no divisors other than one and itself, showing GCD(a + b, a#) = 1 must hold. For the reverse condition given by

GCD(a + b, a#) = 1 ⇒ a + b ∈ P Since all composite numbers between a and 2a must be the product of prime numbers less than or equal to a, it follows that when a + b is coprime to a#, that number is not divisible by any lesser prime numbers. This forces a + b ∈ P. Remark 2.3. It is important to note, that in general, b, can take any value up to the value a2 + a. This holds because the ﬁrst number that is coprime to a#, and composite, is the square of the ﬁrst prime greater than a. Since the smallest value this prime can have, in principle, is a + 1, it is clear that a2 + 2a < (a + 1)2. This means that the maximum value for b is a2 + a. 1 In order to predict the number of prime numbers between some value a and a+b, the number of coprime elements to a# up to a + b will be needed. The Mobius Function, see page 98 in [2], will be needed and its properties are as follows.  1, if x = 1  µ(x) = 0, if x is not square-free. (−1)r, where r is the number of distinct primes of x. Recall that a and b sum to a prime only when GCD(a + b, a#) = 1 via Lemma 2.2. This is equivalent to stating that the number of prime numbers between a and a+b are those elements up to a + b and greater than a which are coprime to a#.

Theorem 2.4. The number of primes between a and a+b, denoted by π(a, a+b), where a, b ∈ N P a+b and b ≤ a(a + 1), is given by π(a, a + b) = m|a# µ(m) m − 1. Proof. The totient function is related to the Mobius Function, see page 250 in [1], by X µ(m) φ(a#) = a# and m ∈ m N m|a# and gives the total number of coprime elements of a#. Under Lemma 2.2 and Remark 2.3, a + b ∈ P only when GCD(a + b, a#) = 1 where b ≤ a(a + 1). The number of coprime elements of a# up to a + b is given by taking the partial sum a+b X X µ(m) X a + b = µ(m) m m n=1 m|a# m|a# Since this function gives all of the coprime values to a# up to a+b, it is necessary to subtract off the coprime values less than or equal to a. Since a# is simply the product of all prime numbers up to the value a, the only c < a where GCD(c, a#) = 1 is when c = 1, since any composite or prime less than a will share a factor with a#. With this, the exact number of primes in range P a+b of a and a + b, written as π(a, a + b), is given by π(a, a + b) = m|a# µ(m) m − 1 Theorem 2.5. There exists a possible formula for finding the next prime number Proof. Theorem 2.4 shows that when π(a, a+b) = 1, that a+b is the first new prime larger than P a+b a. This allows for m|a# µ(m) m = 2. Since the sum is related to the totient function by P a+b φ(a#) φ(a#) m|a# µ(m) m = (a+b) a# +c for some c ∈ R, where (a+b) a# +c = 2. A simplification where c0 = 2 − c, shows that π(a) Y 1 a + b = c 0 (1 − 1 ) i=1 pi .

3. Discussion and Future Work The Riemann Hypothesis has many equivalent variations. As an example, the Riemann Hypothesis is equivalent√ to the statement that for suﬃciently large values of x, it follows that π(x) ∼ Li(x) + O( x ln(x)), see page 47 in [3]. Finding the general conditions where π(a, a + b) = 1 would essentially lead to a parametrization for the prime numbers. This would be very useful for aiding in a proof of the Riemann Hypothesis.

acknowledgements This work could not have been possible without the numerous hours spent discussing these topics with Farley Ferrante and Dr. Robert Daly. 2 References [1] H. M. Edwards. Riemann’s Zeta Function. Dover Publications, INC, Mineola, NY, 1974, MR 0466039. [2] J. J. Tattersall Elementary Number Theory in Nine Chapters. Cambridge University Press, ISBN 978-0- 521-85014-8, 2005. [3] The Riemann Hypothesis: A Resource for the Aﬃcionado and Virtuoso Alike Peter Borwein, Stephen Choi, Brendan Rooney, and Andrea Weirathmueller Springer, ISBN 978-0-387-72125-5, 2008 Jason R. South Dallas, Texas email: [email protected]