An Exploration of The History of the Prime Numbers Theorem

This essay provides a comprehensive overview of the history of the Prime Numbers Theorem (PNT), starting from its origins in ancient Greece with the Sieve of Eratosthenes. It traces the theorem's evolution, highlighting the contributions of key figures such as Bernhard Riemann, Jacques Hadamard, Atle Selberg, and P. Erdos. The essay details Riemann's hypothesis and its significance to PNT, as well as Hadamard's proof. It further explores Selberg and Erdos's elementary proof and the PNT's relation to the Clay Institute's Millennium Problem. The document emphasizes the ongoing efforts to understand prime number distribution, including the unresolved Riemann hypothesis, showcasing the PNT's continued relevance in mathematics.

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Running head: The History of the Prime Numbers Theorem
The History of the Prime Numbers Theorem
Institution
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The History of the Prime Numbers Theorem
Introduction
A number is classified as a prime number if it is a whole number and it has only two
factors which are 1 and itself. A factor on the other hand is a whole number that can evenly
be divided into another number. That is the division quotient should be a whole number.
Some examples of prime numbers are 5,13 and 29. The other numbers are classified as
composite numbers. Prime numbers have several applications in mathematics which makes
them a key part of mathematics. For instance, in the field of Algebra prime numbers are
applied to solve proofs. An example being the group theory (Ingemar and Karol, 2017). In
addition, the connection of prime numbers with Riemann zeta function makes them
applicable during analytical tasks. In this document will review the history of prime numbers
theorem and the role of some of the greatest contributors to the theorem.
The History of prime numbers theorem
The Prime numbers theorem (PNT) was introduced to illustrate the asymptotic
distribution of the prime numbers within the positive integers. The theorem formalizes the
insinuation that prime numbers become rare as they increase in size through the
quantification of the occurrence rate. All this are summarized by the PNT statement: “For
large values of x, π(x) is approximately equal to x/ln(x)”.
The first time the PNT was introduced in mathematics was by ancient Greek
mathematicians who were trying to study perfect and amicable numbers. In the 200 BC an
algorithm for calculating prime numbers known as a sieve of Eratosthenes was designed by
the Greek Eratosthenes. From here it took a significant amount of time before important
developments were made in this area. This come in the 17th Century when Fermat proved a
speculation previously made by Albert Girard that all prime numbers in the form 4 n+1can be
rewritten as a summation of four squares. This technique provided a new way of factoring
large numbers (Rudman, 2007).
People involved in the development of prime numbers theorem
Prime numbers just like other mathematic concept took the effort of several
individuals before it could be proven and applied in the current mathematical field. This list
involved those who developed the theories, those who proved the theories as well as the
mathematicians who managed to gauge the applications of the theories to other mathematical
models.

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The History of the Prime Numbers Theorem
Below is a list of some of the great mathematicians who contributed to the theorem
and the roles that they played.
Riemann
In 1859 Bernhard Riemann proposed a hypothesis which was termed as the Riemann
hypothesis. This was a conjecture that suggested that the Riemann zeta function has zeros at
the negative even integers as well as complex numbers having a real part ½. This hypothesis
was aimed at providing a proof to the work of Gauss which developed the prime numbers
theorem (Morrissey and Triplett, 2009). This gave room for a deep study of prime numbers
distribution as well as other generalizations which were considered very vital in solving
problems under pure mathematics (Devlin, 2013). Considering that no one knows exactly
how many prime numbers exist, the proof by Reimann enables calculation of any prime
number. With this it was possible to gauge if a number is prime or not. This made it possible
to decompose large numbers and make use of them in everyday life. Reimann was a student
of Gauss and Dirichlet and his contribution together with that of other come to define the
applications of prime numbers (Cusack, 2016).
Hadamard
The development in the prime numbers theorem took significant steps during the 19th
century. This transformation was initiated by Gauss. Through his study of log tables and
prime numbers he came to identify the prime numbers theorem. Afterwards, the Riemann
work made great contributions to it before it was proven by Jacques Solomon Hadamard
using the ideas presented earlier on by Riemann (Connes, 1998). Hadamard was a French
citizen who lived between 1865-1963. His major contribution in the development of the
prime numbers theorem was its proof that as n approaches infinity π (n) approaches n
lnn
where π (n)represent all positive prime numbers less than n . His mathematical paper titled
“determination of the Number of Primes Less than a given number” was greatly recognized
and even saw him being awarded the Grand Prix des Sciences for the contributions he made
in the development of prime numbers (Encyclopadia Britannica, 2018).
Selberg and Erdos
To prove the PNT (Prime Numbers Theorem) Atle Selberg and P. Erdo managed to
come up with an elementary proof of the PNT in the year 1948. The use of elementary was

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The History of the Prime Numbers Theorem
meant to replace applications of complex variables, Fourier analysis and other similar non-
elementary techniques in the PNT. This showed that; ϑ ( x )
x approaches 1 when xbecomes
infinite
ϑ ( x ) =∑
p ≤ x
log p ≤logx ∑
p ≤ x
1=π ( x ) logx
This is what comes to define the role of Selberg and Erdo in the development of the
PNT (Huard, 1969).
The relationship between Riemann hypothesis and the Clay Institute’s millennium problem.
In 2010 the Clay Institute of mathematics stated seven problems which were
considered a mystery in mathematics. A solution to any of the problems would be rewarded
one million US dollars. The Riemann hypothesis was on that list. the hypothesis states the
complete set of nontrivial zeros of the analytical continuation of the Riemann zero function
possess a real part of ½. Should this hypothesis be approved or be disproved then the number
theory will be massively affected. For the distribution of prime numbers, the impact will be
even more severe. The approval of the hypothesis has remained a problem for a century now
(Devlin, 2013).
Conclusion
The PNT has undergone an upgrade over the years. Despite the tremendous
improvement in the need to seek a solution to the prime numbers distribution the issue is still
not closed. The Reimann’s hypothesis which was greatly relied upon by Hardeman in
approving the PNT is itself yet to be approved. This indicates the magnitude to which
mathematicians still need to go to settle the issue of prime numbers distribution.

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The History of the Prime Numbers Theorem
References
Ingemar,B. and Karol, Z. (2017). Geometry of quantum states : an introduction to quantum
entanglement. United Kingdom: Cambridge University Press.
Connes, A. (1998). Trace formula in noncommutative geometry and the zeros of the Riemann
zeta function. Cornell University Library.
Devlin, K. J. (2013). The Millennium Problems: The Seven Greatest Unsolved Mathematical
Puzzles of Our Time. New York: Basic Books.
Encyclopadia Britannica. (2018, May 03). Jacques-Salomon Hadamard FRENCH
MATHEMATICIAN. Retrieved from Encyclopadia Britannica:
https://www.britannica.com/biography/Jacques-Salomon-Hadamard.
Huard, J. G. (1969). An Elementary Proof of the Prime Number Theorem. The University of
Maine.
Cusack, P. (2016). Riemann Hypothesis Clay Institute Millennium Problem Solution.
Journal of Applied & Computational Mathematics.
Rudman, P. S. ( 2007). How Mathematics Happened: The First 50,000 Years. Amherst, NY:
Prometheus.
Morrissey, B. and Triplett, A. (2009). Euler’s Zeta Function and the Riemann Hypothesis.
Texas: Texas Woman’s University.