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On the Prime Number Theorem

Schoennett, Zofia LU (2025) In Bachelor's Theses in Mathematical Sciences MATK11 20251
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
The main subject of this thesis is the proof of the Prime Number Theorem. It revolves around the Riemann zeta function, whose continuation to a meromorphic function in the complex plane is discussed. We employ auxiliary functions, including the Chebyshev function, to simplify the proof of the theorem to showing an equivalent statement.
Popular Abstract
The Prime Number Theorem describes the asymptotic distribution of prime numbers. It was discovered through careful analysis of tables of factors, from which one can tabulate the density of primes in various sets. The theorem is crucial to understanding their distribution. In particular, it allows us to easily estimate the amount of prime numbers in a given interval with large enough endpoints.

There are many uses for primes. They constitute a common subject of study in number theory. Furthermore, relatively large prime numbers, that is to say with several hundred decimal digits, are widely employed in cryptography. The RSA public-key cryptosystem relies on encryption and decryption keys created with primes to secure data transmission.... (More)
The Prime Number Theorem describes the asymptotic distribution of prime numbers. It was discovered through careful analysis of tables of factors, from which one can tabulate the density of primes in various sets. The theorem is crucial to understanding their distribution. In particular, it allows us to easily estimate the amount of prime numbers in a given interval with large enough endpoints.

There are many uses for primes. They constitute a common subject of study in number theory. Furthermore, relatively large prime numbers, that is to say with several hundred decimal digits, are widely employed in cryptography. The RSA public-key cryptosystem relies on encryption and decryption keys created with primes to secure data transmission. Since only the intended receiver can understand the message, the RSA system is widely employed for protecting personal data.

It is epistemologically curious to consider how the Prime Number Theorem can be proven. Not only is the distribution of primes erratic with no closed formula for it, there exists an infinite amount of them. It thus seems exceedingly challenging to find a proof, and indeed it took mathematicians almost a century. The connection to the zeta function, given by Euler's product formula, is a vital element of the proof that enabled its formulation. Not only does it facilitate showing the theorem is true by providing a simpler, equivalent statement, it showcases a deep connection between analytic and algebraic properties of prime numbers. (Less)
Please use this url to cite or link to this publication:
author
Schoennett, Zofia LU
supervisor
organization
course
MATK11 20251
year
type
M2 - Bachelor Degree
subject
keywords
prime numbers, prime, Riemann, Zeta function, Chebyshev
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4175-2025
ISSN
1654-6229
other publication id
2025:K9
language
English
id
9208819
date added to LUP
2025-10-28 14:41:29
date last changed
2025-10-28 14:41:29
@misc{9208819,
  abstract     = {{The main subject of this thesis is the proof of the Prime Number Theorem. It revolves around the Riemann zeta function, whose continuation to a meromorphic function in the complex plane is discussed. We employ auxiliary functions, including the Chebyshev function, to simplify the proof of the theorem to showing an equivalent statement.}},
  author       = {{Schoennett, Zofia}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{On the Prime Number Theorem}},
  year         = {{2025}},
}