07-26-2026, 01:47 AM
The prime number theorem: Analytic and elementary proofs
BY Ciaran O’Rourke
Summary
This master's thesis, The Prime Number Theorem: Analytic and Elementary Proofs by Ciarán O'Rourke (Maynooth University, 2013), presents a clear and comprehensive study of one of the central results in number theory: the Prime Number Theorem (PNT), which states that the number of primes less than or equal to (x) is asymptotically (x/\log x). The thesis develops three distinct proofs of the theorem.
It begins with Chebyshev's theorem, establishing essential upper and lower bounds for the prime-counting function that lay the groundwork for later arguments. It then presents a classical analytic proof, relying on complex analysis, Cauchy's residue theorem, and the properties of the Riemann zeta function, following the methods of Hadamard and de la Vallée Poussin.
Next, it explores an elementary proof based entirely on number-theoretic techniques, including Selberg's formulas, Möbius inversion, Dirichlet convolution, and Abel summation, demonstrating that the PNT can be proved without complex analysis. Finally, the thesis concludes with Newman's remarkably short proof, which combines the Laplace transform with analytic continuation to provide a more concise analytic argument.
Throughout, the author emphasizes the intuition and historical development behind each approach, illustrating how different branches of mathematics converge to explain the asymptotic distribution of prime numbers and highlighting the enduring significance of the Prime Number Theorem in modern mathematics.
THESIS [PDF]
BY Ciaran O’Rourke
Summary
This master's thesis, The Prime Number Theorem: Analytic and Elementary Proofs by Ciarán O'Rourke (Maynooth University, 2013), presents a clear and comprehensive study of one of the central results in number theory: the Prime Number Theorem (PNT), which states that the number of primes less than or equal to (x) is asymptotically (x/\log x). The thesis develops three distinct proofs of the theorem.
It begins with Chebyshev's theorem, establishing essential upper and lower bounds for the prime-counting function that lay the groundwork for later arguments. It then presents a classical analytic proof, relying on complex analysis, Cauchy's residue theorem, and the properties of the Riemann zeta function, following the methods of Hadamard and de la Vallée Poussin.
Next, it explores an elementary proof based entirely on number-theoretic techniques, including Selberg's formulas, Möbius inversion, Dirichlet convolution, and Abel summation, demonstrating that the PNT can be proved without complex analysis. Finally, the thesis concludes with Newman's remarkably short proof, which combines the Laplace transform with analytic continuation to provide a more concise analytic argument.
Throughout, the author emphasizes the intuition and historical development behind each approach, illustrating how different branches of mathematics converge to explain the asymptotic distribution of prime numbers and highlighting the enduring significance of the Prime Number Theorem in modern mathematics.
THESIS [PDF]
┌────────────────────────────────┐
│ KONSTANTINOS MICHAILIDIS │
└────────────────────────────────┘
│ KONSTANTINOS MICHAILIDIS │
└────────────────────────────────┘