Introduction to number theory lecture 48 Proof of the prime number theorem

Item: Introduction to number theory lecture 48 Proof of the prime number theorem
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMathematics PBHNumber theory

🎙 Richard E Borcherds 👥 82K 📅 April 14, 2022 ⏱ 17 min 👁 12K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

prime number theoremRiemann zeta functionNewman's Tauberian theoremChebyshev functionanalytic number theory

Summary

This lecture, part of Berkeley’s Math 115 course, provides a sketch of the proof of the prime number theorem (PNT), which states that the number of primes less than x is asymptotically x/log x. The proof is broken into five steps. First, it establishes that the Riemann zeta function has no zeros on the line Re(s)=1, using a clever product of zeta functions with exponents from binomial coefficients, leading to a contradiction if a zero existed. Second, it invokes Newman’s Tauberian theorem (without proof) to show that a certain integral involving the Chebyshev function psi(x) converges. Third, it demonstrates that the integral from 1 to infinity of (psi(x)-x)/x^2 dx converges, using the previous step. Fourth, it shows that psi(x) is asymptotic to x, using a lemma about increasing functions. Finally, it deduces the PNT by showing that psi(x) is approximately log x times pi(x), using the fact that log x is nearly constant for large x and that prime powers are negligible. The lecture concludes with a reference to Zagier’s short proof of the PNT.

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Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a high-value, rigorous outline of the proof of the prime number theorem, a central result in analytic number theory. The argumentation is clear and logically structured, with each step building on the previous. The use of the zeta function’s Euler product and the clever product to prove the absence of zeros is well-explained. The lecturer also highlights the key ideas behind Newman’s Tauberian theorem and the asymptotic analysis of psi(x). The presentation is suitable for an advanced undergraduate audience with a background in complex analysis.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, with a clear and correct mathematical argument. The lecturer references standard sources: the textbook by Niven, Zuckerman, and Montgomery, and Zagier’s paper on a short proof of the PNT. The title accurately reflects the content. The lecture is part of a well-structured course, and the lecturer is a respected mathematician, enhancing credibility.

160 words

Title / Content Match

The title accurately describes the content: a lecture on the proof of the prime number theorem.

Quality & Reliability

9/10

The lecture is given by a renowned mathematician (Richard Borcherds) and is part of a university course. It presents a rigorous sketch of the proof of the prime number theorem, with clear logical steps and references to standard literature. The content is mathematically sound and well-structured.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Zagier's paper on a short proof of the prime number theorem — Referenced as a source for a short proof of the prime number theorem, including Newman's Tauberian theorem.
Berkeley Math 115 course playlist — Playlist containing all lectures of the course.

Concurring Sources

Zagier's paper on a short proof of the prime number theorem — Provides a complete proof of the PNT, consistent with the sketch presented.

Contribution & Novelties

This lecture provides a concise and clear sketch of the proof of the prime number theorem, emphasizing the key ideas without getting bogged down in technical details. It is particularly valuable for students learning analytic number theory, as it connects the main steps and highlights the role of the zeta function and Tauberian theorems. The lecturer’s pedagogical approach makes the proof accessible while maintaining mathematical rigor.

Pour aller plus loin :

Prime number theorem - Wikipedia — Provides a comprehensive overview and historical context.
Riemann zeta function - Wikipedia — Essential background on the zeta function and its properties.
Newman’s Tauberian theorem - Wikipedia — Explains the theorem used in the proof.
Chebyshev function - Wikipedia — Details on the function psi(x) and its role.

125 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both information-dense and technically rigorous. The balance between quantity and quality of information is excellent, and the technical level is appropriate for the intended audience.

Reliability 9/10