C3.8 Analytic Number Theory (2026-27)
Main content blocks
- Lecturer: Profile: Tom Sanders
The Riemann \(\zeta\)-function for \(ℜ(s) > 1\). Euler's proof of the infinitude of primes. \(\zeta\) and the von Mangoldt function.
Schwartz functions on \(\mathbf{R}\), \(\mathbf{Z}\), \(\mathbf{R}/\mathbf{Z}\) and their Fourier transforms. *Inversion formulas and uniqueness*. The Poisson summation formula. The meromorphic continuation and functional equation of the \(\zeta\)-function. Poles and zeros of \(\zeta\) and statement of the Riemann hypothesis. Basic estimates for \(\zeta\).
The classical zero-free region. Proof of the prime number theorem. Implications of the Riemann hypothesis for the distribution of primes.
Section outline
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This sheet gives some practice/revision with basic analytic estimates and the O() notation. It will not be marked and there will be no class on it but I am happy to go over questions after lectures.
The sheet is available as C3.8-sheet0.pdf. Full solutions will be made available by 30th October 2026.
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This is sheet 1 for the first class. It covers material up to the end of Section 2 of the notes.
The sheet is available as C3.8-sheet1.pdf (without solutions), or as C3.8-sheet1-ACsolutions.pdf which has solutions to Sections A and C. Full solutions will be made available after the classes.
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Sheet 2 covers material up to the end of Section 3 of the notes and reinforces some of the earlier material. As usual, it is initially available in two forms: with no solutions, and with solutions to Sections A and C.
The sheet is available as C3.8-sheet2.pdf (without solutions), or as C3.8-sheet2-ACsolutions.pdf which has solutions to Sections A and C. Full solutions will be made available after the classes.
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Sheet 3 covers material up to the end of Section 7 of the notes.
The sheet is available as C3.8-sheet3.pdf (without solutions), or as C3.8-sheet3-ACsolutions.pdf which has solutions to Sections A and C. Full solutions will be made available after the classes.
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Sheet 4 covers material from the whole course but with an emphasis on the later sections.
The sheet is available as C3.8-sheet4.pdf (without solutions), or as C3.8-sheet4-ACsolutions.pdf which has solutions to Sections A and C. Full solutions will be made available after the classes.
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Registration start: Monday, 5 October 2026, 12:00 PMRegistration end: Friday, 6 November 2026, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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