The Riemann zeta function

A graduate course on the analytic theory of the Riemann zeta function: Dirichlet series, prime-counting estimates, the Gamma function and the functional equation, entire-function products, growth and zeros, the prime number theorem and explicit formulas, mean values, critical-line zeros and short intervals.

1. Dirichlet series and Euler products
2. Counting primes by elementary means: Chebyshev and Mertens
3. The Gamma function and Stirling's formula
4. Poisson summation, theta, and the functional equation
5. Entire functions of order one and the Hadamard product of ξ
6. Growth in the critical strip: convexity and the Lindelöf hypothesis
7. Exponential sums and a subconvex bound
8. Nonvanishing on the line σ = 1 and the zero-free region
9. The prime number theorem
10. The Riemann–von Mangoldt formula
11. Perron's formula and the explicit formula for ψ(x)
12. The prime number theorem with the classical error term
13. Growth bounds and wider zero-free regions
14. The Riemann hypothesis and its standard equivalents
15. Oscillation of the error term
16. Mean values of Dirichlet polynomials and of zeta on the critical line
17. Zeros on the critical line: Hardy's theorem
18. The Riemann hypothesis implies the Lindelöf hypothesis
19. Zero-density estimates and primes in short intervals

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Original text CC0 1.0. The adaptations identified in lessons 14, 17 and 19 retain CC BY 4.0; the notices are in the downloads.