Explicit formulas and positivity

The explicit formulas of prime number theory and the reformulations of the Riemann hypothesis they lead to. Weil's explicit formula for ζ with general test functions (including functions with jumps) and for Dirichlet L-functions, Guinand's summation formula, Weil's positivity criterion in the compact-support form (each test function sees finitely many primes), the archimedean distribution and positivity for small support, Li's criterion, the heat flow of ξ and the de Bruijn–Newman constant, Montgomery's pair correlation theorem, and the random-matrix predictions with their evidence. It continues the F1 course's lesson on Weil's proof, which proves smoothed explicit formulas and Weil's criterion for Gaussian-type test functions, and it rests on NT-ZETA and NT-DIRL.

1. The explicit formula with general test functions
2. Explicit formulas for Dirichlet L-functions and primes in progressions
3. Guinand's formula
4. Weil's positivity criterion
5. The archimedean place: W_∞ and positivity for small support
6. Li's criterion
7. Pair correlation of zeros: Montgomery's theorem
8. The random matrix predictions and their evidence

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