Elkies N.D. Introduction to analytic number theory

Harvard: N.D. Elkies, 2015. — 163 p.Introduction: What is analytic number theory?
Distribution of primes before complex analysis: classical techniques (Euclid, Euler); primes in arithmetic progressions via Dirichlet characters and
L-series; Cebyˇsev’s estimates on ˇ π(x).
Distribution of primes using complex analysis: ζ(s) and L(s, χ) as functions of a complex variable, and the proof of the Prime Number Theorem and
its extension to Dirichlet; blurb for Cebotarev density; functional equations; ˇthe Riemann hypothesis, extensions, generalizations and consequences.
Selberg’s quadratic sieve and applications.
Analytic estimates on exponential sums (van der Corput etc.); prototypical applications: Weyl equidistribution, upper bounds on |ζ(s)| and |L(s, χ)| on vertical lines, lattice point sums.
Lower bounds on discriminants, conductors, etc. from functional equations; geometric analogue: how many points can a curve of genus g → ∞
have over a given finite field?
Analytic bounds on coefficients of modular forms and functions; applications to counting representations of integers as sums of squares, etc.