Analytic Number Theory
Complex analysis applied to prime distribution and L-functions.
Analytic Number Theory. Complex analysis applied to prime distribution and L-functions.
Foundations and canonical references
The standard treatments of analytic number theory approach the subject from complementary angles. Davenport, Multiplicative Number Theory (2000) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Iwaniec, Analytic Number Theory (2004) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.
Open methodological questions for analytic number theory include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.
Prerequisites
Sources
- textbook · primary · 2000Multiplicative Number Theorydavenport-2000
- textbook · primary · 2004Analytic Number Theoryiwaniec-2004, kowalski-2004
In context
Where this topic sits in the prerequisite graph. Click any node to jump.
Explore
- 01
Riemann Zeta Function
Analytic continuation, functional equation, and zero-free regions.
- 02
Prime Distribution
Prime number theorem, sieve methods, and gaps between primes.
- 03
Sieve Methods
Brun, Selberg, and large sieves; bounded gaps.
- 04
L-Functions
Dirichlet L-functions, automorphic L-functions, and the Selberg class.
- 05
Exponential Sums
Weyl, Vinogradov, and bounds via Bombieri–Iwaniec.
- 06
Multiplicative Functions and the Anatomy of Integers
Erdős–Kac, Granville–Soundararajan pretentiousness.
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