Mathematics / advanced
Analytic number theory
Use complex analysis, multiplicative structure, and harmonic methods to study primes and arithmetic averages.
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Inside this module
Module 01 advanced
Analytic number theory
Use complex analysis, multiplicative structure, and harmonic methods to study primes and arithmetic averages.
- 01
Arithmetic functions and Dirichlet convolution
Organize divisor identities with convolution, multiplicativity, and Möbius inversion.
24 min - 02
Partial summation and Abelian estimates
Transfer cancellation in partial sums into estimates for weighted arithmetic sums.
25 min - 03
Dirichlet series and Euler products
Translate divisor convolution into analytic products inside a justified half-plane.
26 min - 04
Riemann zeta function and analytic continuation
Continue zeta beyond its defining series while keeping poles, gamma factors, and special values distinct.
28 min - 05
Zeros and explicit formulas
Relate prime-power counts to zeta zeros with precise endpoint and truncation conventions.
31 min - 06
Dirichlet characters and L-functions
Encode multiplicative residue classes while distinguishing a character’s modulus from its conductor.
29 min - 07
Primes in arithmetic progressions
Use character orthogonality to separate eligible residue classes and scope their prime-counting asymptotics.
28 min - 08
Perron formula and contour shifting
Invert a Dirichlet series with endpoint weights and keep every contour error under control.
32 min - 09
Sieve methods and Brun–Titchmarsh
Count survivors of divisibility filters and apply a uniform upper bound with its correct range.
30 min - 10
Exponential sums and Weyl differencing
Reduce polynomial oscillations by differencing and state cancellation bounds with a leading-coefficient approximation.
31 min