Arithmetic Functions

Riemann Zeta function
Approximate Functional Equations
Poisson Summmation, Voronoi Summation Formulae
Mean Square Estimates
Fourth Power Moment
Twisted Moments
Higher Moments
Mean values over short intervals
Omega Results
Zero free regions
Zeta Sums, Vinogradov-Koborov
Zero Density estimates
Zeroes on the critical line

The Sieve of Eratosthenes-Legendre
The Brun Sieve
Selberg Sieve
Bombieri's Sieve
Rosser-Iwaniec Sieve
The Semi-Linear Sieve
The Linear Sieve
Chen Sieve
The Large Sieve
The Vaughan Identity, Linnik, Heath-Brown's Identities
Exponential Sums over primes
Primes in Short Intervals
Primes in almost all intervals
Asymptotic Sieve
Primes in a^2+b^4
Primes of form x^3+2y^3
Level of Distribution for Arithmetic sequences
Squarefree Sieve
Almost prime Sieve

Circle Method
Arithmetic Combinatorics
Cauchy-Davenport theorem
the Erdos Heilbronn conjecture, Schnirerlmann density
Mann's theorem
sumset inequalities
Balog-Szemeredi Gowers theorem
Sum-Product theorem
Sum-Product over the Reals, Szemeredi-Trotter, Crossing Numbers
Sidon sets
Perfect Difference sets, Sum-Free sets, Schur's theorem
van der Waerden's theorem
Roth's theorem
Croot-Lev-Pach-Ellenberg-Gijswijit theorem
Szemeredi's regularity lemma
Szemeredi's theorem on arithmetic progressions
Bohr sets, APs and subspaces in sumsets
Croot-Sisask sampling method
Freiman homomorphisms, Chang's lemma
Additive structure in Fourier coefficients
Gauss sums for small multiplicative subgroups
Waring's problem in integers
Hindman's theorem
Gowers's theorem on product-free sets