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This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).

However, it turns out that if one of the sets, say A, is sufficiently “uniform” or “pseudorandom”, then one can always solve this Goldbach-type problem, regardless of what the other two sets are. This type of fact is often established by Fourier-analytic means (or by closely related techniques, such as spectral theory), but let me give a heuristic combinatorial argument to indicate why one would expect this type of phenomenon to occur.

quares Primes Lagrange’s four square theorem: For every positive integer N, there exists a pattern in of the form . Vinogradov’s theorem: For every sufficiently large integer N, there exists a pattern in of the form . Fermat’s two square theorem: For every prime number , there exists a pattern in of the form . Even Goldbach conjecture: For every even number , there exists a pattern in of the form . Fermat’s four square theorem: There does not exist any pattern in of the form with . Green-Tao theorem: For any , there exist infinitely many patterns in of the form with . Pell’s equation: There are infinitely many patterns in of the form . Sophie Germain conjecture: There are infinitely many patterns in of the form .

We extend two well-known results in additive number theory, S\'ark\"ozy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restriction-type Fourier analytic argument of Green and Green-Tao.