= Analytic number theory
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<Analytic number theory> uses <real analysis> and <complex analysis> to study <integers>, <primes>, and <arithmetic functions>. It encodes discrete counting problems in sums, integrals, and <Dirichlet series>, then extracts information from convergence, oscillation, and singularities.
For example, the <Euler product> $\zeta(s)=\prod_p(1-p^{-s})^{-1}$ links the <Riemann zeta function> with all <primes>. Its logarithmic derivative has coefficients given by the <Von Mangoldt function>, turning prime counting into an analytic problem. The <Prime number theorem> is a central outcome: $\pi(x)\sim x/\log x$. The method is useful because analytic transformations can control an enormous discrete sum without enumerating every term.
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