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Dirichlet’s theorem on primes in arithmetic progressions
Dirichlet's theorem on primes in arithmetic progressions
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2021-09-14
Chebotarev density theorem is a generalization of Unsorted/Dirichlet’s theorem on primes in arithmetic progressions.
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Riemann Hypothesis
Compare to Dirichlet’s theorem on primes in arithmetic progressions. Fix an arithmetic progressions \(A = A(a, d) = \left\{{a, a+d, a+2d,\cdots}\right\}\) for \(a,d\) coprime and define \(\pi_A(x) \coloneqq\left\{{p\in A {~\mathrel{\Big\vert}~}p\leq x, p\text{ prime }}\right\}\); then GRH implies an asymptotic estimate in \(x\): \begin{align*} \pi_A(x) \sim \phi(d)^{-1}\int_2^x {1\over \log t}\,dt+ { \mathsf{O}} \qty{x^{{1\over 2} + {\varepsilon}}} \end{align*}
- L function