Euclid proved that there are infinitely many primes via a clever algebraic argument that most of us are familiar with. Euler proved that $\lim_{s \to 1^+} \sum_p \frac{1}{p^s} = +\infty$ which gives us analytic proof of Euclid's result. Is this is the first known result in analytic number theory?
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asked Jan 20, 2021 at 1:35
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4$\begingroup$ Euler's result is the first one I can think of! (which isn't saying much) $\endgroup$– AsvinCommented Jan 20, 2021 at 1:50
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3$\begingroup$ The first analytic number theorist is clearly Dirichlet. $\endgroup$– reunsCommented Jan 20, 2021 at 1:56
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3$\begingroup$ Just to be precise, Euler did not deal in such a careful way with limits as you express his result. He had no variable $s$ in the limit but used $s = 1$ throughout, so he only "proved" $\sum_p 1/p = \infty$.. He sensed that $\sum_{p \leq x} 1/p$ might grow like $\log\log x$ because he wrote $\sum_p 1/p = \log \log \infty$, but he did not work with truncated expressions either. $\endgroup$– KConradCommented Jan 20, 2021 at 4:00
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4$\begingroup$ To address Dirichlet vs. Euler, a label like "first analytic number theorist" has been applied to both of them. Euler not only proved the result in the post (at a level of rigor accepted in his time), but also found the Euler product for the zeta-function, a form of the functional equation for the zeta-function and $L$-function of the nontrivial character mod $4$ (at integers), and proved results about the partition function using what we'd call $q$-products. Iwaniec and Kowalski, in the introduction of their book, describe Euler's work as "the beginning of analytic number theory" (contd.) $\endgroup$– KConradCommented Jan 20, 2021 at 4:32
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4$\begingroup$ and Dirichlet as "the true father of analytic number theory". $\endgroup$– KConradCommented Jan 20, 2021 at 4:32
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