Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \qquad \text{ for } \sigma > 1 \text{ and } s= \sigma + it$$ ...
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How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
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I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are excluded: Books by mathematical ...
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What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
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How does $\sum_{p<x} p^{-s}$ grow asymptotically for $\text{Re}(s) < 1$?

Note the $p < x$ in the sum stands for all primes less than $x$. I know that for $s=1$, $$\sum_{p<x} \frac{1}{p} \sim \ln \ln x ,$$ and for $\mathrm{Re}(s) > 1$, the partial sums ...