Erdos conjectured that there are $\gg_\epsilon x^{1-\epsilon}$ carmichael numbers up to $x$, whereas Shanks, based on calculations, was skeptical as to whether one might even find an $x$ up to which there are $x^{1/2}$ Carmichael numbers. In this article we show that they were both correct, in that by understanding the structure of Carmichael numbers one sees why there will only be a lot of Carmichael numbers for very large $x$.

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We give best possible upper bounds on $|L(1,\chi)|$ for characters $\chi$ of given order $k$, given only Burgess's Theorem and the knowledge one is summing a multiplicative function whose $k$th power is 1.

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Although Kummer's conjecture that the first factor of the class number $h_1(p)$ of the $p$th cyclotomic field is $\sim G(p):= 2p(p/4\pi^2)^{(p-1)/4}$ is wrong (assuming two widely believed conjectures), it has been shown that it is almost always true. Here we show that for any rational $\beta$, we have $h_1(p)\sim e^\beta G(p)$ for $\sim c_\beta x/(\log x)^{A(\beta)}$ primes $\leq x$.

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A survey of the arithmetic consequences of the $abc$-conjecture.

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A survey of the connections between zeros of zeta functions and quantum chaos, written for a general scientific audience.

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In English: Prime Possibilities and Quantum Chaos, Emissary