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