### 2002 Publications

#### Two contradictory conjectures concerning Carmichael numbers (with Carl Pomerance) Mathematics of Computation, 71 (2002), 873-881.

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 $$x$$.

Article

#### Upper bounds for $$|L(1,\chi)|$$ (with K. Soundararajan) Quarterly Journal of Mathematics (Oxford), 53 (2002), 265-284.

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.

Article

#### Unit fractions and the class number of a cyclotomic field (with Ernie Croot) Journal of the London Mathematical Society, 66 (2002), 579-591.

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 $$\leq x$$.

Article

#### It's as easy as abc (with Tom Tucker) Notices of the American Mathematical Society, 49 (2002), 1224-1231.

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

Article

#### Nombres premiers et chaos quantique Gazette des Mathematiciens , 97 (2002), 29-44.

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

Article
In English: Prime Possibilities and Quantum Chaos, Emissary