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Nov 26 |
answered | Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$? |
Nov 10 |
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Ideal Class Number
Dedekind's proof is in his book on algebraic numbers. This was translated into English (Richard Dedekind - Theory of Algebraic Integers - translated by John Stillwell, CUP 1996). The proof uses the pigeon hole principle, which is hardly surprising. |
Nov 10 |
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Ideal Class Number
@David: You can use the Poisson summation formula to give a Mellin transform identity for the Dedekind zeta function, from which the nature of the singularity at $s = 1$ is clear. Harold M. Stark wrote an expository paper in which he sketched a proof of the Dirichlet Unit Theorem by this approach, among other things. The application of the Poisson summation formula is called the Hecke theta formula, and is central to Hecke's proof of the functional equation of the Dedekind zeta function. |
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Jul 20 |
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Careers advice for Ph.D.s without current postdocs or university jobs
One way to begin a career outside academia in the USA is to take actuarial exams. The first couple of exams don't require much more than a background in undergraduate mathematics and statistics. Then they get more specialized and harder. It used to be said that with five under your belt, you could get a job in an insurance company, but I am not sure whether that holds any longer. Actuaries historically had very safe and fairly well paid jobs. People need insurance in good times and bad. How one gets into actuarial work in continental Europe or the UK I don't know. |