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answered Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$?
Nov
10
comment 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
comment 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
comment 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.
Jul
4
comment Is Li(x) the best possible approximation to the prime-counting function?
Actually, no version of the Prime Number Theorem is needed to establish that no rational function of x and log(x) can be a better approximation to $\pi(x)$ than Li(x). The last result of Chebyshev's first (and less well known) paper on prime number number theory is that no algebraic function of x and log(x) can be a better approximation than Li(x). The result is independent of the PNT and is established by means of the Euler product formula and the nature of the singularity of $\zeta(\sigma)$ at $\sigma = 1$ as a function of a real variable.