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seen Feb 24 '11 at 15:49

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awarded  nt.number-theory
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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.
Oct
21
awarded  Yearling
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.
Jun
7
asked number fields generated by units of number fields
Jun
5
comment Why does undergraduate discrete math require calculus?
Our standard calculus course is oriented towards rigorous proofs, so it is unrealistic to expect all students to be well enough prepared for it in the first semester. We start from the completeness property of $\mathbb{R}$ and prove the major theorems, such as the Extreme Value Theorem, the Intermediate Value Theorem, the Riemann integrability of monotone functions, the Riemann integrability of continuous functions, and the Fundamental Theorem of Calculus. By the way, the place where I work is not that big, and we cannot have many tailor-made calculus courses, like in the USA.
Jun
5
comment Why does undergraduate discrete math require calculus?
Well, if I say where I work, I might as well not have a pseudonym at all. The introductory discrete math course is offered because of demand from the computer science department. The CS students are required to take calculus, but not in their first year. The CS students are also required to take an introductory probability and statistics course, and calculus is a prerequisite for that. Basically, we have to adapt our earliest courses to the needs of other departments. We also have a less demanding, cookbook calculus course, for students outside the hard sciences, like biology.