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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. |