Sunday, 3 May 2009

nt.number theory - Power series f such that f(zetapn1)=1/pn for almost all ngeq0

As Brian pointed out, the principal parts problem always has a solution on the open unit disk, and Lazard's 1962 article "Les zéros des fonctions analytiques d’une variable sur un corps valué complet" gives a nice proof which is also rather explicit.



Your problem has additional symmetries so one can be a bit more explicit, as follows.



Let H be the set of power series holomorphic on the open unit disk, let varphi:HtoH be the map defined by varphi(f)(X)=f((1+X)p1) and let d be defined by d(f)(X)=(1+X)df/dX. Note that dvarphi=pvarphid. You can check that if you have a function f such that varphi(f)pf=(1+X/2)log(1+X)/X, then f(0)neq0 and f(zetapn1)=pnf(0) so this answers your problem.



Therefore you need to be able to solve an equation of the form varphi(f)pf=g. By taking d of both sides this gives varphi(df)df=dg/p. Now you can solve an equation of the form varphi(a)a=b if binXcdotH by writing a=sumngeq0varphin(b), which should converge in H for its Fréchet topology. We have d((1+X/2)log(1+X)/X)inXcdotH (this is what the (1+X/2) was put in for), and once you know df, you get f by integrating and adjusting the constant.

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