Today I will be giving a talk at the Maine-Quebec Number Theory conference. Each year that I attend this conference, I marvel at how friendly and inviting an environment it is --- I highly recommend checking the conference out (and perhaps modelling other conferences after it).
The theme of my talk is about spectral poles and their contribution towards asymptotics (especially of error terms). I describe a few problems in which spectral poles appear in asymptotics. Unlike the nice simple cases where a single pole (or possibly a few poles) appear, in these cases infinite lines of poles appear.
For a bit over a year, I have encountered these and not known what to make of them. Could you have the pathological case that residues of these poles generically cancel? Could they combine to be larger than expected? How do we make sense of them?
The resolution came only very recently.1 1In fact, I had originally intended to give this talk as a plea for advice and suggestions in considering these questions. But then I happened to read work from Ingham in the 1920s, carrying an idea that was new to me. The talk concludes with this idea. It's not groundbreaking --- but it's new to me.
I will later write a dedicated note to this new idea (involving Dirichlet integrals and Landau's theorem in this context), but for now --- here are the slides for my talk.
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Comments (2)
2019-10-07 Peter Humphries
Omega results of this form using this method are presented quite nicely in Chapter 15 of Montgomery and Vaughan; they give applications towards sign changes of the Chebyshev psi function and of partial sums of the Mobius function.
2019-10-17 davidlowrydud
Thanks Peter! You're right --- and it's a nice reference. I had no idea.