Spaces of Dirichlet Series

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 Note: this was originally a thread on my twitter

Let's talk about a weird connection between functional analysis and number theory: Dirichlet Series!
 (Full Disclosure: I'm not a number theorist, not even an analytic number theorist, but spaces of  functions are my jam.)

The most famous Dirichlet series is the Riemann zeta function, but we can generalize by changing the  coefficients:

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To help get our hands on these, let's consider a very simple case where we don't have to be distressed about convergence: \(2^{-s}\). Looking at how this maps the plane helps me think about things like periodicity (which gets complicated when looking at a full DS).

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Complex TAYLOR series converge in disks in \(\mathbb{C}\). Dirichlet series converge in right half planes of \(\mathbb{C}\). Show this by showing that if a DS converges at a point \(s_0\), it converges in a sector, then union those to get a half plane. 

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So instead of a radius of convergence like we get for Taylor series, we get an *abcissa of convergence*

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An example is the Riemann zeta function vs the alternating Riemann zeta function:

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This means it's also useful to have an abcissa of absolute convergence, and in fact, there are abcissae corresponding to various types of convergence.

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Lots of folks spent lots of time examining the relationships between these abcissae. In particular, Bohr showed that the abcissa of bounded convergence is the same as the abcissa of uniform convergence.

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(There are important technical notes about boundedness/uniform convergence because the abcissae are defined as infimums, but I'm going to brush that under the rug.)

Bohr also showed that there is a nice relationship between the abcissa of bounded convergence and the abcissa of absolute  convergence, and Bohnenblust and Hille showed that this inequality is sharp.

    

Ok, neat, but we're supposed to be talking about SPACES of functions; this is functional analysis after all. So let's go back and remember Hilbert spaces, and in particular, the Hardy space of the disk.

The Hardy space norm and inner product defined in terms of the coefficients


We're requiring square summable coefficients, so why don't we do the same thing to build ourselves a space of Dirichlet  series? And since it's kind of like the Hardy space, we can give it similar notation.
(Fun fact: people really do call this "curly-H 2") 

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This space is very nifty! In particular, one can use Cauchy-Schwarz to show that the abcissa of absolute convergence of any function in curly-H^2 is less than or equal to one half. So we have all the nice convergences in that half plane!  

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(It can also be shown that you can't get bigger than that half plane and that there is a particular choice of coordinates that gives abcissa=1/2.)

This also shows that the point evaluations are bounded for any point in our half plane, which means that we have a reproducing kernel!

writing out the definition of the reproducing kernel with the inner product  

And here's something exciting! The reproducing kernel is formed from the Riemann zeta function!

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Here's another neat thing: remember how we built this space just by saying "ok, make the coeffecients square summable"? Well, with the Hardy space of the disk, it turned out that the inner product could be defined in terms of an integral as well.

The Hardy space inner product defined as an integral mean  

So let's open up another infinite dimensional can of worms and talk about the Bohr correspondence between Dirichlet series on the half plane and Taylor series on the infinite dimensional polydisk.

The trick of the Bohr correspondence is using prime factorization and thinking of each prime factor as giving you a complex variable:

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Then you think of setting up the inner product/norm with integration on the polytorus! (This gets into some neat stuff about the Kronecker flow and ergodicity.) Bayart did an extension of this idea to talk about non-Hilbert spaces of Dirichlet series.

Here are some useful references:

The multiplier problem for the case of Dirichlet series from a lovely paper by Hedenmalm, Lindqvist, and Seip (which reinvigorated the  study of Dirichlet series in a modern context) \https://people.kth.se/~haakanh/publications/hedenmalm-lindqvist-seip-1.pdf
An open questions paper from 2016 https://arxiv.org/pdf/1601.01616.pdf
Weighted Hardy spaces of Dirichlet series  https://www.ams.org/journals/tran/2004-356-03/S0002-9947-03-03452-4/S0002-9947-03-03452-4.pdf 

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