mixedmath

I wrote a series of notes on some aspects of the theoretical behavior and zeros of Dirichlet series in the extended Selberg Class. There are a few different ways of extending the Selberg Class, but here I mean Dirichlet series \begin{equation*} L(s) = \sum_{n \geq 1} \frac{a(n)}{n^s} \end{equation*} with a functional equation of the shape $s \mapsto 1 - s$, satisfying

1. A Ramanujan–Petersson bound on average, meaning that $\sum_{n \leq N} \lvert a(n) \rvert^2 \ll N^{1 + \epsilon}$ for any $\epsilon > 0$.
2. $L(s)$ has analytic continuation to $\mathbb{C}$ to an entire function of finite order.
3. $L(s)$ satisfies a functional equation of the typical self-dual shape.

There is no assumption of an Euler product.

Although I write these notes for general Dirichlet series in the extended Selberg class, I was really thinking about Dirichlet series associated to half-integral weight modular forms.

Links and Summaries of each Note

1. The first note sets the stage, defines the relevant series, and establishes fundamental results to be used later. Jensen's inequality and Jensen's theorem are given, as are generic convexity bounds for these Dirichlet series.

   The first note also contains a proof of a new fact to me:^{1 1}but only new to me. I based my presentation of this fact on notes from Hardy from a century ago. if a Dirichlet series has a zero in its domain of absolute convergence, then it has infinitely many, and these zeros are almost periodic.

2. The second note is relatively short and shows that these Dirichlet series are in fact entire of order $1$. Then it establishes weak zero-counting results based only on this order of growth.

   These are foundational ideas, and in essence are no different than analysis for $\zeta(s)$ or typical $L$-functions in the Selberg class.

3. The third note describes a theorem of Potter from 1940, proving Lindelöf-on-average (in the $t$-aspect) on certain vertical lines, depending on the degree. This suggests that for Dirichlet series associated to half-integral weight modular forms, the Lindelöf Hypothesis might be true even though the Riemann Hypothesis is false.

4. The fourth note proves that one hundred percent of zeros of Dirichlet series associated to half-integral weight modular forms lie within $\epsilon$ of the critical line, for any $\epsilon > 0$.

   For these proofs, the standard proofs that I've seen elsewhere for Selberg $L$-functions don't quite apply. Fundamentally, this is because zeros are counted using some form of the argument principle, and for Selberg $L$-functions with Euler product, the logarithmic derivative of the $L$-functions is both easier to understand (because of the Euler product) and gives convenient access to sum up changes in the argument. But in practice the actual methods I use were mostly lightly modified from the vast, extensive literature on various techniques applied to study $\zeta(s)$ (before better techniques arose using the Euler product in more sophisticated ways).

   I note that I do not know how to prove a lower bound for the percentage of zeros lying directly on the critical line for series without an Euler product. It is possible to apply a generalized form of the classical argument of Hardy to show that there are infinitely many zeros on the line, but I haven't managed to modify any of the various results for lower bounds.

I will also note that with Thomas Hulse, Mehmet Kiral, and Li-Mei Lim, I've computed many examples of zeros of half-integral weight forms and I conjecture that 100 percent of their zeros lie directly on the critical line. (But there are many, many zeros not on the critical line). I described some of those computations in this talk.

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