Introduction to Series
In this series of notes, I examine zeros of a very general class of Dirichlet series. Ultimately I have a class of Dirichlet series in mind, which are those Dirichlet series coming from half-integral weight modular cuspforms. I've described these series in various talks before (see for example Notes from a talk on half-integral weight Dirichlet series), but I don't focus on them yet.
I've learned that there is a variety of very classical techniques that can be modified to work with more general Dirichlet series than typically studied. I describe what I've learned in these notes.
In this preliminary note, I largely set the stage and introduce some of the definitions and foundational results that I'll use later.
$\renewcommand{\Re}{\operatorname{Re}}$
Extended Selberg Class
We focus on an extended Selberg Class of Dirichlet series, which I'll refer to as the $\widetilde{S}$ class. These are Dirichlet series \begin{equation}L(s) = \sum_{n \geq 1} \frac{a(n)}{n^s} \end{equation} with a functional equation, normalized so that the functional equation is of the shape $s \mapsto 1 - s$. We require that this Dirichlet series has the following properties:
- The coefficients $a(n)$ satisfy a Ramanujan–Petersson bound on average, meaning that $\sum_{n \leq N} \lvert a(n) \rvert^2 \ll N^{1 + \epsilon}$ for any $\epsilon > 0$.
- $L(s)$ has an analytic continuation to $\mathbb{C}$ to an entire function of finite order.
- $L(s)$ satisfies a functional equation of the form \begin{equation} \Lambda(s) := L(s) Q^s \prod_{\nu = 1}^N \Gamma( \alpha_\nu s + \beta_\nu)= \omega \overline{\Lambda(1 - \overline{s})}, \end{equation}where $Q$ and $\alpha_\nu$ are positive real numbers, $\beta_\nu$ are complex numbers with nonnegative real part, and $\lvert \omega \rvert = 1$.
These properties are similar to (but weaker than) the typical Selberg class of entire Dirichlet series. Again, by convention I will exclude the trivial series $L \equiv 0$ from $\widetilde{S}$.
CN Class of Dirichlet Series
I will also sometimes refer to the generic class of Dirichlet series appearing in the seminal works of Chandrasekharan and Narasimhan (1962 and 1964). This is the same class of Dirichlet series that Taniguchi, Thorne, and I study in our paper Uniform bounds for lattice point counting and partial sumsof zeta functions. I'll refer to these Dirichlet series as the CN class.
We start with a general pair of Dirichlet series $\phi(s)$ and $\psi(s)$, related by a functional equation of the form \begin{equation} \Delta(s) \phi(s) = \Delta(\delta - s) \psi(\delta - s). \end{equation}Here, $\delta > 0$ (typically we normalize so that $\delta = 1$) and \begin{equation} \Delta(s) = \prod_{\nu=1}^N \Gamma(\alpha_\nu s + \beta_\nu), \end{equation}for some $N \geq 1$, $\alpha_\nu > 0$, and $\beta_\nu \in \mathbb{C}$. We take our two Dirichlet series to be of the form \begin{align} \phi(s) &= \sum_{n \geq 1} \frac{a(n)}{\lambda_n^s}, \\ \psi(s) &= \sum_{n \geq 1} \frac{b(n)}{\mu_n^s}, \end{align}for strictly increasing positive sequences ${ \lambda_n }$ and ${ \mu_n }$. We may assume without loss of generality that $\lambda_1 = 1$, perhaps by multiplying by $\lambda_1^s$ (and suitably adjusting $\mu_n$ in the functional equation).
A complete set of data for these Dirichlet series is $(\phi, \psi, \Delta, \delta)$, though I will typically say simply that $(\phi, \psi)$ is in the CN class.
We also assume that these series are of practical interest. They are each to converge absolutely in some half-plane and to share the same analytic continuation in the sense that there is some meromorphic function $F(s)$ such that $F(s) = \Delta(s) \phi(s)$ for $\Re s \gg 1$, $F(s) = \Delta(\delta - s) \psi(\delta - s)$ for $\Re s \ll -1$, and such that $F(s)$ is holomorphic and offinite order outside of a sufficiently large disk centered at the origin. This is a long-winded way of stating that the two Dirichlet series meromorphically continue into each other in the expected way, but also includes an assumption that our completed Dirichlet series has only finitely many poles. I also assume that they aren't constant or identically $0$.
Although this looks a bit odd, note that this includes all ''ordinary'' Dirichlet series that we come across (with the exception of objects like Artin $L$-functions, unless we assume the Artin conjecture). Specifically, we allow the ''conductor'' in a typical automorphic functional equation to be absorbed into the generalized denominators $\mu_n$ and $\lambda_n$.
As with $\widetilde{S}$, there are no assumptions about the existence of an Euler product or bounds on the shape of the coefficients, aside from bouunds implied by the existence of half-plances of absolute convergence and meromorphic continuation.
An example showing different behavior
The extended Selberg class $\widetilde{S}$ of Dirichlet series contain all $L$-functions that are expected to satisfy a Riemann Hypothesis. But they also contain other Dirichlet series. A primary difference with the typical Selberg class is that neither CN or $\widetilde{S}$ Dirichlet series are required to have Euler products. Instead, they're defined by analytic data instead of arithmetic data.
Taking linear combinations of Selberg class $L$-functions with compatible functional equations ruins Euler products, but preserves most other nice properties. Davenport and Heilbronn studied one such linear combination. Let $\chi = \chi_5(2, \cdot)$ be the unique Dirichlet character mod $5$ with $\chi(2) = i$. For a particular constant $\theta$, Davenport and Heilbronn introduce the Dirichlet series \begin{equation}L_5(s) = \frac{1 - i\theta}{2} L(s, \chi) + \frac{1 + i \theta}{2} L(s, \overline{\chi}) \end{equation}and show that it satisfies the functional equation \begin{equation} \Lambda_5(s) := L_5(s) \Gamma(\tfrac{s + 1}{2}) (5 / \pi)^{s/2} = \Lambda_5(1 - s). \end{equation}But they also show that this Dirichlet series has zeros in the half-plane of absolute convergence, $\Re s > 1$. They also show that there are infinitely many zeros on the critical line.
The point of this example is that these classes include more Dirichlet series. It is not possible to prove the Riemann Hypothesis by studying only these classes, because generically the Riemann Hypothesis seems to be false for these classes of Dirichlet series. Nonetheless, we'll see that often the Riemann Hypothesis is almost true (for some definition of almost).
Almost Periodicity
Having an expression as a Dirichlet series implies structure. One source of this structure comes from a family of results sometimes collectively called almost periodicity.
An analytic function $f(s)$, defined at least on a vertical strip $A < \Re s < B$, is called almost periodic if, for $\epsilon > 0$, and any $\alpha, \beta$ with $A < \alpha < \beta < B$, there exists a length $\ell$ (depending on essentially every other variable, $f$, $\alpha$, $\beta$, and $\epsilon$), such that every interval $(t_1, t_2 = t_1 + \ell)$ of length $\ell$ contains an almost period of $f$, i.e. there exists a number $\tau \in (t_1, t_2)$ such that \begin{equation} \lvert f(\sigma + it + i\tau) - f(\sigma + it) \rvert < \epsilon \quad \text{for any} \quad \alpha \leq \sigma \leq \beta, t \in \mathbb{R}. \end{equation}
This isn't quite a period, since the ''equality'' only holds up to $\epsilon$; and also because this doesn't hold for multiples of the same ''period length'', but instead the ''equality'' is only guaranteed once per ''period length''. So it's not quite equal, and it's not quite regular.
Bohr showed that every Dirichlet series is almost periodic in its half-plane of absolute convergence. (I haven't read his paper. I'm more familiar with this topic from Besicovitch's monograph on Almost Periodic Functions, published by Dover). The essence of the proof is that Dirichlet series are essentially exponential series with well-behaved coefficients.
Bohr studied almost periodicity in relation to the Riemann Hypothesis. It might be that he thought he had an approach to resolve the Riemann Hypothesis through almost periodicity. Some of the results that he showed include: if $\chi$ is a non-principal Dirichlet character, then RH for $L(s, \chi)$ is equivalent to the almost periodicity of $L(s, \chi)$ in the half-plane $\Re s > \frac{1}{2}$ (larger than its region of absolute convergence). Interesting!
There is a classical argument that strengthens almost periodicity to exact repetition (though as is often the case, I couldn't actually find this written down anywhere and don't know what this argument is called. I have seen people refer to a classical result on almost periodic functions with Rouche's Theorem before, and that's as close as I got).
Suppose that $f$ is an almost periodic function in a strip $A < \Re s < B$. If $f(s) = a$ has a solution in the strip, then there are infinitely many solutions, and the number of solutions $s = \sigma + it$ with $\lvert t \rvert < T$ is $\gg T$.
If $f$ is almost periodic, then so is $f(s) - a$. Without loss of generality we consider only the case when $f(s) = 0$.If $f$ is identically zero, then we are done. So suppose that $f$ is not identically zero.
Let $s_0$ be a zero of $f(s)$ in the strip. There exists some $\eta_0 > 0$ such that the circle $C = \{ s : \lvert s - s_0 \rvert = \eta_0 \}$ is contained within the strip $A < \Re s < B$ and such that $f$ has no zeros on $C$. (We are now using the assumption that $f$ is not identically zero).
Let $\epsilon = \inf_{s \in C} \lvert f(s) \rvert$ be the minimum value of $f$ on the circle $C$. By almost periodicity, there exists a length $\ell$ such that all intervals of length $\ell$ contains $\tau$ such that \begin{equation} \lvert f(s + i\tau) - f(s) \rvert < \epsilon/2 \quad \qquad (\forall s \in C). \end{equation}
Recall Rouché's Theorem, which in one form states that for two holomorphic functions $g$ and $h$ on a region $\Omega$, if $\lvert g(s) - h(s) \rvert < \lvert h(s) \rvert$ on $\partial \Omega$, then $g$ and $h$ have the samenumber of zeros in $\Omega$.
Then here, as \begin{equation} \lvert f(s + i\tau) - f(s) \rvert< \inf_{w \in C} \lvert f(w) \lvert \leq \lvert f(s) \rvert \qquad (\forall s \in C), \end{equation} Rouche's Theorem shows that $f$ has the same number of zeros (i.e. at least one) in $C$ and $C + i\tau$.
The density result also follows from almost periodicity, as one such $\tau$ exists in every interval of length $\ell$.$\square$
As an immediate corollary, if a Dirichlet series $L$ has at least one zero in the domain of absolute convergence, then it has infinitely many.
Suppose $L(s)$ is a Dirichlet series, absolutely convergent for $\Re s > 1$, such that there is a zero $L(\sigma + i\gamma) = 0$ with $\sigma > 1$. Then $L(s)$ has infinitely many zeros $s$ with $\Re s > 1$.
It follows that the Dirichlet series $L_5(s)$ introduced by Davenport and Heilbronn above has $\gg T$ zeros with imaginary part bounded by $T$ located outside the critical strip.
Technical Ingredients
We will repeatedly apply a few foundational results and bounds. We gather them here.
Suppose $f$ is an integrable function on $[a, b]$ and let $\phi$ be a convex function. Then \begin{equation} \phi\Big( \frac{1}{b - a} \int_a^b f(x) dx\Big) \leq \frac{1}{b - a} \int_a^b \phi(f(x)) dx. \end{equation}The opposite inequality is true if $\phi$ is concave.
We call $\phi$ convex if for every point $(x_0, \phi(x_0))$ on the graph of $\phi$, there is a line $y = \alpha(x - x_0) + \phi(x_0)$ (which is the tangent line for smooth $\phi$) such that $\phi(x) \geq \alpha(x - x_0) + \phi(x_0)$ for all $x$ in the domain of $\phi$.
Let $x_0 = \frac{1}{b-a} \displaystyle \int_a^b f(x) dx$. Integrating the inequality \begin{equation} \phi(f(x)) \geq \alpha(f(x) - x_0) + \phi(x_0) \end{equation}gives \begin{equation} \int_a^b \phi(f) dx \geq \alpha (\int_a^b f(x) dx - (b-a) x_0) + (b-a) \phi(x_0). \end{equation}Note the first term on the RHS cancels due to the definition of $x_0$, so we find that \begin{equation} \int_a^b \phi(f) dx \geq(b-a) \phi(x_0) = (b-a) \phi \Big( \frac{1}{b-a} \int_a^b f(x) dx \Big), \end{equation}as desired.
If $f$ were concave, then the initial inequality above is exactly reversed.$\square$
Let $f:\mathbb{C} \longrightarrow \mathbb{C}$ be a holomorphic function in $\lvert s \rvert \leq R$ with no zeros on $\lvert s \rvert = R$, and such that $f(0) \neq 0$. Then \begin{equation} \frac{1}{2\pi} \int_0^{2\pi} \log \lvert f(Re^{i \theta}) \rvert d \theta= \log \lvert f(0) \rvert + \int_0^R \frac{n(r)}{r} dr, \end{equation}where $n(r)$ is the number of zeros of $f(s)$ inside the circle $\lvert s \rvert = r$.
Recalling that $\Re \log z = \log \lvert z \rvert$, we rewrite the LHS as \begin{align*} \frac{1}{2\pi} \int_0^{2\pi} &\log \lvert f(Re^{i \theta}) \rvert d \theta \\&= \frac{1}{2\pi} \int_0^{2\pi} \Re \log f(Re^{i \theta}) d \theta \\&= \frac{1}{2\pi} \int_0^{2\pi} \Re \Big( \int_0^R \frac{\frac{df}{dr}}{f} dr + \log f(0) \Big)d \theta \\&= \log \lvert f(0) \rvert + \frac{1}{2\pi} \int_0^{2\pi} \Re \Big( \int_0^R \frac{\frac{df}{dr}}{f} dr \Big)d \theta \\&= \log \lvert f(0) \rvert + \frac{1}{2\pi} \Re \int_0^{2\pi} \Big( \int_0^R \frac{f'(re^{i \theta})}{f(re^{i \theta})} e^{i \theta} dr \Big)d \theta \\&= \log \lvert f(0) \rvert + \Re \int_0^R \frac{1}{r} \frac{1}{2 \pi i} \Big( \int_{\lvert s \rvert = r} \frac{f'(s)}{f(s)} ds \Big)dr \\&= \log \lvert f(0) \rvert + \int_0^R \frac{n(r)}{r} dr. \end{align*} Note that $n(r) = 0$ for $r$ sufficiently small, as $f(0) \neq 0$, and thus this last integral converges.$\square$
For a Dirichlet series $L \in \widetilde{S}$, define \begin{equation} \mu(\sigma; L) = \limsup_{\lvert t \rvert \to \infty} \frac{\log \lvert L(\sigma + it) \rvert}{\log \lvert t \rvert}. \end{equation}This is the exponent of the (polynomial) growth in $t$ on the line $\Re s = \sigma$. As is classically known, this is a convex function of $\sigma$.
The average Ramanujan–Petersson bound shows that $\mu(\sigma; L) = 0$ for $\sigma > 1$. The order of growth for $\sigma < 0$ is ruled by the functional equation. Write the functional equation as \begin{equation}L(s) = \Delta(s) \overline{L(1 - \overline{s})} \end{equation}with \begin{equation} \Delta(s):= \omega Q^{1 - 2s} \prod_{\nu = 1}^N \frac{\Gamma(\alpha_\nu(1 - s) + \overline{\beta_\nu})}{\Gamma(\alpha_\nu s + \beta_\nu)}. \end{equation}For reference, we note the following form of Stirling's formula which considers uniformity.
Fix any $\delta > 0$. Let $\Omega$ denote the region of the complex plane consisting of $s$ with $\lvert s \rvert \geq \delta$, and for $s$ confined to the sector $\lvert \arg s \rvert < \pi - \delta$. Then we have that \begin{equation} \Gamma(s)= \sqrt{2 \pi} s^{s - \frac{1}{2}} e^{-s} (1 + O(\lvert s \rvert^{-1})) \end{equation}uniformly for all $s \in \Omega$.
Then a tedious computation with Stirling's formula, applied to each of the Gamma functions in $\Delta(s)$, gives the following.
For $t \geq 1$, uniformly in $\sigma$ (for $\sigma$ restricted to any finite interval), we have that \begin{align*} \Delta(\sigma + it) &= \omega(\alpha Q^2 t^{d_L})^{\frac{1}{2} - \sigma - it} \\ &\exp \Big( it d_L + \frac{i \pi (\beta - d_L)}{4} \Big) \Big(1 + O \big(\tfrac{1}{t}\big) \Big), \end{align*} where \begin{align*} \alpha &= \prod_{\nu = 1}^N \alpha_\nu^{2 \alpha_\nu}, \\ \beta &= 2 \sum_{\nu = 1}^N (1 - 2 \beta_\nu), \\ d_L &= 2\sum_{\nu = 1}^N \alpha_\nu. \end{align*}
The number $d_L$ is typically called the degree of $L(s)$ when $L$ is in the Selberg class — in perhaps every arithmetic application, $d_L$ is an integer. Combined with the functional equation, this shows that for $\sigma < 0$, we have that \begin{equation} \mu(\sigma; L) = \big( \tfrac{1}{2} - \sigma \big) d_L. \end{equation}
As $L(s)$ is of finite order, the Phragmén-Lindelöf convexity principle applies and we obtain upper bounds in the critical strip.
Let $L \in \widetilde{S}$. In any fixed strip $A < \Re s < B$, uniformly in $\sigma$ as $\lvert t \rvert \to \infty$, we have that \begin{equation}L(\sigma + it) \asymp \lvert t \rvert^{(\frac{1}{2} - \sigma)d_L} \lvert L(1 - \sigma + it) \rvert, \end{equation}and \begin{equation} \mu(\sigma; L) \leq \begin{cases}0 & \sigma > 1, \\\\ \tfrac{d_L}{2}(1 - \sigma) & 0 \leq \sigma \leq 1, \\\\(\tfrac{1}{2} - \sigma) d_L & \sigma < 0. \end{cases} \end{equation}
Notice that this gives the typical convexity bound $\mu(\tfrac{1}{2}; L) \leq \frac{1}{4} d_L$, or rather \begin{equation}L(\tfrac{1}{2} + it) \ll \lvert t \rvert^{d_L/4 + \epsilon}. \end{equation}
Additional comments
See Iwaniec and Kowalski Analytic Number Theory or Titchmarsh and Heath-Brown's treatise on the Riemann zeta function for the background material sourced for this note. In later notes, we'll revisit these topics and reference back here.
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