# Asymptotic number of zeros for Dirichlet series with functional equation

I think the usual proof for the asymptotic number of zeros of the Riemann zeta function $$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}Im(\rho)\ \in\ [0,T]\\ Re(\rho) \ \in\ (-1,2)\end{array}\right\} = \frac{ T\ln T}{2\pi}-\frac{1+\log 2\pi}{2\pi}T+\mathcal{O}(\ln T)$$ works for any Dirichlet series in the extended Selberg class $S^{\#}$ (for the Selberg class it is a well-known result), that is Dirichlet series having a functional equation with gamma factors, and being analytic except possibly a pole at $s=1$. More precisely :

• $F(s) = \sum_{n=1}^\infty a_n n^{-s}$ converges absolutely on $Re(s)>1$, and $F(s) (s-1)^m$ is entire of finite order,

• with $\gamma(s) = Q^s\prod_{j=1}^{k} \Gamma(\omega_j s+\mu_j), \omega_j > 0$ and $\Phi(s) = \gamma(s)F(s) : \quad \Phi(s) = \xi\, \overline{\Phi(1-\overline{s})}$

Then $$N_F(T) = \frac{d_F}{2\pi} T \ln T+\frac{c_F}{2\pi} T+\mathcal{O}(\ln T), \qquad d_F=\sum_{j=1}^m \omega_j, \quad c_F = \ln |Q|^2-1-\ln 2 d_F$$ where $N_F(T) = \#\left\{\rho : \ F(\rho)=0, \begin{array}{l}Im(\rho)\ \in\ [0,T]\\ Re(\rho) \ \in\ (-\delta_F,1+\delta_F)\end{array}\right\}$ is the number of zeros, and $\delta_F$ is chosen such that $\sum_{n=2}^\infty |a_n| n^{-1-\delta_F} < |a_1|$ i.e. $\text{arg}(F(1+\delta_F+it) = \mathcal{O}(1)$

Questions :

• Do you have a reference confirming this ? And if I missed something, what additional hypothesis on $F(s)$ are needed ?

• That the asymptotic number of zeros depends on the functional equation not on the Euler product, what does it tell us, about $\zeta(s)$ and the L-functions ?

The asymptotics for the zeros allow us to write $\frac{F'}{F}(s) = \sum_{|Im(\rho)-t)| < A} \frac{1}{s-\rho}+\mathcal{O}(\ln t)$ in the critical strip, and then to look at how different constraints (Euler product, growth rate estimates for $F,F',1/F$) interact with those density of zeros. Assuming the GRH, we are probably also allowed to make some general statements about the number of zero crossings of $\Phi(1/2+it)$, and to link it to some properties of modular forms on $Re(\tau) =0$.

• What happens if I add to $S^{\#}$ the constraint that there is some $l$ such that $\frac{1}{\zeta(\sigma)^l} < |F(s)| \le \zeta(\sigma)^l$ for every $Re(s)=\sigma > 1$ ?

(it could mean that $F(s)$ has an Euler product of the form $\prod_{j=m}^l \prod_p (1-\alpha_{j}(p)p^{-s})^{-1}$ where $|\alpha_j(p)|\le 1$)

The von Mangoldt-type formula for this class of L-functions was alredy stated without proof by Selberg himself in his paper:

• Atle Selberg, "Old and new conjectures and results about a class of Dirichlet series" (1989)

The precise statement is

$$N(T)=\frac{d}{2\pi}T\log T+cT+O(\log T)$$

where $d$ is the degree of $F$ and $c$ is some constant depending on $F$.

Note that it is stated for what we know call $S^{\#}$, not for $S$.

The proof is essentially the same as for $\zeta(s)$. You can see a sketch on Li Zheng's survey on the Selberg class.

I'm not sure what to make of the second or third question, but I'm confident that the answer to the second is "nothing". Perhaps you can elaborate on what dependence you expected between $N(T)$ and the Euler product.

• ok then do you have a reference, a link ? Oct 28 '16 at 18:38
• @user1952009 Sure. I've edited to include it. Oct 28 '16 at 18:40
• I can't find Selberg's paper, and in Li Zheng's concise survey it is only about $S$, not $S^{\#}$ Oct 28 '16 at 18:45
• @user1952009 Perhaps you have better luck finding the 2nd volume of Selberg's collected works. I haven't been able to find the Amalfi Proceedings (where it first appeared) myself. Oct 28 '16 at 18:52
• publications.ias.edu/selberg has the Amalfi lecture(s). Oct 28 '16 at 21:47

Also, there is a paper by Bombieri and Hejhal, from 1995 in Duke Math. J., that shows that under a modest GRH linear combinations of the two unramified ideal-class character L-functions of $\mathbb Z[\sqrt{-5}]$ have lots of off-the-line zeros. One point is that the two have the same Gamma factors, and the same functional equation, so any linear combination does, as well. Yes, the same argument-principle discussion tells the asymptotics of the number of zeros up to height $T$, and it is the same for linear combinations...

In fact, similar linear combinations and integrals of L-functions give Epstein zeta functions, which Potter-Titchmarsh and others have shown have many off-line zeros. The asymptotic zero-counts to height $T$ are as expected, but quite a lot of zeros are off-the-line.

So, for example, the functional equation alone is insufficient to have any hope of truly all zeros on-the-line. (I gather that it is not impossible that asymptotically 100\% are on-the-line...?)

• of course, I never said there is a RH for those Oct 28 '16 at 22:19
• @user1952009, no, indeed, but/and I have the impression that there is a general folkloric belief in such directions that is too optimistic... so, with the excuse of responding to your question... :) Oct 28 '16 at 22:26
• But looking at the articles, it seems there are some deep conjectures about those, about the density of zeros $Re(\rho) \ge \sigma$ for $\sigma \ge 1/2$. Oct 29 '16 at 19:35
• @user1952009, yes, indeed, there appear to be subtleties about even the zeros of linear combinations of Euler products, etc. Also, in case one hadn't happened to think of it or hear about it, Landau and Hecke and Siegel and others noted Euler products with nice functional equations under $s\to 1-s$) that have no zeros on-the-line (e.g., the classic $\zeta(2s)\zeta(2s-1)$). Various other natural "periods" of Eisenstein series (e.g., along $O(n)$ on $O(n,1)$) tend to have zeros farther off the critical line as $n\to+\infty$. Oct 29 '16 at 20:28
• Yes. That's what I meant implicitly in my question : $S^{\#}$ is (more or less) the set of Mellin transforms of modular/automorphic forms. For example $F(s) = \zeta(s)\zeta(s+2k-1)$ is the Mellin transform of $E_{2k}(ix)-1$ the Eisenstein series, and $F(2k s)\in S^{\#}$. Clearly its zeros are in $Re(s)\in (0,1/2k) \cup (1-1/2k,1)$. Oct 29 '16 at 20:58

See the theorem 2.15 in the survey of the Selberg class by Li ZHENG.

https://www.researchgate.net/publication/265103975_A_CONCISE_SURVEY_OF_THE_SELBERG_CLASS_OF_L-FUNCTIONS