The Jensen-P\'{o}lya program for various L-functions

TL;DR

This paper extends the Jensen-Pólya program to various L-functions, showing that their Jensen polynomials are hyperbolic for all but finitely many cases, providing evidence for the generalized Riemann hypothesis.

Contribution

It generalizes previous results from the Riemann Xi-function to a broader class of L-functions, supporting the generalized Riemann hypothesis.

Findings

Jensen polynomials for L-functions are hyperbolic except finitely many cases.

Supports the generalized Riemann hypothesis.

Extends Pólya's program to new classes of functions.

Abstract

P\'{o}lya proved in 1927 that the Riemann hypothesis is equivalent to the hyperbolicity of all of the Jensen polynomials of degree $d$ and shift $n$ for the Riemann Xi-function. Recently, Griffin, Ono, Rolen, and Zagier proved that for each degree $d \geq 1$ all of the Jensen polynomials for the Riemann Xi-function are hyperbolic except for possibly finitely many $n$. Here we extend their work by showing the same statement is true for suitable $L$-functions. This offers evidence for the generalized Riemann hypothesis.

Equations

$$\Xi (z):=\frac{1}{2}\left(-z^2-\frac{1}{4}\right){\pi }^{\frac{iz}{2}-\frac{1}{4}}\Gamma \left(-\frac{iz}{2}+\frac{1}{4}\right)\zeta \left(-iz+\frac{1}{2}\right),$$

$${\Xi }_1(x)=8\cdot \Xi \left(i\sqrt{x}\right)=:\sum _{n\ge 0}\frac{\gamma (n)}{n!}\cdot x^n.$$

$$J_a^{d,n}(X):=\sum _{j=0}^d\left(\genfrac{}{}{0ex}{}{d}{j}\right)a(n+j)X^j.$$

$$\underset{n⟶\infty }{\lim }{J}_{\gamma }^{d,n}(\alpha (n)X+\beta (n))=H_d(X),$$

$$\log \left(\frac{a(n+j)}{a(n)}\right)=A(n)j-\delta (n{)}^2{j}^2+o(\delta (n{)}^d)\text{as }n⟶\infty$$

$$\Lambda (s)=N^{\frac{s}{2}}{\int }_0^{\infty }\left[f(t)-f(\infty )\right]t^s\frac{dt}{t},$$

$$f(t)=\alpha (0)+\sum _{n\ge n_0}\alpha (n)e^{-\pi nt},$$

$$f\left(\frac{1}{Nt}\right)=ϵN^{\frac{k}{2}}{t}^{k}f(t),$$

$$\Xi (z):=\{\begin{array}{ll}\left(-z^2-\frac{k^2}{4}\right)\Lambda \left(\frac{k}{2}-iz\right)& \text{if }\Lambda (s)\text{has a pole at }s=k\\ \Lambda \left(\frac{k}{2}-iz\right)& \text{otherwise}.\end{array}$$

$${\Xi }_1(x):=\Xi (i\sqrt{x})=:\sum _{n\ge 0}\frac{\gamma (n)}{n!}{x}^n,$$

$$\gamma (n)=(-1{)}^n\frac{n!}{(2n)!}\cdot {\Xi }^{(2n)}(0).$$

$${\Xi }_1(x):=\frac{\Xi (i\sqrt{x})}{\sqrt{x}}=:\sum _{n\ge 0}\frac{\gamma (n)}{n!}{x}^n,$$

$$\gamma (n)=i^{2n+1}\frac{n!}{(2n+1)!}\cdot {\Xi }^{(2n+1)}(0).$$

$$\begin{array}{rl}\Lambda (s)& =N^{\frac{s}{2}}\prod _{j=1}^J{\Gamma }_{\mathbb{R}}(s)\prod _{m=1}^M{\Gamma }_{\mathbb{C}}(s)\cdot L(s)\\ & =N^{\frac{s}{2}}{\int }_0^{\infty }\left[f(t)-f(\infty )\right]t^s\frac{dt}{t},\end{array}$$

$$\Lambda (s)=\frac{(ϵs-s+k)f(\infty )}{s(s-k)}+{\int }_{\frac{1}{\sqrt{N}}}^{\infty }\left(f(t)-f(\infty )\right)\left(ϵN^{\frac{k-s}{2}}{t}^{k-s}+N^{\frac{s}{2}}{t}^s\right)\frac{dt}{t}.$$

$${\Lambda }^{(n)}(s)$$

$$+{\int }_{\frac{1}{\sqrt{N}}}^{\infty }(f(t)-f(\infty ))\left(N^{\frac{s}{2}}{t}^s+(-1{)}^nϵN^{\frac{k-s}{2}}{t}^{k-s}\right){\left(\frac{1}{2}\log (N)+\log (t)\right)}^n\frac{dt}{t}.$$

$${\Lambda }^{(n)}\left(\frac{k}{2}\right)=\frac{2^{n+1}f(\infty )n!((-1{)}^{n+1}-ϵ)}{k^{n+1}}+F(n)$$

$${\Xi }^{(n)}(0)=\{\begin{array}{ll}(-i{)}^n\frac{8\left(\genfrac{}{}{0ex}{}{n}{2}\right)F(n-2)-k^{2}F(n)}{4}& \text{if }\Lambda (s)\text{has a pole at}s=k\\ (-i{)}^{n}F(n)& \text{otherwise},\end{array}$$

$$F(n)=\frac{1}{2^n}{\int }_{\frac{1}{\sqrt{N}}}^{\infty }(f(t)-f(\infty ))N^{\frac{k}{4}}{t}^{\frac{k}{2}-1}(1+(-1{)}^nϵ){\left(\log (N)+2\log (t)\right)}^{n}dt$$

$$F(n)$$

$$\times e^{\frac{k}{4}(L-\log (N))-\frac{2n}{L}-\frac{k}{2}+1}\left(1+\frac{b_1}{n}+\frac{b_2}{n^2}+\cdots \right)(n⟶\infty ),$$

$$g(t)=\alpha (n_0)e^{-\pi n_{0}t}{N}^{\frac{k}{4}}(1+(-1{)}^nϵ)t^{\frac{k}{2}-1}(\log (N)+2\log (t){)}^n.$$

$$n=\frac{1}{2}\left(\pi n_{0}a-\frac{k}{2}+1\right)(\log (N)+2\log (a)).$$

$$n=\frac{1}{2}\left(\pi n_0{e}^{\frac{L-\log (N)}{2}}-\frac{k}{2}+1\right)L.$$

$$\frac{g((1+\lambda )a)}{g(a)}={\left(1+\frac{2\log (1+\lambda )}{\log (N)+2\log (a)}\right)}^n(1+\lambda {)}^{\frac{k}{2}-1}{e}^{-\pi n_0\lambda a}=e^{-\frac{C{\lambda }^2}{2}}\left(1+A_3{\lambda }^3+A_4{\lambda }^4+\cdots \right),$$

$$A_3$$

$$A_4$$

$$A_5$$

$$A_6$$

$$+n\left(\frac{(10k-50)\epsilon }{90}+\frac{(30k-197){\epsilon }^2}{90}+\frac{(40k-530){\epsilon }^3}{90}-\frac{34{\epsilon }^4}{3}-16{\epsilon }^5-\frac{32{\epsilon }^6}{3}\right).$$

$$\frac{1}{2^n}{\int }_{\frac{1}{\sqrt{N}}}^{\infty }g(t)dt$$

$$=\frac{ag(a)}{2^n}\sqrt{\frac{2\pi }{C}}\left(1+\frac{3A_4}{C^2}+\frac{15A_6}{C^3}+\cdots +\frac{(2i-1)!!A_{2i}}{C^i}+\cdots \right).$$

$$\gamma (n)=\{\begin{array}{ll}(-1{)}^n\frac{n!}{(2n)!}{\Xi }^{(2n)}(0)& \text{if }ϵ=1\\ i^{2n+1}\frac{n!}{(2n+1)!}{\Xi }^{(2n+1)}(0)& \text{if }ϵ=-1,\end{array}$$

Full text

1. Introduction and statement of results

By extending notes of Jensen, Pólya [13] proved that the Riemann hypothesis (RH) is equivalent to the hyperbolicity of the Jensen polynomials for Riemann’s Xi-function. The Riemann Xi-function is the entire function that shifts the zeros of the Riemann zeta-function, $\zeta(s)$, from the line with real part $\frac{1}{2}$ to the real line. It is given by

[TABLE]

where $\Gamma(s)$ is the gamma function. We can consider a change of variable and define the coefficients $\gamma(n)$ by the Taylor expansion of this new function:

[TABLE]

Pólya originally proved that RH is equivalent to $\Xi_{1}$ having an infinite product expansion of the form $\Xi_{1}(x)=ce^{\sigma x}\prod_{n\geq 1}\left(1+\frac{x}{x_{n}}\right)$, where $c$ is a constant, $\sigma\geq 0$, $x_{n}\in\mathbb{R}^{+}$, and $\sum x_{n}^{-1}<\infty$. This condition can be encoded by the hyperbolicity of Jensen polynomials.

We say that a polynomial $f\in\mathbb{R}[X]$ is hyperbolic if all of its roots are real. Given a sequence $a:\mathbb{N}\longrightarrow\mathbb{R}$ and positive integers $d$ and $n$, the associated Jensen polynomial of degree $d$ and shift $n$ is defined by

[TABLE]

RH is equivalent to the hyperbolicity of $J_{\gamma}^{d,n}(X)$ for all $d$ and $n$ and where $\gamma$ is given in equation ($1.2$) as the Taylor coefficients of $\Xi_{1}(x)$ [5, 6, 13]. The historical context of this approach to RH and a commentary on the results of [8] is given in [2]. Due to the difficulty of proving RH, research before [8] focused on establishing hyperbolicity for all shifts $n$ for small $d$. Work of Csordas, Norfolk, and Varga and Dimitrov and Lucas [4, 6] shows that $J_{\gamma}^{d,n}(X)$ is hyperbolic for all $n$ when $d\leq 3$. In [8], Griffin, Ono, Rolen, and Zagier prove that for any $d\geq 1$, $J_{\gamma}^{d,n}(X)$ is hyperbolic with at most finitely exceptions $n$. They prove this by showing that for a fixed $d$,

[TABLE]

where $H_{d}(X)$ is the $d$-th Hermite polynomial and $\alpha(n)$ and $\beta(n)$ are certain sequences. The Hermite polynomials are known to have real distinct roots so $J_{\gamma}^{d,n}(X)$ must also eventually have real distinct roots. In fact, Griffin, Ono, Rolen, and Zagier show that there is a family of sequences whose Jensen polynomials share the same property.

Definition 1.1**.**

A real sequence $a:\mathbb{N}\longrightarrow\mathbb{R}$ is Hermite-Jensen if there exists sequences of positive real numbers $\{A(n)\}$ and $\{\delta(n)\}$ with $\delta(n)$ tending to zero, which satisfy

[TABLE]

for some $d\geq 1$ and all $0\leq j\leq d$.

Remark*.*

In [8] the authors give a more general statement about the asymptotic behavior needed for the Jensen polynomials of a sequence to converge to other families of polynomials.

In order to show that the Taylor coefficients of Riemann’s Xi-function are Hermite-Jensen, an arbitrary precision asymptotic formula for the derivatives $\Xi^{(2n)}(0)$ was found in [8]. To extend the results in [8] we show that any good Dirichlet series is Hermite-Jensen.

Definition 1.2**.**

A Dirichlet series $L(s)=\sum_{n\geq 1}a(n)n^{-s}$ is good if the following hold.

1. (1)

$L(s)$ has a completed form, $\Lambda(s)$ given in equation (2.1), that has an integral representation of the form

[TABLE]

where the function $f(t)$ has the form

[TABLE]

where $f(\infty)=\alpha(0)$. 2. (2)

The function $f(t)$ satisfies

[TABLE]

where $\epsilon\in\{\pm 1\}$ which gives rise to an analytic continuation and a functional equation $\Lambda(s)=\epsilon\Lambda(k-s)$ for some $k\in\mathbb{Q}$. 3. (3)

The coefficients of $\Lambda(s)$ are real.

For a good Dirichlet series $L(s)$, we define

[TABLE]

If $\Lambda(s)=\Lambda(k-s)$, then we define

[TABLE]

where

[TABLE]

If $\Lambda(s)=-\Lambda(k-s)$, then define

[TABLE]

where

[TABLE]

Theorem 1.3**.**

Suppose that $L(s)$ is a good Dirichlet series. Then $J_{\gamma}^{d,n}(X)$ is hyperbolic with at most finitely many exceptions $n$ for each fixed $d\geq 1$.

Remark*.*

This offers evidence for the generalized Riemann Hypothesis (GRH).

Remark*.*

Notice that in order to study the Jensen polynomials associated to $\gamma(n)$ we must understand the derivatives of $\Xi(z)$ at $z=0$, or equivalently the derivatives of $\Lambda(s)$ at $s=\frac{k}{2}$. In order to prove Theorem 1.3 we will prove an asymptotic formula with arbitrary precision for these derivatives.

Remark*.*

All good $L$-series satisfy the Gaussian Unitary Ensemble (GUE) random matrix prediction in derivative aspect. Dyson, Montgomery, and Odlyzko [7, 10, 12] conjectured that the non-trivial zeros of the Riemann zeta function and other suitable $L$-functions are distributed like the eigenvalues of random Hermitian matrices. These eigenvalues and the roots of the suitably normalized Hermite polynomials, $H_{d}(X)$, as $d\longrightarrow\infty$ both satisfy Wigner’s Semicircular Law (Chapter 3 of [1]). The roots of $J_{\gamma}^{d,0}(X)$, as $d\longrightarrow\infty$, approximate the zeros of $\Lambda\left(\frac{k}{2}-iz\right)$ [13] so these roots are also expected to satisfy Wigner’s Semicircular Law. The derivatives of the completed $L$-function are also predicted to satisfy GUE and higher derivatives correspond to $n$ growing in $J_{\gamma}^{d,n}(X)$ so it is natural to study $J_{\gamma}^{d,n}(X)$ as $n\longrightarrow\infty$. For a good $L$-function the $J_{\gamma}^{d,n}(X)$ converge to the Hermite polynomials which satisfy GUE in degree aspect. This is what is meant by the statement that good $L$-functions satisfy GUE in derivative aspect.

The following corollaries give some examples of Hermite-Jensen Dirichlet series.

Corollary 1.4**.**

Dirichlet $L$-functions for real primitive self-dual characters are good.

Corollary 1.5**.**

Let $f\in S^{new}_{2k}(\Gamma_{0}(N))$ be a weight $2k$ modular newform on $\Gamma_{0}(N)$, then the modular $L$-function associated to $f$ is good.

Corollary 1.6**.**

The Dedekind zeta-function for a number field is good.

In each of these cases we prove an arbitrary precision asymptotic formula for the derivatives of the completed $L$-series at its central value. We do this to show that these $L$-series are Hermite-Jensen, but these results are also of independent interest. This paper is organized in the following way. In Section 2 we will prove Theorem 1.3. In Section 3 we will prove the three corollaries. This section will include the asymptotic formulae for the derivatives of the $L$-functions mentioned above.

Acknowledgements

The author would like to thank Larry Rolen, Michael Griffin, and Ken Ono for helpful discussions related to this work.

2. Asymptotics for $\Xi^{(n)}(0)$.

Let $L(s)=\sum_{n\geq 1}a(n)n^{-s}$ be a good Dirichlet series. We thus know that $L(s)$ has a completed form

[TABLE]

where $\Gamma_{\mathbb{R}}(s):=\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)$, $\Gamma_{\mathbb{C}}(s):=2(2\pi)^{-s}\Gamma(s)$, and $\Gamma(s):=\int_{0}^{\infty}e^{-t}t^{s-1}dt$ is the usual gamma function. Because of the transformation properties of $f(t)$, we split the integral at $\frac{1}{\sqrt{N}}$ to arrive at

[TABLE]

We have the following expression for the derivatives of $\Lambda(s)$:

[TABLE]

At $s=\frac{k}{2}$ and $z=0$ we have

[TABLE]

and

[TABLE]

where

[TABLE]

for all $n\geq 0$. The large asymptotics of $\Lambda^{(n)}\left(\frac{k}{2}\right)$ and $\Xi^{(n)}(0)$ are obtained from the following theorem.

Theorem 2.1**.**

The function $F(n)$ defined in (2.5) is given to all orders in $n$ by the asymptotic expansion

[TABLE]

where $L=L(n)\approx 2\log\left(\frac{n\sqrt{N}}{\log(n\sqrt{N})}\right)$ is the unique positive solution of the equation $n=\frac{1}{2}\left(\pi n_{0}e^{\frac{L-\log(N)}{2}}-\frac{k}{2}+1\right)L$ and each coefficient $b_{k}$ belongs to $\mathbb{Q}(L)$, the first value being $b_{1}=\frac{2(31L^{4}+189L^{3}+542L^{2}+744L+496)}{3(L+2)^{3}}$.

Proof of Theorem 2.1.

We approximate the integrand in (2.5) by the function

[TABLE]

From now on we let $n$ be fixed and will omit it from our notations. We have that $t\frac{d}{dt}(\log(g(t)))=\frac{2n}{\log(N)+2\log(t)}-\pi n_{0}t+\frac{k}{2}-1$, so $g(t)$ assumes its unique maximum at $t=a$ where $a$ is the solution in $\left(\frac{1}{\sqrt{N}},\infty\right)$ of

[TABLE]

For convenience, we define $L=\log(N)+2\log(a)$ so we have

[TABLE]

We can then use Lambert’s $W$ function to asymptotically solve this equation. Lambert’s $W$ function is defined as the solution to $z=W(z)e^{W(z)}$. It has the nice property that $Y=Xe^{X}$ if and only if $X=W(Y)$. If we take a branch cut to restrict $W$ to be real valued, then we have that the principal branch has a Taylor series around [math] given by $W(x)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}x^{n}$. For large $x$, $W(x)$ is asymptotic to $W(x)=\ln(x)-\ln\ln(x)+\frac{\ln\ln(x)(\ln\ln(x)-2)}{\ln^{2}(x)}+O\left(\left(\frac{\ln\ln(x)}{\ln(x)}\right)^{3}\right)$ [3]. Therefore, we have $L\approx 2\log\left(\frac{n\sqrt{N}}{\log(n\sqrt{N})}\right)$. We now follow [8] and apply the saddle point method. The Taylor expansion of $g(t)$ around $t=a$ is given by

[TABLE]

where $C=n\left(\frac{\varepsilon}{2}+\varepsilon^{2}\right)+\frac{k}{8}-\frac{1}{4}$, $\varepsilon=\frac{1}{\log(N)+2\log(a)}=L^{-1}$, and the $A_{i}$ are polynomials of degree $\lfloor i/3\rfloor$ in $n$ with coefficients in $\mathbb{Q}[\varepsilon]$. This expansion was found by expanding $\log(g((1+\lambda)a))-\log(g(a))$ in $\lambda$. The linear term vanises by choice of $a$ and the quadratic term is $-\frac{C\lambda^{2}}{2}$. The coefficients of the higher powers of $\lambda$ are all linear expressions in $n$ with coefficients in $\mathbb{Q}[\varepsilon]$. Exponentiating this expansions gives our expression for $g((1+\lambda)a)/g(a)$. The important behavior is that the dominant term of each $A_{i}$ comes primarily from the exponential of the cubic term of the logarithmic expansion. The first few $A_{i}$ are

[TABLE]

We plug in $t=(1+\lambda)a$ to arrive at the asymptotic expansion

[TABLE]

This expression and the one in Theorem 2.1 are interpreted as asymptotic expansions. These series do not converge for a fixed $n$, but we can truncate the approximation at $O(n^{-A})$ for some $A>0$, and as $n\longrightarrow+\infty$ this approximation becomes true to the given precision. We substitute the formulas for $C$ and $A_{i}$ in terms of $n$ in order to obtain the statement in the theorem. We also replace $F(n)$ by the integral over $g(t)$ with only the $A_{2i}$ with $i\leq 3k$ contributing to $b_{k}$. The same asymptotic formula will hold with this replacement because the ratio of $g(t)$ and the integrand of $F(n)$ is equal to $1+O(n^{-K})$ for any $K>0$ for $t$ near $a$.

∎

3. Proof of Theorem 1.3

Our goal is to show that $\{\gamma(n)\}$ satisfies the growth conditions of Definition 1.1. Recall from Section 1 that

[TABLE]

where $\gamma(n)$ are the Taylor coefficients of $\Xi_{1}(x)$. Therefore, if we have

[TABLE]

and

[TABLE]

as in the example above, then

[TABLE]

We will show that the $\gamma(n)=\frac{n!}{m!}\widehat{F}(m)\cdot\left(1+O\left(\frac{1}{n^{2-\varepsilon}}\right)\right)$ for $m=2n-2,2n-1,2n$, or $2n+1$ form a Hermite-Jensen sequence. Recall that

[TABLE]

Using Stirling’s approximation $r!=\sqrt{2\pi r}\left(\frac{r}{e}\right)^{r}\cdot\left(1+\frac{1}{12r}\right)\cdot\left(1+O(1/r^{2})\right)$, we have

[TABLE]

Recall that $L(m)$ and $b_{1}(m)$ are given in Theorem 2.1 and $C(m)=m\left(\frac{L(m)^{-1}}{2}+L(m)^{-2}\right)+\frac{k}{8}-\frac{1}{4}$. $L(m)$ can be viewed as a holomorphic and non-vanishing function for ${\rm{Re}}(m)>0$, so we have a Taylor expansion in $j$ for the ratio $L(m+2j)/L(m)$ given by

[TABLE]

which converges when $|j|<\frac{m}{2}$, so we will assume this throughout the proof. If we let $J=\frac{\lambda m}{2}$ for some $-1<\lambda<1$, then the asymptotic $L(m)\approx\log\left(\frac{\sqrt{N}m}{\log(\sqrt{N}m)}\right)$ gives the limit

[TABLE]

This implies that $\ell_{r}(m)=o\left(\left(\frac{2}{m}\right)^{r}\right)$. If we expand

[TABLE]

in $j$ then we find $\ell_{1}(m)=\frac{8}{4m(L/2+1)+L^{2}(k/2-1)}=\frac{2}{C\cdot L^{2}}$ and $\ell_{2}(m)=\frac{-(L/2+2)(m+kL/4-L/2)}{C^{3}\cdot L^{5}}$, where $L=L(m)$ and $C=C(m)$. We will also define

[TABLE]

and

[TABLE]

We have the limits

[TABLE]

which imply $c_{r}(m)=o\left(\left(\frac{2}{m}\right)^{r}\right)$ and $\beta_{r}(m)=o\left(\left(\frac{2}{m}\right)^{r+1}\right)$. Using the expansion for $\mathcal{L}(j;m)$ and the expression for $\ell_{1}(m)$ we can find that $c_{1}(m)=\frac{L+2}{C\cdot L^{2}}-\frac{m(L+4)}{C^{2}\cdot L^{4}}$. Define $R_{\gamma}(j;m):=\frac{\widehat{\gamma}(n+j)}{\widehat{\gamma}(n)}$, then after some manipulations we have

[TABLE]

By equation (3.4), for $j$ fixed and as $m\longrightarrow\infty$ (and thus $n\longrightarrow\infty$), we have

[TABLE]

Notice that the first factor in $R(j;m)$ is the $j$th power of $\frac{enL(m)^{2}}{2^{2}m^{2}}$. This factor will essentially be $e^{A(n)}$. We will now look at the expansion

[TABLE]

We again let $J=\frac{m\lambda}{2}$ for $-1<\lambda<1$, then we have

[TABLE]

which tells us that $g_{r}(m)=O\left(\left(\frac{m}{2}\right)^{1-r}\right)$. We can use our previous expansions and the formula for $R_{\gamma}(j;m)$ to find

[TABLE]

We can use the formulas for $\ell_{1}(m)$ and $\ell_{2}(m)$ to simplify these to

[TABLE]

We now let

[TABLE]

These functions satisfy the conditions of Definition 1.1 for the sequence $\{\gamma(n)\}$. The fact that $\delta(n)\longrightarrow 0$ follows from the asymptotics given above and the precision of $O\left(\frac{1}{n^{2-\varepsilon}}\right)$ satisfies the necessary growth conditions given in Definition 1.1.

4. Proofs of Corollaries

4.1. Dirichlet $L$-functions

4.1.1. Proof of Corollary 1.4

Let $\chi$ be a Dirichlet character modulo $N>1$. Then we define the Dirichlet L-function as

[TABLE]

for ${\rm{Re}}(s)>1$. If we let $\chi$ be the trivial character, then our $L$-function is the Riemann zeta function. This case was handled in [8]. Next, recall the twisted theta function

[TABLE]

where $\nu=0$ if $\chi$ is even and $\nu=1$ if $\chi$ is odd. The twisted theta function satisfies the functional equation

[TABLE]

where $\tau(\chi)$ is a Gauss sum and $\bar{\chi}$ is the dual character. We will focus on real primitive self-dual characters so we have $\chi=\bar{\chi}$ and $\tau(\chi)=i^{\nu}\sqrt{N}$. Define the completed Dirichlet $L$-function by

[TABLE]

Using equation (4.3) and the fact that $\chi$ is a real primitive self-dual character, we have the following functional equation

[TABLE]

The completed Dirichlet $L$-function has the required integral representation, functional equation, and real coefficients so it is good.

4.1.2. Derivatives at central values and Dirichlet Jensen polynomials

We want to study the derivatives of $\Lambda(\chi,s)$ which are given by

[TABLE]

At the central value $s=\frac{1}{2}$ we have

[TABLE]

Because the Dirichlet $L$-functions fit into our framework we have the following theorem which gives an arbitrary precision asymptotic formula for these derivatives.

Theorem 4.1**.**

Assume the notation above. The large $n$ asymptotics for $\Lambda^{(n)}\left(\chi,\frac{1}{2}\right)$ and $\Xi^{(n)}(\chi,0)$ are given to all orders by the asymptotic expansion

[TABLE]

where $L=L(n)\approx 2\log\left(\frac{nN}{\log(nN)}\right)$ is the unique positive solution to $n=\frac{1}{2}\left(\pi e^{\frac{L-\log(N^{2})}{2}}+\frac{3}{4}-\frac{\nu}{2}\right)L$ and each coefficient $b_{k}$ belongs to $\mathbb{Q}(L)$, the first value being $b_{1}=\frac{L^{4}+9L^{3}+32L^{2}+24L+16}{24(L+2)^{3}}$.

Example**.**

Let $\chi_{4}$ be the odd Dirichlet character of modulus $4$. Using the two-term approximation $\widehat{F}(n)$ given in equation (3.2) we give some approximations $\widehat{\gamma}_{\chi_{4}}(n)$ in the table below.

[TABLE]

In the previous section we showed that the Dirichlet $L$-function $L(\chi,s)$ is good. Dirichlet $L$-functions have a pole at $s=1$ if $\chi$ is principal so we define

[TABLE]

and

[TABLE]

where

[TABLE]

By Theorem 1.3 or by using the asymptotic expansion above we know that if $d\geq 1$, then $J_{\gamma_{\chi}}^{d,n}(X)$ is hyperbolic with at most finitely many exceptions $n$.

Example*.*

To exemplify Corollary 1.4 we will again consider the odd Dirichlet character of modulus $4$. Let $L(n)$ be the unique solution to $n=\frac{L(n)}{2}\left(\pi e^{\frac{L(n)-\log(16)}{2}}+\frac{1}{4}\right)$ and set $L=L(2n)$. Also set

[TABLE]

The following table demonstrates for $d=2$ and $d=3$ that $\widehat{J}^{d,n}_{\gamma_{\chi_{4}}}(X)=J^{d,n}_{\gamma_{\chi_{4}}}\left(\frac{\delta(n)X-1}{e^{A(n)}}\right)$ converges to $H_{d}(X)$ as $n\longrightarrow\infty$. The polynomials have been normalized so that their leading coefficients are $1$.

[TABLE]

4.2. Modular $L$-functions

4.2.1. Proof of Corollary 1.5

Let $f\in S_{k}\left(\Gamma_{0}(N)\right)$ be an even weight newform with real coefficients and write $f(z)=\sum_{n\geq 1}a(n)e^{2\pi inz}$. Assume that $f$ is normalized so that $a(1)=1$. We focus newforms with trivial character. Define the L-function associated to $f$ by

[TABLE]

for ${\rm{Re}}(s)>1+\frac{k}{2}$. Define the completed modular $L$-function by

[TABLE]

We have the transformation property

[TABLE]

which gives rise to the functional equation

[TABLE]

where $\epsilon_{f}\in\{\pm 1\}$ is the eigenvalue of $f$ under the Atkin-Lehner involution. The completed modular $L$-function $\Lambda(f,s)$ has the required integral representation, the modular properties of $f(z)$ gives a functional equation, and the coefficients are real so $L(f,s)$ is good.

4.2.2. Derivatives at central values and modular Jensen polynomials

Similarly to the Dirichlet $L$-function case, the $n$th derivative takes the form

[TABLE]

At the central value $s=\frac{k}{2}$ we have

[TABLE]

The following theorem gives an arbitrary precision asymptotic formula for these derivatives at central values.

Theorem 4.2**.**

Assume the notation above. Large $n$ asymptotics for $\Lambda^{(n)}\left(f,\frac{k}{2}\right)$ and $\Xi^{(n)}(f,0)$ is given to all orders by the asymptotic expansion

[TABLE]

*where $L=L(n)\approx 2\log\left(\frac{n\sqrt{N}}{\log(n\sqrt{N})}\right)$ is the unique solution of the equation

$n=\frac{1}{2}\left(\pi e^{\frac{L-\log(N)}{2}}-\frac{k}{2}+1\right)L$ and each coefficient $b_{k}$ belongs to $\mathbb{Q}(L)$, the first value being $b_{1}=\frac{L^{4}+9L^{3}+32L^{2}+24L+16}{24(L+2)^{3}}$.*

We have showed that the modular $L$-function $L(f,s)$ and does not have a pole so define

[TABLE]

Depending on the sign of the functional equation we define the Taylor coefficients by

[TABLE]

where

[TABLE]

By Theorem 1.3 or from the asymptotic expansion above we have that if $d\geq 1$, then$J_{\gamma_{f}}^{d,n}(X)$ is hyperbolic with at most finitely many exceptions $n$.

4.3. Dedekind zeta-functions

4.3.1. Proof of Corollary 1.6

The Dedekind zeta-function case will require some setup and notation. We will mostly follow the notation in [11]. Let $K$ be a number field of degree $j$ and $\mathcal{O}_{K}$ its ring of integers. Denote the embeddings by $\sigma_{1},\dots,\sigma_{r_{1}},\rho_{1},\overline{\rho}_{1},\dots,\rho_{r_{2}},\overline{\rho}_{r_{2}}$ where there are $r_{1}$ real embeddings and $r_{2}$ pairs of complex embeddings so that $r_{1}+2r_{2}=j$. Denote the class group of $K$ by $Cl(K)$. Let ${\bf{C}}=\prod_{\tau}\mathbb{C}$ and ${\bf{R}}=\left[\prod_{\tau}\mathbb{C}\right]^{+}=\{z\in{\bf{C}}:z=\overline{z}\}$ be the Minkowski space of $K$ where $\overline{z}=\overline{(z_{\tau})}=(\overline{z}_{\overline{\tau}})$ is the usual complex conjugation and $\tau$ runs over the $j$ embeddings. We define the trace and norm by

[TABLE]

and have a Hermitian scalar product given by

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We will also require the spaces

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and

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in order to define the two homomorphisms

[TABLE]

Let $\mathfrak{p}=\{\sigma,\overline{\sigma}\}$ denote a conjugacy class of embeddings (so $\mathfrak{p}$ has one or two elements depending on whether the embedding is real or complex) and observe that there is an isomorphism between ${\bf{R}}_{+}^{*}$ and $\prod_{\mathfrak{p}}\mathbb{R}_{+}^{*}$. We now have a Haar measure, which we denote by $\frac{dy}{y}$, that corresponds to the product measure $\prod_{\mathfrak{p}}\frac{dt}{t}$ where $\frac{dt}{t}$ is the usual Haar measure on $\mathbb{R}_{+}^{*}$. We can now define a suitable generalization of the gamma function by

[TABLE]

The Dedekind zeta-function for $K$ is given by

[TABLE]

for ${\rm{Re}}(s)>1$ where $N(\mathfrak{a})=[\mathcal{O}_{K}:\mathfrak{a}]$ is the norm of the ideal $\mathfrak{a}$. For each $B\in Cl(K)$ we define the partial zeta function by

[TABLE]

We therefore have

[TABLE]

We define the completed partial Dedekind zeta-function by

[TABLE]

where $d_{K}$ is the discriminant of $K$ and $g$ is some theta function that we will not specify now. The image of the unit group $\mathcal{O}_{K}^{*}$ under the mapping $|\ |:{\bf{R}}^{*}\longrightarrow{\bf{R}}_{+}^{*}$, which we will denote by $|\mathcal{O}_{K}^{*}|$, is contained in the norm-one hypersurface

[TABLE]

We obtain a direct decomposition ${\bf{R}}_{+}^{*}=S\times\mathbb{R}_{+}^{*}$ by writing

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for any $y\in{\bf{R}}_{+}^{*}$. We will need to choose a fundamental domain $F$ for the action of the group

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on $S$. The $\log$ map $\log:{\bf{R}}_{+}^{*}\longrightarrow{\bf{R}}_{\pm}$ takes $S$ to the trace-zero space

[TABLE]

and by Dirichlet’s unit theorem the group $|\mathcal{O}_{K}^{*}|$ is taken to a complete lattice $G$ in $H$. We may choose $F$ to be the preimage of any fundamental mesh of the lattice $2G$. Now using this decomposition we have that

[TABLE]

where $B$ is the class of $\mathfrak{a}^{-1}$ and

[TABLE]

In the above equation $w_{K}$ is the number of roots of unity in $K$, $d^{*}x$ is the appropriate Haar measure such that $d^{*}x\times\frac{dt}{t}=\frac{dy}{y}$, and the theta function is defined by

[TABLE]

where $d_{\mathfrak{a}}=|N(\mathfrak{a})|^{2}|d_{K}|$ is the absolute value of the discriminant of $\mathfrak{a}$. Using the properties of the theta function it is not difficult to show

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and

[TABLE]

where $F^{-1}$ is again a fundamental domain, $\mathfrak{d}_{K}$ is the different ideal of $K$, and $R(K)$ is the regulator of $K$. Note that $(\mathfrak{a}\mathfrak{d}_{K})^{-1}$ is the dual lattice of $\mathfrak{a}$ and that $f(\infty)$ does not depend on the fundamental domain or ideal choice so we will supress notation whenever possible. We now define the completed Dedekind zeta-function by

[TABLE]

where $h_{K}$ is the class number of $K$ and if $B_{i},1\leq i\leq h_{k}$ are ideal classes, then $B_{i}$ is the class of $\mathfrak{a}_{i}^{-1}$. This shows that we have the functional equation

[TABLE]

The completed Dedekind zeta-function has suitable integral representation, functional equation, and real coefficients so $\zeta_{K}(s)$ is good.

4.3.2. Derivatives at central values and Dedekind Jensen polynomials

The $n$th derivative of the completed Dedekind zeta-function has the form

[TABLE]

At the central value $s=\frac{1}{2}$ we have

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In order to state the asymptotic expansion we need to find the first nonzero coefficient of each $f(\mathfrak{a}_{i}t)$. Let $\epsilon$ be a unit with norm $1$, then the smallest nonzero exponent in $f(\mathfrak{a}_{i},t)$ is given by

[TABLE]

Let

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then the expansion of $f(\mathfrak{a}_{i},t)$ begins

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We will let $C_{i}=\frac{2^{r_{1}+r_{2}-1}R(K)}{w_{K}}M_{\mathfrak{a}_{i}}$ and

[TABLE]

in order to simplify the next theorem.

Theorem 4.3**.**

Assume the notation above, then we have

[TABLE]

and $F_{i}(n)$ is given to all orders by the asymptotic expansion

[TABLE]

where $L_{i}=L_{i}(n)\approx j\log\left(\frac{n}{\log(n)}\right)$ is the unique solution of the equation $n=\left(\frac{m_{\mathfrak{a}_{i}}d_{\mathfrak{a}_{i}}^{-\frac{1}{j}}}{j}\pi e^{\frac{L_{i}}{j}}+\frac{3}{4}\right)L_{i}$ and each coefficient $b_{i,k}$ belongs to $\mathbb{Q}(L_{i})$.

We have shown that $\zeta_{K}(s)$ is good so define

[TABLE]

and

[TABLE]

where the Taylor coefficients are given by

[TABLE]

The derivatives $\Xi^{(2n)}(0)$ are given by

[TABLE]

and so we can use the above asymptotic expansion above or Theorem 1.3 to show that if $d\geq 1$, then $J_{\gamma_{K}}^{d,n}(X)$ is hyperbolic with at most finitely many exceptions $n$.