On certain zeta functions associated with Beatty sequences

TL;DR

This paper introduces a new zeta function related to Beatty sequences, analyzing its analytic continuation and singularities depending on the relation of a parameter to a specific lattice.

Contribution

It defines and studies a novel zeta function associated with Beatty sequences, detailing its analytic properties and singularities based on the parameter's lattice membership.

Findings

Analytic continuation to half-plane depending on lattice membership

Simple pole at s=1 when r is in the lattice

No singularities in certain regions for r outside the lattice

Abstract

Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence ${\mathcal B}(\alpha):=(\lfloor\alpha m\rfloor)_{m\in{\mathbb N}}$. If $r$ is an element of the lattice ${\mathbb Z}+{\mathbb Z}\alpha^{-1}$, then $Z_\alpha^\sharp(r,q;s)$ continues analytically to the half-plane $\{\sigma>-1/\tau\}$ with its only singularity being a simple pole at $s=1$. If $r\not\in{\mathbb Z}+{\mathbb Z}\alpha^{-1}$, then $Z_\alpha^\sharp(r,q;s)$ extends analytically to the half-plane $\{\sigma>1-1/(2\tau^2)\}$ and has no singularity in that region.

Equations

$$\zeta (z,q;s)\text{vbox}..=\sum _{n=0}^{\infty }\frac{\mathbf{e}(zn)}{(n+q{)}^s},$$

$$\zeta (z,q;1-s)=(2\pi {)}^{-s}\Gamma (s)\left(\mathbf{e}(\frac{1}{4}s-zq)\zeta (-q,z;s)+\mathbf{e}(-\frac{1}{4}s+zq)\zeta (q,1-z;s)\right)$$

$$\mathcal{B}(\alpha )\text{vbox}..=(\lfloor \alpha m\rfloor {)}_{m\in \mathbb{N}},$$

$$Z_{\alpha }(r,q;s)\text{vbox}..=\sum _{n\in \mathcal{B}(\alpha )}\frac{\mathbf{e}(rn)}{(n+q{)}^s},$$

$$Z_{\alpha }^{♯}(r,q;s)\text{vbox}..=e^{\pi ir}{Z}_{\alpha }(r,q;s)+e^{-\pi ir}{Z}_{\alpha }(-r,1-q;s).$$

$$Z_{\alpha }(r,q;s)+Z_{\beta }(r,q;s)+q^{-s}={\alpha }^{-1}\zeta (r,q;s)+{\beta }^{-1}\zeta (r,q;s).$$

$${\zeta }^{♯}(r,q;s)\text{vbox}..=e^{\pi ir}\zeta (r,q;s)+e^{-\pi ir}\zeta (-r,1-q;s).$$

$$Z_{\alpha }^{♯}(r,q;s)-{\alpha }^{-1}{\zeta }^{♯}(r,q;s)$$

$$Z_{\alpha }^{♯}(0,\frac{1}{2};s)\text{vbox}..=2Z_{\alpha }(0,\frac{1}{2};s)=\sum _{n\in \mathcal{B}(\alpha )}\frac{2}{(n+\frac{1}{2}{)}^s}$$

$${\zeta }^{♯}(0,\frac{1}{2};s)\text{vbox}..=2\zeta (0,\frac{1}{2};s)=\sum _{n=0}^{\infty }\frac{2}{(n+\frac{1}{2}{)}^s}=(2^{s+1}-2)\zeta (s),$$

$$Z_{\alpha }(0,\frac{1}{2};s)\text{vbox}..=\sum _{n\in \mathcal{B}(\alpha )}\frac{1}{(n+\frac{1}{2}{)}^s}$$

$$F_{\alpha}^{\sharp}(0,\tfrac{1}{2};s)=\pi^{-s/2}\Gamma(s/2)\big{(}2Z_{\alpha}(0,\tfrac{1}{2};s)-\alpha^{-1}(2^{s+1}-2)\zeta(s)+2^{s}\big{)}$$

$$\langle\hskip-3.0pt\langle t\rangle\hskip-3.0pt\rangle\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\min_{n\in\mathbb{Z}}|t-n|=\min\big{(}\{t\},1-\{t\}\big{)}\qquad(t\in\mathbb{R}).$$

$$t=\{\begin{array}{ll}[[t]]+⟨⟨t⟩⟩& if\{t\}⩽\frac{1}{2},\\ [[t]]-⟨⟨t⟩⟩& if\{t\}>\frac{1}{2}.\end{array}$$

$$D(M)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup_{\mathcal{I}\subseteq[0,1)}\bigg{|}\frac{V(\mathcal{I},M)}{M}-|\mathcal{I}|\,\bigg{|},$$

$$\tau\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sup\big{\{}t\in\mathbb{R}:\liminf\limits_{n\to\infty}~{}n^{t}\langle\hskip-3.0pt\langle\gamma n\rangle\hskip-3.0pt\rangle=0\big{\}}.$$

$$D_{\gamma ,\delta }(M)\ll M^{-1/(\tau +\epsilon )},$$

$${\Theta }_{v,w}(u)\text{vbox}..=\mathbf{e}(\frac{1}{2}vw)\sum _{n\in \mathbb{Z}}{e}^{-\pi (n+v{)}^{2}u}\mathbf{e}(wn)(u>0).$$

$${\Theta }_{v,w}(u)=u^{-1/2}{\Theta }_{w,-v}(u^{-1}).$$

$$f(x)\text{vbox}..=e^{-\pi (x+v{)}^{2}u}\mathbf{e}(wx)(x\in \mathbb{R}).$$

$$\hat{f}(x)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\nolimits{}_{-\infty}^{\infty}f(y)\mathbf{e}(xy)dy=\nolimits{}_{-\infty}^{\infty}e^{g(x,y)}dy,$$

$$g(x,y)\text{vbox}..=-\pi (y+v{)}^{2}u+2\pi i(x+w)y.$$

$$g(x,y)=-\pi (y+v-i(x+w)u^{-1}{)}^{2}u-2\pi iv(x+w)-\pi (x+w{)}^2{u}^{-1},$$

$$\hat{f}(x)=e^{-\pi(x+w)^{2}u^{-1}}\mathbf{e}(-v(x+w))\nolimits{}_{-\infty}^{\infty}e^{-\pi(y+v-iwu^{-1}-ixu^{-1})^{2}u}dy.$$

$$\nolimits{}_{-\infty}^{\infty}e^{-\pi(y+v)^{2}u}dy=\nolimits{}_{-\infty}^{\infty}e^{-\pi y^{2}u}dy=u^{-1/2}\nolimits{}_{-\infty}^{\infty}e^{-\pi y^{2}}dy=u^{-1/2}.$$

$$\hat{f}(x)=u^{-1/2}{e}^{-\pi (x+w{)}^2{u}^{-1}}\mathbf{e}(-v(x+w))(x\in \mathbb{R}).$$

$$\sum _{n\in \mathbb{Z}}f(n)=\sum _{n\in \mathbb{Z}}\hat{f}(n),$$

$$1_{\alpha }(n)\text{vbox}..=\{\begin{array}{ll}1& ifn=\lfloor \alpha m\rfloor for\; somem\in \mathbb{N},\\ 0& otherwise.\end{array}$$

$$Z_{\alpha }(r,q;s)=\sum _{n=1}^{\infty }\frac{1_{\alpha }(n)\mathbf{e}(rn)}{(n+q{)}^s}.$$

$${\mathcal{X}}_{\gamma }(t)\text{vbox}..=\{\begin{array}{ll}1& if\{t\}\in (0,\gamma ),\\ \frac{1}{2}& ift\in \mathbb{Z}ort\in \gamma +\mathbb{Z},\\ 0& otherwise.\end{array}$$

Full text

To the memory of Tom Apostol

1. Introduction and statement of results

The Lipschitz-Lerch zeta function is defined in the half-plane $\{\sigma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\Re s>1\}$ by an absolutely convergent series

[TABLE]

where $\mathbf{e}(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{2\pi it}$ for all $t\in\mathbb{R}$, and by analytic continuation it extends to meromorphic function on the whole $s$-plane. The function $\zeta(z,q;s)$ was introduced by Lipschitz [20] for real $z$ and $q>0$; see also Lipschitz [21]. It also bears the name of Lerch [19], who showed that for $\Im z>0$ and $q\in(0,1)$ the functional equation

[TABLE]

holds; this is called Lerch’s transformation formula. For an interesting account of the analytic properties of the Lipschitz-Lerch zeta function and related functions, we refer the reader to the work of Lagarius and Li [15, 16, 17, 18]; see also Apostol [2].

If $z\in\mathbb{Z}$, then $\zeta(z,q;s)=\zeta(0,q;s)$ is the Hurwitz zeta function; in this case, $\zeta(z,q;s)$ has a simple pole at $s=1$ but no other singularities in the $s$-plane. On the other hand, if $z\in\mathbb{R}\setminus\mathbb{Z}$ or $\Im z>0$, then $\zeta(z,q;s)$ is an entire function of $s$.

For a given real number $\alpha>0$, the homogeneous Beatty sequence associated with $\alpha$ is the sequence of natural numbers defined by

[TABLE]

where $\left\lfloor\cdot\right\rfloor$ denotes the floor function: $\left\lfloor t\right\rfloor$ is the greatest integer $\leqslant t$ for any $t\in\mathbb{R}$. Beatty sequences appear in a wide variety of unrelated mathematical settings, and their arithmetic properties have been extensively explored in the literature; see, for example, [1, 3, 4, 5, 6, 8, 9, 11, 12, 22, 23, 28] and the references therein.

In this paper, we introduce and study a variant of the Lipschitz-Lerch zeta function that is naturally associated with the Beatty sequence $\mathcal{B}(\alpha)$. Specifically, let us denote

[TABLE]

where $r\in\mathbb{R}$ and $q\in(0,1)$. For technical reasons, the work in this paper is focused on properties of the function

[TABLE]

Further, we assume that $\alpha>1$ is irrational and of finite type (see §2.2). Note that for rational $\alpha$, the Beatty sequence $\mathcal{B}(\alpha)$ is a finite union of arithmetic progressions, and therefore the analytic properties of $Z_{\alpha}(r,q;s)$ and $Z_{\alpha}^{\sharp}(r,q;s)$ can be gleaned from well known properties of the Lipschitz-Lerch zeta function.

The series (1.1) converges absolutely the half-plane $\{\sigma>1\}$, uniformly on compact regions, hence $Z_{\alpha}(r,q;s)$ is analytic there; this implies that $Z_{\alpha}^{\sharp}(r,q;s)$ is analytic in $\{\sigma>1\}$ as well.

Since $\mathcal{B}(\alpha)$ is a set of density $\alpha^{-1}$ in the set of natural numbers, it is reasonable to expect that $Z_{\alpha}(r,q;s)$ is closely related to the function $\alpha^{-1}\zeta(r,q;s)$. This belief is strengthened by the fact that if $\alpha^{-1}+\beta^{-1}=1$, then the set of natural numbers can be split as the disjoint union of $\mathcal{B}(\alpha)$ and $\mathcal{B}(\beta)$, so we have

[TABLE]

Naturally, one might also expect that $Z_{\alpha}^{\sharp}(r,q;s)$ is closely related to the function $\alpha^{-1}\zeta^{\sharp}(r,q;s)$, where

[TABLE]

As it turns out, such expectations are erroneous. Our first theorem establishes that $Z_{\alpha}^{\sharp}(r,q;s)$ has a simple pole at $s=1$ whenever $r$ is an element of the lattice $\mathbb{Z}+\mathbb{Z}\alpha^{-1}$; this lattice is a dense subset of $\mathbb{R}$. By contrast, the function $\alpha^{-1}\zeta^{\sharp}(r,q;s)$, being a linear combination of Lipschitz-Lerch zeta functions, can only have a pole when $r$ is an integer.

Theorem 1.1.

Let $\alpha>1$ be an irrational number of finite type $\tau$. Suppose that $r=k\alpha^{-1}+\ell$ for some integers $k$ and $\ell$, and let $q\in(0,1)$. Then the function

[TABLE]

continues analytically to the half-plane $\{\sigma>-1/\tau\}$ with a simple pole at $s=1$ and no other singularities. The residue at $s=1$ is $2(-1)^{\ell}\alpha^{-1}$ when $k=0$, and it is $2(-1)^{\ell}(\sin(\pi k\alpha^{-1}))/(\pi k)$ for $k\neq 0$.

In particular, taking $r\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=0$ and $q\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tfrac{1}{2}$ above, we have

[TABLE]

and

[TABLE]

where $\zeta(s)$ is the Riemann zeta function studied by Riemann [24] in 1859. Since $\zeta(s)$ has a simple pole at $s=1$ with residue one, from Theorem 1.1 we deduce the following corollary.

Corollary 1.2.

Let $\alpha>1$ be irrational of finite type $\tau$. The Dirichlet series

[TABLE]

converges absolutely in $\{\sigma>1\}$ and extends analytically to $\{\sigma>-1/\tau\}$, where it has a simple pole at $s=1$ with residue $2\alpha^{-1}$ and no other singularities.

When $r\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=0$ and $q\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\tfrac{1}{2}$ our proof of Theorem 1.1 shows that the function

[TABLE]

is analytic at $s=0$. Since $\Gamma(s/2)$ has a pole at $s=0$, one sees that $Z_{\alpha}(0,\tfrac{1}{2};s)$ takes the same value at $s=0$ regardless of the choice of $\alpha$.

Corollary 1.3.

For every irrational $\alpha>1$ of finite type, $Z_{\alpha}(0,\tfrac{1}{2};0)=-\tfrac{1}{2}$.

Our second theorem is complementary to Theorem 1.1; it establishes that $Z_{\alpha}^{\sharp}(r,q;s)$ does not have a pole at $s=1$ whenever the real number $r$ is not contained in the lattice $\mathbb{Z}+\mathbb{Z}\alpha^{-1}$.

Theorem 1.4.

Let $\alpha>1$ be an irrational number of finite type $\tau$. Let $q\in(0,1)$, and suppose that $r$ is a real number not of the form $k\alpha^{-1}+\ell$ with $k,\ell\in\mathbb{Z}$. Then the function (1.2) continues analytically to the half-plane \big{\{}\sigma>1-1/(2\tau^{2})\big{\}} with no singularities in that region.

2. Preliminaries

2.1. General notation

Throughout the paper, we fix an irrational number $\alpha>1$ of finite type $\tau=\tau(\alpha)$ (see §2.2), and we set $\gamma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\alpha^{-1}$. Note that $\gamma$ has the same type $\tau$.

As stated earlier, we write $\mathbf{e}(t)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=e^{2\pi it}$ for all $t\in\mathbb{R}$. We use $\left\lfloor t\right\rfloor$ and $\{t\}$ to denote the greatest integer not exceeding $t$ and the fractional part of $t$, respectively. The notation $\langle\hskip-3.0pt\langle t\rangle\hskip-3.0pt\rangle$ is used to represent the distance from the real number $t$ to the nearest integer; in other words,

[TABLE]

For every real number $t$, we denote by $\llbracket t\rrbracket$ the integer that lies closest to $t$ if $t\not\in\tfrac{1}{2}+\mathbb{Z}$, and we put $\llbracket t\rrbracket\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\left\lfloor t\right\rfloor$ if $t\in\tfrac{1}{2}+\mathbb{Z}$. Then

[TABLE]

In what follows, any implied constants in the symbols $O$, $\ll$ and $\gg$ may depend on the parameters $\alpha,r,q,\varepsilon$ but are independent of other variables unless indicated otherwise. For given functions $F$ and $G$, the notations $F\ll G$, $G\gg F$ and $F=O(G)$ are all equivalent to the statement that the inequality $|F|\leqslant c|G|$ holds with some constant $c>0$.

2.2. Discrepancy and type

The discrepancy $D(M)$ of a sequence of (not necessarily distinct) real numbers $(a_{m})_{m=1}^{M}$ contained in $[0,1)$ is given by

[TABLE]

where the supremum is taken over all intervals $\mathcal{I}$ in $[0,1)$, $V(\mathcal{I},M)$ is the number of positive integers $m\leqslant M$ such that $a_{m}\in\mathcal{I}$, and $|\mathcal{I}|$ is the length of $\mathcal{I}$.

The type $\tau=\tau(\gamma)$ of a given irrational number $\gamma$ is defined by

[TABLE]

Using Dirichlet’s approximation theorem, one sees that $\tau\geqslant 1$ for every irrational number $\gamma$. The theorems of Khinchin [10] and of Roth [25, 26] assert that $\tau=1$ for almost all real numbers (in the sense of the Lebesgue measure) and all irrational algebraic numbers $\gamma$, respectively; see also [7, 27].

Given an irrational number $\gamma$, the sequence of fractional parts $(\{n\gamma\})_{n=1}^{\infty}$ is known to be uniformly distributed in $[0,1)$ (see [14, Example 2.1, Chapter 1]). In the case that $\gamma$ is of finite type, the following more precise statement holds (see [14, Theorem 3.2, Chapter 2]).

Lemma 2.1.

Let $\gamma$ be a fixed irrational number of finite type $\tau$. Then, for every $\delta\in\mathbb{R}$ the discrepancy $D_{\gamma,\delta}(M)$ of the sequence $(\{\gamma m+\delta\})_{m=1}^{M}$ satisfies the bound

[TABLE]

where the implied constant depends only on $\gamma$ and $\varepsilon$.

2.3. Functional equations of theta functions

Lemma 2.2.

For any real numbers $v,w$ let

[TABLE]

Then

[TABLE]

Proof.

Let $u>0$ be fixed, and put

[TABLE]

The Fourier transform of $f$ is given by

[TABLE]

where

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

Using Cauchy’s theorem to shift the line of integration vertically, we see that the integral in (2.5) is equal to

[TABLE]

Consequently,

[TABLE]

Applying the Poisson Summation Formula

[TABLE]

we immediately deduce the functional equation (2.4). ∎

2.4. The pulse wave

Let $\mathbf{1}_{\alpha}$ denote the indicator function of $\mathcal{B}(\alpha)$; that is,

[TABLE]

In this notation we have

[TABLE]

Let $\mathcal{X}_{\gamma}$ be the periodic function defined by

[TABLE]

Since $\mathcal{X}_{\gamma}$ is periodic of bounded variation, and $\mathcal{X}_{\gamma}(t)=\tfrac{1}{2}(\mathcal{X}_{\gamma}(t^{+})+\mathcal{X}_{\gamma}(t^{-}))$ for all $t\in\mathbb{R}$, its Fourier series converges everywhere, and we have

[TABLE]

where the Fourier coefficients are given by

[TABLE]

For any irrational $\alpha>1$, it is easy to see that a natural number $n$ lies in the Beatty sequence $\mathcal{B}(\alpha)$ if and only if $\{-n\gamma\}\in(0,\gamma)$. Using this characterization, the indicator function $\mathbf{1}_{\alpha}$ given by (2.6) satisfies

[TABLE]

From now on, we regard $\mathbf{1}_{\alpha}$ as a function on all of $\mathbb{Z}$ by defining the value $\mathbf{1}_{\alpha}(n)$ at an arbitrary integer $n$ via the relation (2.8).

Using our hypothesis that $\alpha$ is of finite type, the relation (2.8) can be made more explicit; namely, for any positive real number $K$, we have the estimate

[TABLE]

for any given $\varepsilon>0$. Indeed, for each nonzero integer $n$ let

[TABLE]

Using standard estimates for exponential sums (see, e.g., Korobov [13]) we have

[TABLE]

Since $\gamma$ is of type $\tau$, this implies that the bound

[TABLE]

holds, and therefore

[TABLE]

Bounding $\sum_{k>K}\widetilde{}\mathcal{X}_{\gamma}(-k)\mathbf{e}(kn\gamma)$ in a similar manner, we deduce (2.9) in the case that $n\neq 0$. When $n=0$, we have by (2.8):

[TABLE]

Writing

[TABLE]

we have $S_{K}(0;u)\ll\langle\hskip-3.0pt\langle\gamma\rangle\hskip-3.0pt\rangle^{-1}=\alpha\ll 1$, and therefore

[TABLE]

This yields (2.9) in the case that $n=0$.

3. The proofs

For all $u>0$ we denote

[TABLE]

and we also put

[TABLE]

It is easy to see that

[TABLE]

holds for $u>0$, and the relations

[TABLE]

are immediate. Indeed, (3.2) follows from the fact that the polynomial $(n+\tfrac{1}{2})^{2}$ is invariant under the map $n\mapsto-n-1$. To prove (3.3), we note that (2.8) implies

[TABLE]

hence $\mathbf{1}_{\alpha}(n)=\mathbf{1}_{\alpha}(-n-1)$ for all $n\in\mathbb{Z}$.

Next, recall that

[TABLE]

for every positive real number $\mu$. Taking into account that $\mathbf{1}_{\alpha}(0)=\tfrac{1}{2}$ in view of (2.7) and (2.8), the function

[TABLE]

satisfies the relation

[TABLE]

Therefore, using (3.2) and (3.3) it follows that

[TABLE]

satisfies the relation

[TABLE]

To prove Theorems 1.1 and 1.4, it suffices to show that $F_{\alpha}^{\sharp}(r,q;s)$ continues analytically in an appropriate manner according to whether or not $r$ lies in the lattice $\mathbb{Z}+\mathbb{Z}\alpha^{-1}$.

To simplify the notation, we put

[TABLE]

and write

[TABLE]

where

[TABLE]

In view of (3.1) it is clear that the integral $F_{\infty}(s)$ converges absolutely and uniformly on compact regions of $\mathbb{C}$, hence $F_{\infty}(s)$ continues to an entire function of $s\in\mathbb{C}$. Thus, the analytic continuation of $F(s)$ reduces to that of $F_{0}(s)$.

Let $K$ be a real-valued function such that $K(u)\geqslant 2$ for all $u>0$. For the moment, let $u>0$ be fixed. Using (2.9) along with the bound

[TABLE]

we have

[TABLE]

Reversing the order of summation and recalling (2.3), we see that

[TABLE]

Since $\widetilde{}\mathcal{X}_{\gamma}(0)=\gamma$, and $\Psi(r,q;u)=\mathbf{e}(-\tfrac{1}{2}qr)\Theta_{q,r}(u)$ by (2.3), we derive the estimate

[TABLE]

where

[TABLE]

In particular,

[TABLE]

By the functional equation (2.4) and the definition (2.3),

[TABLE]

Combining this expression with (3.4) we have

[TABLE]

Now, we make the specific choice

[TABLE]

where $L$ is a large positive real number, and $\varepsilon>0$ is fixed (and small). In particular, for all large $u$ we have

[TABLE]

Note that (3.5) takes the form

[TABLE]

provided that $\sigma>-2\theta$.

To proceed further, for each integer $k$ we write

[TABLE]

with $\nu_{k}\in\{\pm 1\}$ as in (2.2). Making the change of variables $n\mapsto n-\llbracket r-k\gamma\rrbracket$ in the inner summation of (3.6), it follows that

[TABLE]

We introduce the notation

[TABLE]

and let $\kappa_{u}$ denote an integer for which

[TABLE]

Then we have

[TABLE]

where

[TABLE]

By (3.7) it follows that the estimate

[TABLE]

holds provided that $\sigma>-2\theta$, where

[TABLE]

Thus, the analytic continuation of $F_{0}(s)$ rests on the analytic properties of the integrals $G_{L}^{(j)}(s)$.

Since $\widetilde{}\mathcal{X}_{\gamma}(k)\ll|k|^{-1}$ and $\langle\hskip-1.7pt\langle r-k\gamma\rangle\hskip-1.7pt\rangle\leqslant\tfrac{1}{2}$ for every $k\neq 0$, the bound

[TABLE]

is obvious. For fixed $L$, this implies that the integral $G_{L}^{(1)}(s)$ converges absolutely for all $s\in\mathbb{C}$, uniformly on compact regions, and hence $G_{L}^{(1)}(s)$ is an entire function of $s$.

To determine the analytic behavior of $G_{L}^{(2)}(s)$, we consider two distinct cases according to the size of $\delta_{u}$.

Case 1: $\delta_{u}\geqslant u^{-1/2}\log u$. By the definition of $\delta_{u}$ (see (3.8)) it follows that $\langle\hskip-3.0pt\langle r-k\gamma\rangle\hskip-3.0pt\rangle\geqslant u^{-1/2}\log u$ for all $k$ such that $|k|\leqslant u^{\theta}L$. Since $\widetilde{}\mathcal{X}_{\gamma}(k)\ll|k|^{-1}$ for every $k\neq 0$, we have

[TABLE]

Note that the bound

[TABLE]

also holds in this case.

Case 2: $\delta_{u}\leqslant u^{-1/2}\log u$. Let $k$ be such that $0<|k|\leqslant u^{\theta}L$ and $k\neq\kappa_{u}$. Then

[TABLE]

where $\nu\in\{\pm 1\}$ as in (2.2), and therefore

[TABLE]

Noting that $|\kappa_{u}-k|\leqslant 2u^{\theta}L$, and using the fact that $\gamma$ is of type $\tau$, we also have

[TABLE]

As $\delta_{u}=o(u^{-1/2+\varepsilon/2})$ as $u\to\infty$, from (3.11) and (3.12) we derive the lower bound

[TABLE]

Arguing as in Case 1, this implies that

[TABLE]

For fixed $L$, the bounds (3.9) and (3.13) together imply that the integral $G_{L}^{(2)}(s)$ converges absolutely for all $s\in\mathbb{C}$, uniformly on compact regions, and therefore $G_{L}^{(2)}(s)$ is an entire function of $s$.

Turning now to the analytic behavior of $G_{L}^{(3)}(s)$, we consider two distinct cases according to whether or not $\delta_{u}$ vanishes on the interval $(1,\infty)$.

First, suppose that $\delta_{u}=0$ for some $u>1$. In view of the definition (3.8), this condition is equivalent to the statement that $r=k\gamma+\ell$ for some (uniquely determined) integers $k$ and $\ell$. In this case, one has $\delta_{u}=0$ and $\kappa_{u}=k$ for all sufficiently large $u$. In particular, for some sufficiently large real number $U_{L}$, one sees that $\Upsilon_{L}^{(3)}(u)=\widetilde{}\mathcal{X}_{\gamma}(k)$ once $u\geqslant U_{L}$. Consequently, if we denote

[TABLE]

then clearly $G_{L}^{(4)}(s)$ converges absolutely for all $s\in\mathbb{C}$, uniformly on compact regions, and so $G_{L}^{(4)}(s)$ is an entire function of $s$. Putting everything together, we have therefore shown that

[TABLE]

where

[TABLE]

is an entire function of $s\in\mathbb{C}$ for any fixed $L$. The sequence $(H_{L}(s))_{L\geqslant 1}$ converges uniformly to $F_{\alpha}^{\natural}(r,q;s)$ on every compact subset of the half-plane $\{\sigma>-2\theta\}$, hence $F_{\alpha}^{\natural}(r,q;s)$ is analytic in the same region. Since $\theta\to 1/(2\tau)$ as $\varepsilon\to 0^{+}$, Theorem 1.1 follows.

Next, we suppose that $\delta_{u}\neq 0$ for all $u>1$. Observe that the map $u\to\delta_{u}$ is a positive nonincreasing step function which tends to zero as $u\to\infty$. Let $u_{1}<u_{2}<\cdots$ be the ordered sequence of real numbers $u_{j}>1$ that have one or both of the following properties:

• $(i)$

$\delta_{u}>\delta_{u_{j}}$ for all $u\in(1,u_{j})$,

• $(ii)$

$\delta_{u_{j}}=u_{j}^{-1/2}\log u_{j}$.

Put $u_{0}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=1$. Note that the sequence $(u_{j})_{j\geqslant 0}$ is countable. For each $j\geqslant 0$, let $\mathcal{I}_{j}$ denote the open interval $(u_{j},u_{j+1})$. To prove Theorem 1.4, it suffices to establish the upper bound

[TABLE]

for every $j\geqslant 0$, where the implied constant in (3.14) may depend on $L$ but is independent of the index $j$. Indeed, since

[TABLE]

the bound (3.14) implies that the integral $G_{L}^{(3)}(s)$ converges absolutely throughout the half-plane $\{\sigma>1-2\theta^{2}\}$, uniformly on compact regions, and thus $G_{L}^{(3)}(s)$ is analytic in that region. Then

[TABLE]

where

[TABLE]

is analytic in the half-plane $\{\sigma>1-2\theta^{2}\}$. The sequence $(H_{L}(s))_{L\geqslant 1}$ converges uniformly to $F_{\alpha}^{\sharp}(r,q;s)$ on every compact subset of $\{\sigma>1-2\theta^{2}\}$, hence $F_{\alpha}^{\sharp}(r,q;s)$ is analytic in that half-plane. Since $\theta\to 1/(2\tau)$ as $\varepsilon\to 0^{+}$, Theorem 1.4 follows.

It remains to establish (3.14). To this end, put

[TABLE]

By the manner in which the sequence $(u_{j})_{j\geqslant 0}$ is constructed (especially, see $(ii)$ above), it follows that every nonnegative integer $j$ lies either in $\Omega^{+}$ or in $\Omega^{-}$. Moreover, (3.10) immediately yields (3.14) in the case that $j\in\Omega^{+}$. Therefore, it remains to show that (3.14) holds for integers $j\in\Omega^{-}$.

Let $j\in\Omega^{-}$ be fixed. For all $u\in\mathcal{I}_{j}$ we have $\delta_{u}<u^{-1/2}\log u$ (by $(ii)$ above) and $\kappa_{u}\neq 0$ (since $\delta_{u}\neq 0$). Using the estimates

[TABLE]

and

[TABLE]

we derive that

[TABLE]

Consequently, to prove (3.14) it is enough to establish the lower bound

[TABLE]

Let $\Delta_{j}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\delta_{u_{j}}$. Using $(i)$ and $(ii)$ above, and taking into account that $\delta_{u}$ is a right-continuous function of $u$ by (3.8), we see that $\delta_{u}=\Delta_{j}$ and $\delta_{u}<u^{-1/2}\log u$ for all $u\in\mathcal{I}_{j}$. Put $k_{j}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\kappa_{u_{j}}$, so that $\langle\hskip-3.0pt\langle r-k_{j}\gamma\rangle\hskip-3.0pt\rangle=\Delta_{j}$, and note that $|k_{j}|\leqslant u_{j}^{\theta}L$. On the other hand, for any integer $k$ with $|k|<u_{j}^{\theta}L$ write $|k|=u^{\theta}L$ with some real number $u<u_{j}$; then

[TABLE]

This argument shows that

[TABLE]

Since $j\in\Omega^{-}$, the argument given in Case 2 implies that there is precisely one integer $k$ that satisfies both inequalities

[TABLE]

(namely, the integer $k=\kappa_{j}$). On the other hand, using (2.1) in combination with the definition of discrepancy and Lemma 2.1, one sees that the number of integers $k$ satisfying (3.17) is

[TABLE]

For large $j$, this leads to a contradiction unless both bounds

[TABLE]

and

[TABLE]

satisfied. We deduce that

[TABLE]

Now let $u\in\mathcal{I}_{j}$. Since

[TABLE]

the integer $k=\kappa_{u}$ satisfies both inequalities in (3.17). Consequently, $\kappa_{u}=k_{j}$, and using (3.16) and (3.18) we have

[TABLE]

This is the required bound (3.15), and our proof of Theorem 1.4 is complete.