---
abstract: 'We study the Dirichlet series $F_b(s)=\sum_{n=1}^\infty d_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-$b$ digits of the integer $n$, and $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$, where $S_b(n)=\sum_{m=1}^{n-1}d_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have continuations to the plane ${\mathbb{C}}$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.'
author:
- Corey Everlove
bibliography:
- 'sumofdigitsbibliography.bib'
title: 'Dirichlet series associated to sum-of-digits functions'
---

[^1]

Introduction
============

There has been a great deal of study of properties of the radix expansions to an integer base $b\geq 2$ of integers $n$. For each integer base $b\geq 2$, every positive integer $n$ has a unique base-$b$ expansion $$n = \sum_{i \geq 0} \delta_{b,i}(n)b^i$$ with digits $\delta_{b,i}\in\{0,1,\dotsc,b-1\}$ given by $$\delta_{b,i} = \Bigl\lfloor \frac{n}{b^i} \Bigr\rfloor - b \Bigl\lfloor \frac{n}{b^{i+1}} \Bigr\rfloor.$$ This paper considers two summatory functions of base $b$ digits of $n$:

1.  The *base-$b$ sum-of-digits function* $d_b(n)$ is $$d_b(n) = \sum_{i\geq 0} \delta_{b,i}(n).$$

2.  The *(base $b$) cumulative sum-of-digits function* $S_b(n)$ is $$S_b(n) = \sum_{m=1}^{n-1} d_b(m).$$ We follow here the convention of previous authors (including [@delange-75] and [@flajolet-94]), with the sum defining $S_b(n)$ running to $n-1$ instead of $n$.

We associate to the functions $d_b(n)$ and $S_b(n)$ the Dirichlet series generating functions $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ and $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}.$$ These Dirichlet series have abscissa of convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively.

This paper studies the problem of the meromorphic continuation to ${\mathbb{C}}$ of Dirichlet series associated to the base-$b$ digit sums $d_b(n)$ and $S_b(n)$. Here we obtain the meromorphic continuation and determine its exact pole and residue structure. The pole structure contains half of a two-dimensional lattice and the residues involve Bernoulli numbers and values of the Riemann zeta function on the line $\operatorname{Re}(s)=0$. A meromorphic continuation of these functions was previously obtained in the thesis of Dumas [@dumas-thesis] by a different method, which specified a half-lattice containing all the poles but did not determine the residues; in fact infinitely many of the residues on his possible pole set vanish.

The asymptotics of $S_b(n)$ have been extensively studied, see Section \[sec-previous-work\]. We mention particularly work of Delange [@delange-75], given below as Theorem \[thm-delange\], which gives an exact formula for $S_b(n)$ in terms of a continuous nondifferentiable function with Fourier coefficients involving values of the Riemann zeta function on the imaginary axis. Using an interpolation of Delange’s formula we formulate a continuous interpolation of $S_b(n)$ in the base parameter $b$, permitting definitions of $d_\beta(n)$ and $S_\beta(n)$ for a real parameter $\beta>1$. We obtain a meromorphic continuation of the associated Dirichlet series $F_{\beta}(s)$ and $G_{\beta}(s)$ to the half-planes $\operatorname{Re}(s)>0$ and $\operatorname{Re}(s)>1$, respectively. We note apparent fractal properties of $d_{\beta}(n)$ as $\beta$ is varied.

Results
-------

Our first results concern the meromorphic continuation of the functions $F_b(s)$ and $G_b(s)$ to the entire complex plane ${\mathbb{C}}$.

\[thm-db\] For each integer base $b\geq 2$, the function $F_b(s) = \sum_{n=1}^\infty d_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $F_b(s)$ consist of a double pole at $s=1$ with Laurent expansion beginning $$F_b(s)=\frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ simple poles at each other point $s=1+2\pi i m / \log b$ with $m\in {\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1+\frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr),$$ and simple poles at each point $s=1-k+2\pi i m /\log b$ with $k=1$ or $k\geq 2$ an even integer and with $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1-k+\frac{2\pi i m}{\log b} \biggr) =  (-1)^{k+1}\frac{b-1}{\log b}\zeta\biggl(\frac{2\pi i m}{\log b}\biggr)\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr)$$ where $B_k$ is the $k$th Bernoulli number.

Theorem \[thm-db\] is proved by first considering the Dirichlet series $\sum \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$ and then exploiting a relation between power series and Dirichlet series to recover $F_b(s)$. The proof is presented in Section \[sec-mero-cont\].

The meromorphic continuation of Dirichlet series attached to $b$-regular sequences, of which our Dirichlet series $F_b(s)$ is a particular example, was studied by Dumas in his 1993 thesis [@dumas-thesis]; this work also showed that the poles of $F_b(s)$ must be contained in a certain half-lattice, strictly larger than the half-lattice here.

A similar method allows us to meromorphically continue the series $G_b(s)$ to the complex plane.

\[thm-sb\] For each integer $b\geq 2$, the function $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $G_b(s)$ consist of a double pole at $s=2$ with Laurent expansion $$G_b(s) = \frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1} + O(1),$$ a simple pole at $s=1$ with residue $$\operatorname*{Res}(G_b(s),s=1) = \frac{b+1}{12},$$ simple poles at $s=2 + 2\pi i m / \log b$ with $m\in{\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 + \frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log b}\biggr)^{-1} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr)$$ and simple poles at point $s=2-k + 2\pi i m / \log b$ with $k\geq 2$ an even integer and $m\in{\mathbb{Z}}$ with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 - k + \frac{2\pi i m}{\log b} \biggr) = \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr) \biggl(\frac{B_k}{k(k-2)!} \biggr) \prod_{j=1}^{k-2}\biggl(\frac{2\pi i m}{\log b} - j\biggr).$$

An interesting feature of the above theorems is the abundance of poles. Since each function $F_b(s)$ and $G_b(s)$ has $\asymp r^{2}$ poles in the disk $\lvert s \rvert<r$, we have the following corollary, which we discuss further in Section \[sec-meromorphic-functions\].

The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$ on ${\mathbb{C}}$.

The Riemann zeta function, the Dirichlet L-functions, and the Dirichlet series generating functions of many important arithmetic functions (such as the Möbius function $\mu(n)$, the von Mangoldt function $\Lambda(n)$, the Euler totient function $\phi(n)$, and the sum-of-diviors functions $\sigma_\alpha(n)$) analytically continue as meromorphic functions of order $1$ on the complex plane. The Dirichlet series $F_b(s)$ and $G_b(s)$ thus have a different analytic character than many other Dirichlet series considered in number theory.

In Section \[sec-non-int\], we use a formula of Delange [@delange-75] for $S_b(n)$ to define continuous real-valued interpolations of the functions $d_b(n)$ and $S_b(n)$ from integer bases $b\geq 2$ to a real parameter $\beta>1$. As before, we associate to these interpolated sum-of-digits functions the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ and $$G_\beta(s) = \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}.$$

We prove that these Dirichlet series each have a meromorphic continuation one unit to the left of their halfplane of absolute convergence. For the function $F_\beta(s)$ we have the following theorem.

For each real $\beta>1$, the function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi)\bigr) - \frac{\beta+1}{4}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i m}{\log \beta}\biggr) =  -\frac{\beta-1}{2\pi i m}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$

For the function $G_\beta(s)$ we have the following theorem.

For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta-1}{2\log \beta}(s-2)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi) - 1\bigr) - \frac{\beta+1}{4}\biggr)(s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i m}{\log \beta}\biggr) = -\frac{\beta-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$

To prove these theorems, we start by obtaining the continuation of the series $G_\beta(s)$ by working directly with its Dirichlet series and then obtain the continuation of $F_\beta(s)$ by studying the relation between these two Dirichlet series.

Previous work {#sec-previous-work}
-------------

There has been much previous work studying the functions $d_b(n)$ and $S_b(n)$. The function $d_b(n)$ exhibits significant fluctuations as $n$ changes to $n+1$. It can only increase slowly, having $d_b(n+1) \leq d_b(n)+1$ but it can decrease by an arbitrarily large amount. The sequence $d_b(n)$ is a $b$-regular sequence in the sense of Allouche and Shallit [@allouche-shallit-92 Ex. 2, Sec. 7] and is a member of the $b$-th arithmetic fractal group $\Gamma_b({\mathbb{Z}})$ of Morton and Mourant [@morton-mourant-89 p. 256]. Chen et al. [@chen-hwang-zacharovas-14] survey results on the sum-of-digits function of random integers, and give many references.

Concerning the cumulative sum-of-digits function, Mirsky [@mirsky-49] proved in 1949 that for any integer base $b\geq 2$, the function $S_b(n)$ has the asymptotic $$S_b(n) = \frac{b-1}{2\log b} n \log n + O(n).$$ In 1968 Trollope [@trollope-68] expressed the error term for the base-$2$ cumulative digit sum $S_2(n)$ in terms of a continuous everywhere nondifferentiable function, the Takagi function—see [@lagarias-12] for a survey of the properties of this function. In 1975 Delange [@delange-75] proved the following formula for $S_b(n)$, expressing the error term as a Fourier series with coefficients involving values of the Riemann zeta function on the imaginary axis.

\[thm-delange\] The cumulative sum-of-digits function $S_b(n)$ satisfies $$\label{eq-delange-formula}
S_b(n) = \frac{b-1}{2\log b} n \log n + h_b\biggl(\frac{\log n}{\log b} \biggr) n$$ where $h_b$ is a nowhere-differentiable function of period $1$. The function $h_b$ has a Fourier series $$h_b(x) = \sum_{k=-\infty}^\infty c_b(k)e^{2\pi i k x}$$ with coefficients $$c_b(k) = -\frac{b-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log b}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log b} \biggr)$$ for $k\neq 0$ and $$c_b(0) = \frac{b-1}{2\log b}\bigl(\log(2\pi) - 1\bigr) - \frac{b+1}{4}.$$

A complex-analytic proof of a summation formula for general $q$-additive functions, of which the base-$q$ sum-of-digits function is an example, was given by Mauclaire and Murata in 1983 [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88]. A shorter complex-analytic proof of in the specific case of $S_2(n)$ was given by Flajolet, Grabner, Kirschenhofer, Prodinger, and Tichy in 1994 [@flajolet-94]. The method of Flajolet et al. is based on applying a variant of Perron’s formula to the Dirichlet series $$\sum_{n=1}^\infty \bigl(d_2(n)-d_2(n-1)\bigr) n^{-s}.$$ Grabner and Hwang [@grabner-hwang-05] study higher moments of the sum-of-digits function by similar complex-analytic methods.

Our formulas for the residues of $F_b(s)$ and $G_b(s)$ involve the Bernoulli numbers. Kellner [@kellner-17] and Kellner and Sondow [@kellner-sondow-17] investigate another relation between sums of digits and Bernoulli numbers, proving that the least common multiple of the denominators of the coefficients of the polynomial $\sum_{i=0}^n n^k$, which can be written in terms of a Bernoulli polynomial, can be expressed as a certain product of primes satisfying $d_p(n+1)\geq p$.

Sum-of-digits Dirichlet series
==============================

First we consider basic properties of the Dirichlet series $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ attached to the base-$b$ digit sum of $n$ and the Dirichlet series $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ attached to the cumulative base-$b$ digit sum. For standard references on the basic analytic properties of Dirichlet series, see the books of Hardy and Riesz [@hardy-riesz] or Titchmarsh [@titchmarsh-tof Ch. IX].

Recall that each ordinary Dirichlet series $\sum a_n n^{-s}$ has an abscissa of conditional convergence $\sigma_c$ such that the Dirichlet series converges and defines a holomorphic function if $\operatorname{Re}(s)>\sigma_c$ and diverges if $\operatorname{Re}(s)<\sigma_c$. Each Dirichlet series also has an abscissa of absolute convergence $\sigma_a$ such that the Dirichlet series converges absolutely if $\operatorname{Re}(s)>\sigma_a$ and does not converge absolutely if $\operatorname{Re}(s)<\sigma_a$. For ordinary Dirichlet series, one always has $\sigma_a-1\leq \sigma_c \leq \sigma_a$, and $\sigma_c=\sigma_a$ if the coefficients $a_n$ are nonnegative reals.

For each integer $b\geq 2$, the Dirichlet series $$\label{eq-fb-def-2}
F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>1$.

A positive integer $n$ has $[\log n / \log b]+1$ digits when written in base $b$, each of which is at most $b-1$, so $$\label{eq-db-estimate}
d_b(n) \leq (b-1) \biggl(\biggl\lfloor\frac{\log n}{\log b} \biggr\rfloor+1\biggr).$$ We then obtain the estimate $$\label{eq-sb-estimate}
 S_b(n) \ll n \log n$$ with an implied constant depending on $b$. This implies that the Dirichlet series has abscissa of absolute convergence at most 1 and therefore defines a holomorphic function for $\operatorname{Re}(s)>1$.

For each integer $b\geq 2$, the Dirichlet series $$\label{eq-gb-def-2}
G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>2$.

The estimate gives $$\sum_{m=1}^n S_b(m) \ll n^2 \log n,$$ which shows that the Dirichlet series converges for $\operatorname{Re}(s)>2$.

It follows from Delange’s formula that $F_b(s)$ and $G_b(s)$ have abscissa of absolute convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively, and this can be proven directly using a more careful estimate of the function $d_b(n)$. We can also obtain the exact values of the abscissas of convergence as a corollary of our theorems on the meromorphic continuation of $F_b(s)$ and $G_b(s)$, since $F_b(s)$ has a pole at $s=1$ and $G_b(s)$ has a pole at $s=2$.

As in previous work on Dirichlet series associated to $q$-additive sequences, it is advantageous to consider the Dirichlet series $$Z_b(s) =\sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$$ obtained by differencing the coefficients of the series $F_b(s)$, setting $d_b(0)=0$. Identity in the following proposition appears in a more general form (for $q$-additive functions) in the work of Mauclaire and Murata [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88] and is stated and proved explicitly for the sum-of-digits series by Allouche and Shallit [@allouche-shallit-90]. We give a more direct proof of this result.

\[prop-differenced-zeta\] For each integer $b\geq 2$, the Dirichlet series $Z_b(s)$ has abscissa of absolute convergence $\sigma_a=1$, abscissa of conditional convergence $\sigma_c=0$, and has a meromorphic continuation to ${\mathbb{C}}$, satisfying $$\label{eq-zb-formula}
Z_b(s) = \frac{b^s-b}{b^s-1}\zeta(s).$$

For bases $b\geq 3$, we have $\lvert d_b(n)-d_b(n-1) \rvert \geq 1$ for all $n$; if $b=2$, we have $\lvert d_b(n)-d_b(n-1)\rvert \geq 1$ for at least all odd $n$. Hence $\sigma_a\geq 1$. We also have $d_b(n)-d_b(n-1)\ll \log n$, so $\sigma_a\leq 1$. The abscissa of conditional convergence $\sigma_c=0$ follows from the bound $$\sum_{m\leq n} \bigl(d_b(m)-d_b(m-1)\bigr) = d_b(n) \ll \log n.$$

The effect of adding 1 on the digit sum in base-$b$ arithmetic depends on the divisibility of $n$ by $b$; in particular, we have $$d_b(n) - d_b(n-1) = 1 - k(b-1)$$ where $k$ is the largest integer such that $b^k \mid n$. We may also express this as $$d_b(n) - d_b(n-1) = \sum_{m\mid n} \alpha(m) \beta(m/n)$$ where $$\alpha(n) = \begin{cases} 1 &\text{if $n=b^k$ for some $k$}\\ 0 &\text{otherwise} \end{cases} \qquad \beta(n)= \begin{cases} 1-b &\text{if $b\mid n$} \\ 1 &\text{if $b \nmid n$} \end{cases}.$$ Then we have, for $\operatorname{Re}(s)>1$, $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} =  \sum_{n=1}^\infty \Bigl( \sum_{m\mid n} \alpha(m) \beta(m/n) \Bigr) n^{-s}$$ Writing the right side as a product of two Dirichlet series, we have $$Z_b(s) = \sum_{n=1}^\infty \alpha(n) n^{-s} \sum_{n=1}^\infty \beta(n) n^{-s} = \sum_{n=1}^\infty b^{-ns} \sum_{n=1}^\infty (1-b) (bn)^{-s}.$$ Summing the geometric series, we obtain $$Z_b(s) = \frac{1}{1-b^{-s}} \bigl( \zeta(s) - bb^{-s} \zeta(s)\bigr) =  \frac{b^s-b}{b^s-1}\zeta(s)$$ as claimed. Equation then provides a meromorphic continuation of $Z_b(s)$ since the right side is meromorphic on ${\mathbb{C}}$.

We will obtain information about the mermorphic continuation of $F_b(s)$ and $G_b(s)$ by considering the relation between these series and the series $Z_b(s)$. For future use, we list the poles of the function $Z_b(s)$.

\[lem-poles-of-ndb-ds\] The function $Z_b(s)$ is meromorphic on ${\mathbb{C}}$, with simple poles at $s=2\pi i m / \log b$ for $m\in{\mathbb{Z}}$. The residue at each pole is $$\operatorname*{Res}\biggl(Z_b(s),s=\frac{2\pi i m}{\log b}\biggr) = - \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ In particular, at $s=0$, the function $Z_b(s)$ has a Laurent expansion beginning $$Z_b(s) = \biggl(\frac{b-1}{2\log b}\biggr)s^{-1} + \biggl(- \frac{b+1}{4} + \frac{b-1}{2\log b} \log(2\pi) \biggr) + O(s).$$

The function $(b^s-b)/(b^s-1)$ has simple poles at $s=2\pi i m /\log b$ for each $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl(\frac{b^s-b}{b^s-1},s=\frac{2\pi i m}{\log b}\biggr) = -\frac{b-1}{\log b}.$$ The Laurent expansion at $s=0$ follows from multiplying the expansions $$\frac{b^s-b}{b^s-1} = -\frac{b-1}{\log b}s^{-1} + \frac{b+1}{2} + O(s)$$ and $$\zeta(s) = -\frac{1}{2} -\frac{1}{2}\log(2\pi)s + O(s^2).$$ The function $\zeta(s)$ has only a simple pole at $s=1$, cancelled by a zero of $(b^s-b)$.

Meromorphic continuation of $F_b(s)$ and $G_b(s)$ {#sec-mero-cont}
=================================================

In this section, we show that for integers $b\geq 2$, the Dirichlet series $F_b(s)$ and $G_b(s)$ have a meromorphic continuation to ${\mathbb{C}}$ and determine the structure of the poles, proving Theorems \[thm-db\] and \[thm-sb\].

Bernoulli numbers
-----------------

Our formulas for the meromorphic continuation of $F_b(s)$ and $G_b(s)$ involve Bernoulli numbers. For standard facts about the Bernoulli numbers and their basic properties, see [@abramowitz-stegun Ch.  23]. For a thorough reference on Bernoulli numbers, their history, and their relation to zeta functions, see [@arakawa-et-al].

The Bernoulli numbers $B_k$ are the sequence of rational numbers defined by the generating function $$\label{eq-bernoulli-gf}
\frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{B_k}{k!}x^k.$$ If $x$ is a complex variable, this series converges for $\lvert x \rvert < 2\pi$.

Note that there are several competing notations for the Bernoulli numbers; with our definition, we have $B_0=1$, $B_1=-1/2$, and $B_2=1/6$. Because the function $$\frac{x}{e^x-1} + \frac{1}{2}x$$ is an even function, we find that $B_{2k+1}=0$ for all $k\geq 1$.

Power series and Dirichlet series
---------------------------------

To prove the meromorphic continuation of $F_b(s)$ and $G_b(s)$, we make use of the following classical relation between Dirichlet series and power series.

\[prop-ps-ds-relation\] Let $\sigma_c$ be the abscissa of conditional convergence of the Dirichlet series $\sum_{n=1}^\infty a_n n^{-s}$. Then $$\Gamma(s) \sum_{n=1}^\infty a_n n^{-s} = \int_0^\infty \Bigl( \sum_{n=1}^\infty a_n e^{-nx} \Bigr) x^{s-1} \, dx$$ for $\operatorname{Re}(s) >\max(\sigma_c,0)$.

See [@montgomery-vaughan eq. 5.23].

Proposition \[prop-ps-ds-relation\] allows us to translate the additive relations between the arithmetic functions $d_b(n)-d_b(n-1)$, $d_b(n)$, and $S_b(n)$, which are easily expressed in terms of power series generating functions, into relations between their associated Dirichlet series.

Meromorphic continuation of $F_b(s)$
------------------------------------

We now prove the meromorphic continuation of the Dirichlet series $F_b(s)$ by combining the relation between the Dirichlet series and power series generating functions of $d_b(n)$ with the relation between the power series generating functions of $d_b(n)$ and $d_b(n)-d_b(n-1)$.

Let $$\label{eq-px-def}
p(x) = \sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) x^n.$$ We note that $$\sum_{n=1}^\infty d_b(n)x^n = \frac{p(x)}{1-x}.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s) F_b(s) = \int_0^\infty \frac{1}{1-e^{-x}} p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. The series expansion $$\frac{x}{1-e^{-x}} = \sum_{k=0}^\infty \frac{(-1)^kB_k}{k!} x^k,$$ which follows from , holds for $\lvert x \rvert<2\pi$. Since $$\Gamma(s)Z_b(s) = \int_0^\infty p(e^{-x})x^{s-1}\, dx,$$ for $\operatorname{Re}(s)>0$, we can write $$\label{eq-first-fb-expansion}
F_b(s) = \sum_{k=0}^K \frac{(-1)^kB_k}{k!} \frac{\Gamma(s-1+k)}{\Gamma(s)} Z_b(s-1+k) + R_K(s)$$ with $$\label{eq-RK-def}
R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \Bigl(\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \Bigr) p(e^{-x})x^{s-2} \, dx.$$ Note that $$\frac{\Gamma(s-1)}{\Gamma(s)} = \frac{1}{s-1}$$ and $$\frac{\Gamma(s-1+k)}{\Gamma(s)} =  (s)(s+1)\dotsm (s+k-2).$$ Since $$\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \ll x^{K+1}$$ as $x\rightarrow 0^+$, the integral in converges and defines a holomorphic function in the region $\operatorname{Re}(s)>1-K$. From Lemma \[lem-poles-of-ndb-ds\] we know that $Z_b(s)$ has simple poles at $s=2\pi ik/\log b$ for $k\in{\mathbb{Z}}$. The $k=0$ term $$\frac{1}{s-1}Z_b(s-1)$$ has a double pole at $s=1$ with Laurent expansion beginning $$\frac{1}{s-1}Z_b(s-1) = \frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ and simple poles at each other point $s=1+2\pi i m /\log b$. Each term $$(-1)^k\frac{B_k}{k!}\prod_{j=0}^{k-2}(s+j) \cdot Z_b(s-1+k)$$ with $k=1$ or with $k$ an even integer with $k\geq 2$ has a simple pole at $s=1-k+2\pi i m/\log b$ for $m\in{\mathbb{Z}}$ with residue $$(-1)^{k}\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr) \cdot \biggl(-\frac{b-1}{\log b}\biggr)\zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ Since $K$ can be taken arbitrarily large, this proves the theorem.

Meromorphic continuation of $G_b(s)$
------------------------------------

We continue the function $G_b(s)$ to the plane in a similar fashion, using the fact that $S_b(n)$ is a double sum of the differences $d_b(n)-d_b(n-1)$ appearing in the series $Z_b(n)$.

Define $p(x)$ by . We make use of the identity of power series $$\frac{x}{(1-x)^2} p(x) = \sum_{n=1}^\infty S_b(n)x^n.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s)G_b(s) = \int_0^\infty \frac{e^{-x}}{(1-e^{-x})^2} p(e^{-x}) x^{s-1} \, dx$$ for $\operatorname{Re}(s)>2$. From and noting that $$\frac{e^{-x}}{(1-e^{-x})^2}=-\frac{d}{dx}\biggl(\frac{1}{e^x-1}\biggr),$$ we have the power series expansion $$\frac{x^2e^x}{(e^x-1)^2} = 1-\sum_{k=2}^\infty \frac{B_k}{k(k-2)!}x^k.$$ Then for a fixed integer $K\geq 2$ can write $G_b(s)$ as $$\label{eq-gbs-expansion}
G_b(s) = \frac{\Gamma(s-2)}{\Gamma(s)}Z_b(s-2) - \sum_{k=2}^K \frac{B_k}{k(k-2)!} \frac{\Gamma(s-2+k)}{\Gamma(s)} Z_b(s-2+k) + R_K(s)$$ with remainder $R_K$ given by $$R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \biggl( \frac{x^2e^{-x}}{(1-e^{-x})^2} - 1 + \sum_{k=2}^K \frac{B_k}{k(k-2)!} x^k \biggr) p(e^{-x}) x^{s-3} \, dx.$$ The function $R_K(s)$ is holomorphic for $\operatorname{Re}(s)>2-K$.

As before, we consider the poles of each term of . The first term $$\frac{1}{(s-1)(s-2)} Z_b(s-2)$$ has a double pole at $s=2$ with Laurent expansion $$\frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1}+ \dotsb,$$ a simple pole at each point $s=2+2\pi i m / \log b$ with $m\neq 0$, and a simple pole at $s=1$ with residue $(b+1)/12$. Each other term $$\frac{B_k}{k(k-2)!} \prod_{j=0}^{k-3} (s+j) \cdot Z_b(s-2+k).$$ has simple poles at $s=2-k+2\pi i m / \log b$ for all $m$.

Instead of relating $G_b(s)$ to the series $Z_b(s)$ as we did in this proof, we could have also used the relation $$\Gamma(s) G_b(s) = \int_0^\infty \frac{e^{-x}}{1-e^{-x}} \biggl(\sum_{n=1}^\infty d_b(n) e^{-xn}\biggr) x^{s-1} \, dx,$$ following the proof of Theorem \[thm-db\] to write $G_b(s)$ in terms of $F_b(s)$.

Order of $F_b(s)$ and $G_b(s)$ as meromorphic functions {#sec-meromorphic-functions}
-------------------------------------------------------

The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions on the complex plane with infinitely many poles on a left half-lattice. We now raise a further question about the analytic properties of these functions. Recall that the *order* of an entire function $f(z)$ on ${\mathbb{C}}$ is $$\rho = \inf \bigl\{ \rho\geq 0 \, \bigm\vert \, f(z) = O\bigl(\exp(\lvert z \rvert^{\rho+\varepsilon})\bigr)\text{ as $\lvert z \rvert\rightarrow\infty$} \bigr\}.$$ An entire function is of *finite order* if $\rho<\infty$.

The order of a meromorphic function is defined as the order of growth of its associated Nevanlinna characteristic function. This definition is equivalent (by [@rubel] Lemma 15.6 and Theorem on p. 91) to the following.

The *order* of a meromorphic function $f(z)$ is the infimum of $\rho\geq 0$ such that $f$ can be written as $f(z) = g(z)/h(z)$ for entire functions $g(z)$ and $h(z)$ of order $\rho$.

Many of the common Dirichlet series of analytic number theory are meromorphic (or entire) functions of order $1$. Examples include the Riemann zeta function $\zeta(s)$, the Dirichlet $L$-functions $L(s,\chi)$, and their relatives; more generally, all Dirichlet series in the Selberg class (as introduced in Selberg [@selberg-92]) are meromorphic functions of order 1.

By Proposition \[prop-differenced-zeta\], the function $Z_b(s)$ is meromorphic of order $1$. The functions $F_b(s)$ and $G_b(s)$, however, must have greater order by the following fact.

Let $f(z)$ be a meromorphic function of order $\rho$ and let $n(r,a)$ be the number of zeros of $f(z)-a$ in the disc $\lvert z \rvert<r$. Then $n(r,a)=O(r^{\rho+\varepsilon})$.

By Theorems \[thm-db\] and \[thm-sb\], the functions $F_b(s)$ and $G_b(s)$ each have $\gg r^2$ poles in the disc $\lvert z \rvert<r$. Hence we have the following corollary.

The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$.

A meromorphic function of order greater than 2 could still have only $O(r^2)$ poles in $\lvert z \rvert<r$, so without further information, we cannot deduce that $F_b(s)$ and $G_b(s)$ have order 2.

Are the functions $F_b(s)$ and $G_b(s)$ mermorphic functions of order exactly $2$?

Such a question has been answered positively in the related setting of strongly $q$-multiplicative functions: the Dirichlet series attached to such functions are entire of order exactly $2$ (see Alkauskas [@alkauskas-04]).

Meromorphic continuation of Dirichlet series for non-integer bases {#sec-non-int}
==================================================================

In this section, we consider the problem of extending the digit sums $d_b(n)$ and $S_b(n)$ from integer bases $b$ to real parameters $\beta>1$. There are a number of possible ways to do this. One natural approach concerns the notion of $\beta$-expansions introduced by Renyi [@renyi-57] and studied at length by Parry [@parry-60]. However, for non-integer values of $\beta$, the $\beta$-expansion of an integer generally has infinitely many digits, so the sum of the digits will generally be infinite. Digit sums related to a different digit expansion with respect to an irrational base were considered by Grabner and Tichy [@grabner-tichy-91].

The approach which we consider in this section is to use the formula of Delange to define the cumulative digit sum $S_\beta(n)$ for real parameters $\beta>1$, from which we can define a digit sum $d_\beta(n)$ by differencing. The resulting functions are continuous in the $\beta$-parameter.

Extension to non-integer bases by Delange’s formula
---------------------------------------------------

We begin by replacing the integer variable $b$ in Theorem \[thm-delange\], which gives a formula for $S_b(n)$ for integer bases $b\geq 2$, by a real parameter $\beta>1$.

\[def-non-integer\] For $\beta\in{\mathbb{R}}$ with $\beta>1$, define a generalized cumulative sum-of-digits function $S_\beta(n)$ by $$\label{eq-sbeta-delange-def}
S_\beta(n) := \frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n,$$ where the function $h_\beta(x)$ is defined by the Fourier series $$\label{eq-hbeta-fourier-series}
h_\beta(x) = \sum_{k=-\infty}^\infty c_\beta(k)e^{2\pi i k x}$$ with coefficients $$c_\beta(k) = -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr)$$ for $k\neq 0$ and $$c_\beta(0) = \frac{\beta-1}{2\log \beta}(\log 2\pi - 1) - \frac{\beta+1}{4}.$$

Define the generalized sum-of-digits function $d_\beta(n)$ for real $\beta>1$ by $$d_\beta(n) := S_\beta(n+1) - S_\beta(n).$$

A plot of $S_\beta(10)$ as a function of $\beta$ for $1\leq\beta\leq 15$ is shown in Figure \[Sb10-plot\]. Note that $S_\beta(n)$ is approximately constant for $\beta\geq 10$.

\[Sb10-plot\] ![A plot of $S_\beta(10)$ for $1\leq\beta\leq 15$, using terms with $\lvert k \rvert\leq 1000$ in the Fourier series for $h_\beta(x)$.](graphSb10c.pdf "fig:")

The function $h_\beta(x)$
-------------------------

In this section, we study properties of the function $h_\beta(x)$ appearing in Definition \[def-non-integer\] as a function of the variable $\beta>1$ and as a function of the variable $x$. When $\beta=b\in{\mathbb{N}}$, Delange showed that $h_b(x)$ is a continuous but everywhere non-differentiable real-valued function of $x$ with period 1.

For each fixed $\beta>1$, the function $h_\beta(x)$ is a real-valued continuous function of $x$ on ${\mathbb{R}}$.

The zeta function satisfies the bound $$\lvert\zeta(it)\rvert\ll t^{1/2+\varepsilon}$$ for $t\in{\mathbb{R}}$ (see for example [@titchmarsh-zeta eq. 5.1.3], so the Fourier coefficients of $h_\beta$ satisfy $$c_\beta(k)= -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr) \ll_\beta k^{-3/2+\varepsilon}.$$ This estimate shows that the Fourier series is absolutely and uniformly convergent for $x\in{\mathbb{R}}$, so gives a continuous function of $x$.

The function $h_\beta(x)$ is real-valued for $x\in{\mathbb{R}}$ since the Fourier coefficients $c_\beta(k)$ satisfy $\overline{c_\beta(k)} = c_\beta(-k)$.

\[fb2-plot\] ![A plot of $h_\beta(2)$ for $1\leq\beta\leq 8$, using terms with $\lvert k \rvert\leq 1000$ in the Fourier series for $h_\beta(x)$.](graphfb2a.pdf "fig:")

\[fbl2-plot\] ![A plot of $h_\beta(\log 2 / \log \beta)$ for $1\leq\beta\leq 8$, using terms with $\lvert k \rvert\leq 1000$ in the Fourier series for $h_\beta(x)$.](graphfbl2a.pdf "fig:")

A plot of $h_\beta(2)$ as a function of the real parameter $\beta$ for $1\leq \beta \leq 8$ is shown in Figure \[fb2-plot\]. From the plot, it also appears that $h_\beta$ might be non-differentiable as a function of the real parameter $\beta$.

For fixed $x\in{\mathbb{R}}$, is the function $h_\beta(x)$ everywhere non-differentiable as a function of the real variable $\beta$?

Meromorphic continuation of $G_\beta(s)$
----------------------------------------

Our proofs of the meromorphic continuation of $F_b(s)$ and $G_b(s)$ for integer bases relied on the identity $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} = \frac{b^s-b}{b^s-1}\zeta(s).$$ If for non-integer $\beta>1$ we define $$Z_\beta(s) \coloneq \sum_{n=1}^\infty \bigl( d_\beta(n) - d_\beta(n-1) \bigr) n^{-s},$$ then $Z_\beta(s)$ is *not* equal to $$\label{eq-zbeta-wrong}
\frac{\beta^s-\beta}{\beta^s-1}\zeta(s)$$ as is not an ordinary Dirichlet series. We must therefore take a different approach.

We first consider the Dirichlet series generating function $$G_\beta(s) := \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}$$ for $\beta\in{\mathbb{R}}$ with $\beta>1$. Since the coefficients satisfy $$S_\beta(n) \asymp n \log n,$$ the Dirichlet series $G_\beta(s)$ has abscissa of absolute convergence $\sigma_a=2$. We show that the function $G_\beta(s)$ can be analytically continued to a larger halfplane.

For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-2)^{-2} + c_\beta(0) (s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i k}{\log \beta}\biggr) = c_\beta(k),$$ where the numbers $c_\beta(k)$ are those in Definition \[def-non-integer\].

Using the definition of $S_\beta$, we have $$\begin{aligned}
G_\beta(s)&=\sum_{n=1}^\infty \Biggl(\frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n\Biggr)n^{-s}\\
&= -\frac{\beta-1}{2\log \beta} \zeta'(s-1) + \sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)}.\end{aligned}$$ The function $\zeta'(s-1)$ is meromorphic on ${\mathbb{C}}$ with only singularity a double pole at $s=2$ with Laurent expansion $\zeta'(s-1)=-(s-1)^{-2} + O(1)$. Using the Fourier series for $h_\beta$, we have $$\begin{aligned}
\sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) \exp\biggl(2 \pi i k \frac{\log n}{\log \beta}\biggr) n^{-(s-1)}\\ &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) n^{-(s-1 - 2\pi i k / \log \beta)}.\end{aligned}$$ This double sum is absolutely convergent, so we may exchange the sums, giving $$\sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} = \sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr).$$ If $\operatorname{Re}(s)>1$, then $$\zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr) \ll k^{1/2+\varepsilon}$$ for any $\varepsilon>0$. On any compact set $K$ in the halfplane $\operatorname{Re}(s)>1$ not containing a point $s=2+2\pi i k / \log \beta)$ for any $k\in{\mathbb{Z}}$, the sum $$\sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr)$$ is uniformly convergent on $K$; if the compact set $K$ contains a point of the form $s=2 + 2\pi i k_0 / \log \beta$, then one term of the sum has a simple pole with residue $c_\beta(k_0)$ while the remaining sum is uniformly convergent.

When $\beta\geq 2$ is an integer, we know that the function $G_\beta(s)$ has a meromorphic continuation to the entire complex plane.

For noninteger $\beta>1$, does the Dirichlet series $G_\beta(s)$ have a meromorphic continuation beyond $\operatorname{Re}(s)>1$?

Meromorphic continuation of $F_\beta(s)$
----------------------------------------

We now consider the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ for real $\beta>1$. We already know that this series has a meromorphic continuation to ${\mathbb{C}}$ when $\beta\geq 2$ is an integer. We show that for each real $\beta>1$, the Dirichlet series $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$.

The function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(c_\beta(0) +\frac{\beta-1}{2\log \beta}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i k}{\log \beta}\biggr) = \biggl(1+\frac{2\pi i k}{\log \beta}\biggr) c_\beta(k).$$

Let $$p(x) = \sum_{n=2}^\infty S_\beta(n)x^n,$$ so that $$\Gamma(s)\bigl(G_\beta(s)-S_\beta(1)\bigr) = \int_0^\infty p(e^{-x}) x^{s-1}\, dx.$$ By our definition of $d_\beta(n)$, we have $$\sum_{n=1}^\infty d_\beta(n) x^n + S_\beta(1)= (x^{-1}-1)p(x).$$ Hence by Proposition \[prop-ps-ds-relation\] we have $$\Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr) = \int_0^\infty (e^x-1)p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. Using the power series expansion Then we write $$\label{eq-fbeta-formula-1}
\Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr)  = \Gamma(s+1)\bigl(G_\beta(s+1)-S_\beta(1)\bigr) + \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx.$$ Dividing by $\Gamma(s)$ and rearranging, we obtain $$\label{eq-fbeta-gbeta-relation}
F_\beta(s) = -S_\beta(1)(s+1) + s G_\beta(s+1) + R(s)$$ where the remainder term $$R(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx$$ is holomorphic in $\operatorname{Re}(s)>0$ since $e^x-1-x\ll x^2$ as $x\rightarrow 0^+$. Since $G_\beta(s+1)$ is meromorphic in $\operatorname{Re}(s)>0$, we find that $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>0$, with poles coming from the poles of $G_\beta(s+1)$. Since $$sG_\beta(s+1) = (s-1)G_\beta(s+1) + G_\beta(s+1),$$ we find that $F_\beta(s)$ has a double pole at $s=1$ with Laurent expansion as given in the theorem. At each other point $s=1+2\pi i m / \log \beta$, $F_\beta(s)$ has a simple pole.

Meromorphic continuation of $F_\beta(s)$ to a larger halfplane would follow from continuation of $G_\beta(s)$ to a larger halfplane; in particular, by using more terms of the power series for $e^x$ in formula , we find that if $G_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c$ for some $c$, then $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c-1$.

Acknowledgements
================

The author thanks Jeffrey Lagarias for many helpful comments.

[^1]: Work partially supported by NSF grant DMS-1701576.
