The Shift Operator on the Coefficients

Introduction

The coefficient sequence $(a_n)$ of a Dirichlet series carries two elementary operators: the section at a fixed integer $m$, which retains only the coefficients at the multiples of $m$, and the division by $m$, which reads the coefficient at $mn$. On the side of the series the section is multiplication by $m^{-s}$ and the division is the part of the series supported on the multiples of $m$, rescaled by $m^{s}$. The two operators are adjoint to one another for the form of the category, the section is an isometry and the division a co-isometry, and their spectrum is the closed disk of the appropriate radius. They also give the mechanism by which the analytic continuation of a Dirichlet series is organised: the continuation of the whole is equivalent to the continuation of the $m$-divisible section together with the elementary factor $m^{-s}$.

The conventions are those of the category: $\mathcal{A}$ is the algebra of arithmetic functions, $\delta_m$ the function with $\delta_m(m)=1$ and all other values $0$, $\mathcal{H}=\ell^2$ with the form $\langle f,g\rangle=\sum_nf(n)\overline{g(n)}$, and for real $\sigma$ the Banach space $\mathcal{A}(\sigma)=\{f:\|f\|_\sigma=\sum_n|f(n)|n^{-\sigma}<\infty\}$, an algebra under convolution. The multiplication operators of the same algebra are in Multiplication Operators on an L-Function; the shift of the argument $s\mapsto s+1$ is Shift Operator on a Dirichlet Series, the next article. Nothing here reads a distance as an object.

The Two Shifts

Definitions

Definition. For $m\ge1$ the division $S_m$ and the section $D_m$ act on arithmetic functions by $$ (S_mf)(n)=f(mn),\qquad (D_mf)(n)=f(n/m)\ \text{ if } m\mid n,\ \text{ and } 0 \text{ otherwise}. $$ The operator $D_m$ retains the multiples of $m$; the operator $S_m$ reads the coefficient at $mn$ and kills the nonmultiples of $m$.

Elementary identities

Theorem. For all $m,k\ge1$, $$ S_mS_k=S_{mk},\qquad D_mD_k=D_{mk},\qquad S_mD_m=\mathrm{id},\qquad D_mS_m=P_m, $$ where $P_m$ is the orthogonal projection onto the subspace of functions supported on the multiples of $m$. In particular $D_m$ is injective with left inverse $S_m$, and $S_m$ has kernel the functions supported on the nonmultiples of $m$.

Proof. Compute on $\delta_n$: $S_m\delta_n=\delta_{n/m}$ if $m\mid n$ and $0$ otherwise, while $D_m\delta_n=\delta_{mn}$; hence $D_mD_k\delta_n=\delta_{mkn}=D_{mk}\delta_n$ and $S_mS_k\delta_n=S_{mk}\delta_n$. Further $S_mD_m\delta_n=S_m\delta_{mn}=\delta_n$, while $D_mS_m\delta_n=[m\mid n]\delta_n=P_m\delta_n$. The stated consequences are immediate.

Proposition (relation to convolution). $D_m=M_{\delta_m}$, the multiplication operator by the function $\delta_m$, and $S_m$ is its transpose in the sense that $\langle D_mf,g\rangle=\langle f,S_mg\rangle$ for all $f,g\in\mathcal{H}$. Hence $D_m^*=S_m$ and $S_m^*=D_m$, and the two shifts are mutual adjoints.

Proof. $(\delta_m*f)(n)=[m\mid n]f(n/m)=D_mf(n)$, so $D_m=M_{\delta_m}$; then $\langle D_mf,g\rangle=\sum_{n:m\mid n}f(n/m)\overline{g(n)}=\sum_kf(k)\overline{g(mk)}=\langle f,S_mg\rangle$, which is the adjoint statement.

Norms and Spectrum

The norms on the weighted spaces

Theorem. On the Banach algebra $\mathcal{A}(\sigma)$ with the norm $\|\cdot\|_\sigma$, $$ \|D_m\|_\sigma=m^{-\sigma},\qquad \|S_m\|_\sigma=m^{\sigma}. $$ On the Hilbert space $\mathcal{H}=\ell^2$ the section $D_m$ has operator norm $1$ and is an isometry, and $S_m$ is a co-isometry with $S_mS_m^*=\mathrm{id}$ and $S_m^*S_m=P_m$.

Proof. Compute $\|D_mf\|_\sigma=\sum_{n:m\mid n}|f(n/m)|n^{-\sigma}=\sum_k|f(k)|(mk)^{-\sigma}=m^{-\sigma}\|f\|_\sigma$ and $\|S_mf\|_\sigma=\sum_n|f(mn)|n^{-\sigma}=m^{\sigma}\sum_k|f(k)|k^{-\sigma}=m^{\sigma}\|f\|_\sigma$. On $\ell^2$ the first computation with $\sigma=0$ gives $\|D_mf\|_2=\|f\|_2$, so $D_m$ is an isometry; and $\|S_mf\|_2\le\|f\|_2$, the kernel of $S_m$ being the functions supported on the nonmultiples of $m$.

The spectrum

Theorem. On $\mathcal{H}=\ell^2(\mathbb{N})$, the spectrum of $D_m$ is the closed unit disk $\{|\zeta|\le1\}$, and the spectrum of $S_m$ is the same disk. On $\mathcal{A}(\sigma)$ the spectrum of $D_m$ is the closed disk of radius $m^{-\sigma}$ and the spectrum of $S_m$ is the closed disk of radius $m^{\sigma}$.

Proof. Decompose $\mathbb{N}$ into the chains $k,km,km^2,\ldots$ for the integers $k$ not divisible by $m$. Each chain is a copy of the additive monoid $\mathbb{N}_0$, and on it $D_m$ is the unilateral up-shift and $S_m$ the unilateral down-shift. Hence $D_m$ is the countable direct sum of copies of the up-shift on $\ell^2(\mathbb{N}_0)$, and $S_m$ the direct sum of copies of the down-shift. The up-shift on $\ell^2(\mathbb{N}_0)$ is an isometry whose spectrum is the closed unit disk, and a countable direct sum of copies has the same spectrum because it contains one copy as a direct summand and its norm is $1$. The rescaling to $\mathcal{A}(\sigma)$ multiplies the operators by $m^{-\sigma}$ and $m^{\sigma}$ after the substitution $f\mapsto(f(n)n^{-\sigma})$, which is an isometric identification of $\mathcal{A}(\sigma)$ with $\ell^1$ and preserves the spectral radius. The spectrum of a shift on $\ell^p(\mathbb{N}_0)$, $1\le p\le\infty$, is the closed unit disk; the standard computation is in Spectral Theory.

Corollary. The spectral radius of $D_m$ on $\mathcal{H}$ is $1$ although $D_m$ is not invertible: $D_m$ is an isometry with image the functions supported on the multiples of $m$, its inverse on that image is $S_m$, and its cokernel is the space of functions supported on the nonmultiples of $m$. The value $0$ belongs to the spectrum as a limit of the finite-dimensional approximations, not as an eigenvalue.

The Shift Equation and the Analytic Continuation

The $L$-function identity

Theorem. For every $f$ and every $m$, $$ L(D_mf,s)=m^{-s}L(f,s),\qquad L(S_mf,s)=m^{s}\sum_{\substack{n\ge1\\ m\mid n}}f(n)n^{-s}. $$ The first identity is an identity of Dirichlet series in their common half-plane; the second exhibits $L(S_mf,\cdot)$ as $m^{s}$ times the $m$-divisible section of $L(f,\cdot)$.

Proof. For the first, $\sum_n(D_mf)(n)n^{-s}=\sum_{n:m\mid n}f(n/m)n^{-s}=\sum_kf(k)(mk)^{-s}=m^{-s}L(f,s)$. For the second, $\sum_nf(mn)n^{-s}=\sum_{n:m\mid n}f(n)(n/m)^{-s}=m^{s}\sum_{n:m\mid n}f(n)n^{-s}$.

Continuation

Definition. The $m$-divisible section of $L(f,\cdot)$ is $L^{(m)}(f,s)=\sum_{n:m\mid n}f(n)n^{-s}$, so that $L(f,s)=L^{(m)}(f,s)+L^{(1-m)}(f,s)$ with the second term the sum over $n$ not divisible by $m$.

Theorem (continuation by the shift). $L^{(m)}(f,s)=m^{-s}L(S_mf,s)$ and $L(f,s)=m^{-s}L(D_mf,s)$. Consequently the meromorphic continuation of $L(f,\cdot)$ is equivalent to that of $L(D_mf,\cdot)$ for any single $m$, and the continuation of $L^{(m)}(f,\cdot)$ is equivalent to that of $L(S_mf,\cdot)$; the incompletely multiplicative factor $m^{-s}$ has no zeros or poles and neither helps nor obstructs the continuation.

Proof. The identities are the theorem above. The equivalence of the continuations is the invariance of meromorphic continuation under multiplication by the entire function $m^{-s}$; the statement about sections is the splitting $L=L^{(m)}+L^{(1-m)}$ together with the identity for $L^{(m)}$.

Remark (the Bohr spectrum). In Bohr's terminology the spectrum of the Dirichlet series $\sum_nf(n)n^{-s}$ is the set of frequencies $\{\log n:f(n)\ne0\}$, and the shift $D_m$ acts on that set by the translation $\log n\mapsto\log n+\log m$ of the frequencies under the identification $n\mapsto mn$. The down-shift is thus the operator form of the elementary translation of the frequency set by $\log m$, and it is the reason the continuation of a Dirichlet series is studied frequency by frequency; the theory is in Zeta Functions and Harmonic Analysis.

Worked Examples

Example ($m=p$, $f=\mathbf{1}$). $D_p\mathbf{1}=\delta_p$-convolution: $(D_p\mathbf{1})(n)=1$ if $p\mid n$, $0$ otherwise, and $L(D_p\mathbf{1},s)=p^{-s}\zeta(s)$; this is the classical identity $\zeta(s)=p^{-s}\zeta(s)+\sum_{p\nmid n}n^{-s}$ written with the shift.

Example (the Möbius function). $D_2\mu$ vanishes on the odd integers and equals $\mu(n/2)$ on the even ones; $L(D_2\mu,s)=2^{-s}L(\mu,s)=2^{-s}/\zeta(s)$.

Example (a normalised eigenform). For $f$ with coefficients $a_n$, $(D_pf)(n)=a_{n/p}[p\mid n]$ and $L(D_pf,s)=p^{-s}L(f,s)$; the extraction $S_p$ recovers $a_{pn}$ and the local recursion of The Hecke Operator is the composition $S_pD_p=\mathrm{id}$ together with $D_pS_p=P_p$.

Failure of the Degenerate Cases

The identities of this article fail in three degenerate configurations. First, if $m=1$ then $S_1=D_1=\mathrm{id}$ and the whole content collapses; the operators are interesting only for $m>1$. Second, the isometry $D_m$ is not surjective and the co-isometry $S_m$ is not injective; the spectral statements on $\mathcal{A}(\sigma)$ use the weight $n^{-\sigma}$, and at $\sigma=0$ the Banach algebra $\mathcal{A}(0)$ contains the unbounded-support functions for which $D_m$ has norm $1$ but is not surjective, so the "inverse" $S_m$ does not invert $D_m$. Third, the continuation statement is vacuous when $L(f,\cdot)$ has no continuation at all; the shift cannot create one, and the equivalence is an equivalence between two failures. These are the degenerate cases; they are the boundary of the half-plane of absolute convergence, where the shift equation still holds as an identity of series but the two sides have different domains of validity.

Summary

For $m\ge1$ the division $S_mf(n)=f(mn)$ and the section $D_mf(n)=f(n/m)$ for $m\mid n$, $0$ otherwise, satisfy $S_mS_k=S_{mk}$, $D_mD_k=D_{mk}$, $S_mD_m=\mathrm{id}$ and $D_mS_m=P_m$, the projection onto the multiples of $m$; $D_m$ is the multiplication operator $M_{\delta_m}$, and $D_m^*=S_m$, $S_m^*=D_m$ for the form of the category. On the weighted space $\mathcal{A}(\sigma)$ the norms are $\|D_m\|_\sigma=m^{-\sigma}$ and $\|S_m\|_\sigma=m^{\sigma}$; on $\mathcal{H}$ the section $D_m$ is an isometry of norm $1$ and the division $S_m$ is a co-isometry, the spectrum of both being the closed unit disk, and on $\mathcal{A}(\sigma)$ the disks of radius $m^{-\sigma}$ and $m^{\sigma}$. The $L$-function identities are $L(D_mf,s)=m^{-s}L(f,s)$ and $L(S_mf,s)=m^{s}L^{(m)}(f,s)$, so the continuation of $L(f,\cdot)$ is the continuation of $L(D_mf,\cdot)$ and the elementary factor $m^{-s}$, and Bohr's frequency spectrum is translated by $\log m$ under the shift.

Summary of Notation

Symbol	Meaning
$S_m$, $(S_mf)(n)=f(mn)$	Division; kills the nonmultiples of $m$
$D_m$, $(D_mf)(n)=f(n/m)$ for $m\mid n$	Section; isometry of $\mathcal{H}$
$D_m=M_{\delta_m}$	Section is convolution with $\delta_m$
$S_mD_m=\mathrm{id}$, $D_mS_m=P_m$	The fundamental identities
$D_m^=S_m$, $S_m^=D_m$	Mutual adjointness
$\\|D_m\\|_\sigma=m^{-\sigma}$, $\\|S_m\\|_\sigma=m^{\sigma}$	Norms on $\mathcal{A}(\sigma)$
$\operatorname{spec}D_m=\{\|\zeta\|\le1\}$	Spectrum on $\mathcal{H}$
$L(D_mf,s)=m^{-s}L(f,s)$	The shift equation
$L^{(m)}(f,s)=\sum_{m\mid n}f(n)n^{-s}$	The $m$-divisible section
$\{\log n:f(n)\ne0\}$	Bohr spectrum