Analytic continuation of Dirichlet series with almost periodic coefficients
Oliver Knill and John Lesieutre
Date: November 9, 2008
We consider Dirichlet series
for fixed irrational
and periodic functions
demonstrate that for Diophantine
is a natural boundary in the Taylor series
, so that the unit circle is the maximal domain of
holomorphy for the almost periodic Taylor series
. We prove that a
an abscissa of convergence
is odd and real analytic
is Diophantine. We show that if
is odd and has bounded
is of bounded Diophantine type
, the abscissa of convergence
is smaller or equal than
. Using a polylogarithm
expansion, we prove that if
is odd and real analytic and
Diophantine, then the Dirichlet series
has an analytic continuation to the entire complex plane.
AMS classification: 11M99, 30D99, 33E20
be a piecewise continuous -periodic function with
Define the function
For irrational , we call this a
Dirichlet series with almost periodic coefficients.
An example is the Clausen function, where
or the poly-logarithm, where
. Another example arises with
, the signed
distance from to the nearest integer. Obviously for
such a Dirichlet series converges uniformly to an analytic limit.
The case that is rational is less interesting, as the
following computation illustrates.
and any odd function , the zeta function
has an abscissa of convergence 0 and allows an analytic continuation
to the entire plane.
, the function
is merely a
multiple of the Riemann zeta function.
is the Hurwitz zeta function. So,
the periodic zeta function is a just finite sum of Hurwitz zeta
functions which individually allow a meromorphic continuation. Each
Hurwitz zeta function is analytic everywhere except at
, where it has
a pole of residue
: the series
abscissa of convergence 0
and allows an analytic extension to the plane.
is the case for example if
is odd, then the periodic
Dirichlet series has abscissa of convergence 0
admits an analytic continuation to the plane.
An other special case
leads to Taylor series
with . Also here, the rational case is well understood:
is rational, the function
has a meromorphic
extension to the entire plane.
The right hand side provides the meromorphic continuation of
We usually assume
because we are interested in the
growth of the random walk in the case and because if
, the abscissa of convergence is in general :
for example, where the Dirichlet series with
is a sum of the standard zeta function and the
Clausen function. The abscissa of convergence is
except for , where the abscissa drops to 0.
generally be considered for any dynamically generated sequence
, where is a homeomorphism of a
compact topological space and is a continuous function and
grows monotonically to .
Random Taylor series associated
with an ergodic transformation were considered in [3,6].
The topic has also been explored in a probabilistic setup,
where are independent random symmetric variables, in which case
is a natural boundary .
Analytic continuation questions have also been studied for other
functions: if the coefficients are generated by finite automata, a
meromorphic continuation is possible .
In this paper we focus on Taylor series and ordinary Dirichlet series.
We restrict ourselves to the case, where the dynamical system is an
on the circle. The minimality
and strict ergodicity of the system will often make the question
independent of the starting point and allow to use
techniques of Fourier analysis and the Denjoy-Koksma inequality. We
are able to make statements
if is Diophantine.
Dirichlet series can allow to get information on the growth of
the random walk
for a -measure
preserving dynamical system if
Birkhoffs ergodic theorem assures
. Similarly as the law of iterated
logarithm refines the law of large numbers in probability theory,
and Denjoy-Koksma type results provide further estimates on the growth rate
in the case of irrational rotations, one can study the growth rate for more
general dynamical systems.
The relation with algebra is as follows: if grows like
then the abscissa of convergence of the ordinary Dirichlet series is
smaller or equal to . In other words, establishing
bounds for the analyticity domain allows via Bohr's theorem to get
results on the abscissa of convergence which give upper bounds
of the growth rate. Adapting the to the situation allows to
explore different growth behavior. The algebraic concept of Dirichlet
series helps so to understand a dynamical concept.
A general Dirichlet series is of the form
this is an ordinary Dirichlet series, while
, it is a Taylor series
. We primarily restrict our attention to these two cases.
We begin by considering the easier problem of Taylor series with
almost periodic coefficients and examine the analytic continuation of
such functions beyond the unit circle. Given a non-constant periodic
function , we can look at the
problem of whether the Taylor series
can be analytically continued beyond the unit circle.
Note that all these functions have radius of convergence
We have already seen in the introduction that if
is rational, the function has a meromorphic
extension to the entire plane. There is an other case where
analytic continuation can be established immediately:
If is a trigonometric polynomial and is arbitrary,
then has a meromorphic extension to the entire plane.
, it is enough to
verify this for
, in which case the series
On the other hand, if infinitely many of the Fourier coefficients for
are nonzero, analytic continuation may not be possible.
Fix . Assume is in for ,
and that all Fourier coefficients of
are nonzero and that is of Diophantine type . Then the
almost periodic Taylor series
can not be continued beyond the unit circle.
and consider the radial limit
The latter sum converges at
, as by the Diophantine
condition, the denominator is bounded below by
, while the
is bounded above by
by the differentiability
. Because the term
it follows that this radial limit is infinite for all
does not admit an analytic continuation to any larger set.
This result is related to a construction of Goursat, which shows
that for any domain
, there exists a function which
as a maximal domain of analyticity [14
]. In contrary to lacunary Taylor series like
which have the unit circle as a natural boundary too, all Taylor
coefficients are in general nonzero in the Taylor series
is defined also outside the unit circle. The subharmonic function
has a Riesz measure
supported on the unit circle which
is dense pure point.
is by no means
necessary. For example, any nonconstant step function
even continuous, but by Szego's theorem (see [14
for power series with finitely many distinct coefficients which do not eventually repeat
can not be analytically extended beyond the unit disk.
The condition that all Fourier coefficients are nonzero may be relaxed
to the assumption that the set of
ranges over the indices of nonzero Fourier coefficients, is dense in
As the last remark may suggest, trigonometric polynomials are not
the only functions whose associated series allow an analytic
continuation beyond the unit circle.
be an arbitrary closed set on the unit
circle. There exists an almost periodic Taylor series which has an
analytic continuation to
but not to any
point of .
, and let
be Diophantine of any type
the unit circle we have
, this sum converges
uniformly on a closed ball around
which does not intersect
(since the denominators of the non-vanishing terms are uniformly
bounded), and thus has an analytic neighborhood around such
. Any point in
lies in a compact neighborhood of
such a point. On the other hand, by the arguments of the preceding
lemma, analytic continuation is not possible in
Cahen's formula for the abscissa of convergence of an ordinary
does not converge .
We will compute the abscissa of convergence
for two classes of functions and so derive bounds on the
random walks which are stronger than those implied by the
The first situation applies to real analytic , where we can invoke the
cohomology theory of cocycles over irrational rotations:
If is real analytic,
Diophantine, then the series for
is analytic for
. In other words, the
abscissa of convergence is 0.
an additive coboundary: the Diophantine
implies that the real analytic function
does not converge,
but stays bounded in absolute value by
Cahen's formula immediately implies that
Lets give a second proof without Cahen's formula.
, we have
The sum is bounded for
. The function
is analytic in
as the limit of a
sequence of analytic functions which converge uniformly on a compact
subset of the right half plane. The uniform convergence follows from
Bohr's theorem (see [4
], Theorem 52).
We do not know what happens for irrational which are not Diophantine
except if is a trigonometric polynomial:
If is a trigonometric polynomial of period 1 with
and is an arbitrary irrational number, then the abscissa of convergence
is a trigonometric polynomial, then
is a coboundary for
In the case of the Clausen function for example and
, we have
. It follows that the series
converges for all trigonometric polynomials
, for all
A special case is the zeta function
which is can be written as
is the polylogarithm
. Integral representations like
(see i.e. [12
]) show the analytic continuation for
It follows that the function
has an analytic continuation to the
entire complex plane for all irrational
is a trigonometric
polynomial. We will use polylogarithms later.
The result in the last section had been valid if satisfies
some Diophantine condition and is real analytic. If the function is
only required to be of bounded variation, then the abscissa of convergence
can be estimated. Lets first recall some definitions:
A real number for
which there exist and satisfying
for all rational is called Diophantine of type . The set of
real numbers of type have full Lebesgue measure for all so that
also the intersection
of all these types have. The variation of a function
, where the supremum is taken over all
If is Diophantine of type and if is of bounded
then the series for
has an abscissa
The Denjoy-Koksma inequality
) implies that
, the sum
Cahen's formula for the abscisse of convergence gives
A weaker result could be obtained directly, without Cahen's formula.
are summable. Then
To compare: if are independent, identically distributed
random variables with mean 0 and finite variance , then
grows by the law of iterated logarithm like
the zeta function converges with probability for
. The following
reformulation of the law of iterated logarithm follows directly from Cahen's formula:
(Law of iterated logarithm)
If is a Bernoulli shift such that
independent identically distributed random variables with finite nonzero variance,
then the Dirichlet series
has the abscissa of convergence .
We do not have examples in the almost periodic case yet, where the abscissa of convergence is
strictly between 0 and like
The Lerch transcendent is defined as
, we get the Lerch zeta function
In the special case we have the polylogarithm
For and general , we have the Hurwitz zeta function, which becomes for
the Riemann zeta function. The following two Lemmas are standard
, the claim is equivalent to
changes this to
The improper integral is
analytic in because
, we have
The Lerch transcendent has for fixed
an analytic continuation to the entire -plane.
In every bounded region in the complex plane, there is a constant such that
For any bounded region
in the complex plane we can find a constant
Similarly, we can estimate
for any integer
allows us to define
: first define
by the recursion (
). Then use the identity (
again to define it in the strip
, then in the strip
The Lerch transcendent is often written as a function of three variables:
it satisfies the functional equation
]. Using the functional equation
to do the analytic continuation is less obvious.
One of the main results in this paper is the following theorem:
For all Diophantine and every real analytic periodic function
, the series
has an analytic continuation to the entire complex plane.
The Fourier expansion of
It produces a polylog expansion of
is real analytic, there exists
For any periodic function , the series
produces for fixed in the region of convergence a new periodic
function in . For fixed it is a
Dirichlet series in . The Clausen function is
and the polylogarithm is
. We may then
consider a new almost periodic Dirichlet series generated by this
function, defined by
The following commutation formula can be useful to extend the
domain, where Dirichlet series are defined:
, we have
which we regard as a periodic function in
. It is continuous in
if evaluated for fixed
Then where our sum converges absolutely (which holds at least for
In fact, the double sum can be expressed as a single sum using the
divisor sum function
We can formulate this as follows: for , we have
For example, evaluating the almost periodic Dirichlet series for the
at is the
same as evaluating the almost periodic Dirichlet series of the periodic
and evaluating it
at . But since is of bounded variation, the Dirichlet
an analytic continuation to all
for example is
defined if is Diophantine of type .
The commutation formula
allows us to define
even so is not in .
The commutation formula generalizes.
The irrational rotation
can be replaced by a
general topological dynamical system on
In the particular case that is the Clausen function ,
the expression () is itself a Fourier series, whose
coefficients are the divisor function .
the value of as a function of is the Fourier transform on
of the multiplicative arithmetic function
For for example, we get
if is the function with Fourier coefficients . In that
case, the Fourier coefficients of
the function has the multiplicative function
(called the index of ) as coefficients.
For , we have
for odd integer , the function
is a Bernoulli polynomial.
For , we get
where is the number of divisors of . These sums converge
. If is a positive odd integer,
is a Bernoulli polynomial (e.g. for , we have
). For (and still ), we have
, regarded as periodic functions of
, may be related by an identity of Ramanujan .
Applying Parseval's theorem to these Fourier series, one can deduce
If fails to be of bounded variation, the previous results do not apply. Still,
there can be boundedness for the Dirichlet series if
appears in the context of KAM theory and was the starting point of our investigations.
is the determinant of a
truncated diagonal matrix representing the Fourier transform of the Laplacian
The function has mean 0 but it is not bounded and
therefore has unbounded variation. Numerical experiments indicate
however that at least for many of constant type,
. We can only show:
and also for general
by the classical Denjoy-Koksma inequality. Choose
The finite orbit
never hits that set and the sum is the same when
with the untruncated
. We get therefore
The rest of the proof is the same as for the classical Denjoy-Koksma
The Denjoy-Koksma inequality is treated in [5,1].
Here is the exposition as found in .
], Lemma 12).
is a periodic approximation of
To see this, divide the circle into
intervals centered at the points
. These intervals have length
and each interval
contains exactly one point of the finite orbit
Renumber the points so that
By the intermediate value theorem, there exists a Riemann sum
for which every
is in an interval
(choosing the point
an lower and
gives an upper bound).
is of constant type then
is bounded and
is Diophantine of type
The general fact
deals with the first term.
The second term is estimated as follows: from
, we have
One knows also
if the continued
eventually. See .
We were able to get entire functions
for rational and Diophantine . What
happens for Liouville if is not a trigonometric polynomial?
What happens for more general ?
For every and we get a function
and Diophantine , where
we observe a self-similar nature of the graph.
Is the Hausdorff dimension of the graph of not an integer?
One can look at the problem for more general dynamical systems. Here is an example:
for periodic Dirichlet series generated by an ergodic translation on a two-dimensional
torus with a vector
are irrational, the series is
In the case , this leads to the Denjoy-Koksma type problem to estimate the growth
rate of the random walk
which is more difficult due to the lack of a natural continued
fraction expansion in two dimensions. In a concrete example like
, the question is, how fast the sum
grows with irrational
. Numerical experiments
indicate subpolynomial growth that
would hold for
Diophantine and suggest the abscissa of convergence of the
. This series is of some historical interest since Riemann knew
in 1861 (at least according to Weierstrass ) that
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Asymptotics of a dynamical random walk in a random scenery: I. law of
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Judy Halchin and Karl Petersen.
Random power series generated by ergodic transformations.
Transactions of the American Mathematical Society, 297
G.H. Hardy and M. Riesz.
The general theory of Dirichlet's series.
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Sur la conjugaison différentiable des difféomorphismes du
cercle à des rotations.
Analytic continuation of random series.
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The Riemann Zeta-Function.
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M.Mendés France J.-P. Allouche and J. Peyriére.
Automatic Dirichlet series.
Journal of Number Theory, 81:359-373, 2000.
Metal-insulator transition for the almost Mathieu operator.
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Some random series of functions.
D.C. Heath and Co, Rahtheon Education Co, Lexington, MA, 1968.
Polylogarithms and Riemann's -function.
Physical Review E, 56, 1997.
Note sur la function
Acta Math., 11:19-24, 1887.
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B. M. Wilson.
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