# Analytic continuation of Dirichlet series with almost periodic coefficients

Oliver Knill and John Lesieutre

Date: November 9, 2008

### Abstract:

We consider Dirichlet series for fixed irrational and periodic functions . We demonstrate that for Diophantine and smooth , the line is a natural boundary in the Taylor series case , so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series . We prove that a Dirichlet series has an abscissa of convergence if is odd and real analytic and is Diophantine. We show that if is odd and has bounded variation and 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 is Diophantine, then the Dirichlet series has an analytic continuation to the entire complex plane.

AMS classification: 11M99, 30D99, 33E20

# Introduction

Let be a piecewise continuous -periodic function with Fourier expansion . 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.

For periodic and any odd function , the zeta function has an abscissa of convergence 0 and allows an analytic continuation to the entire plane.

Proof. Write

where 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 has the abscissa of convergence 0 and allows an analytic extension to the plane. So, if , which is the case for example if is odd, then the periodic Dirichlet series has abscissa of convergence 0 and admits an analytic continuation to the plane.

When , the function is merely a multiple of the Riemann zeta function.

An other special case leads to Taylor series with . Also here, the rational case is well understood:

If is rational, the function has a meromorphic extension to the entire plane.

Proof. If define . Then

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 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.

Zeta functions can more 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 the line is a natural boundary [11]. Analytic continuation questions have also been studied for other functions: if the coefficients are generated by finite automata, a meromorphic continuation is possible [8].

In this paper we focus on Taylor series and ordinary Dirichlet series. We restrict ourselves to the case, where the dynamical system is an irrational rotation 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.

# Almost periodic Taylor series

A general Dirichlet series is of the form

For this is an ordinary Dirichlet series, while for , it is a Taylor series with . 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 because .

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:

Lemma 1   If is a trigonometric polynomial and is arbitrary, then has a meromorphic extension to the entire plane.

Proof. Since , it is enough to verify this for , in which case the series sums to .

On the other hand, if infinitely many of the Fourier coefficients for are nonzero, analytic continuation may not be possible.

Proposition 2   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.

Proof. Write

Fix some and consider the radial limit

The latter sum converges at , as by the Diophantine condition, the denominator is bounded below by , while the numerator is bounded above by for by the differentiability assumption on . Because the term diverges for , it follows that this radial limit is infinite for all . Consequently, does not admit an analytic continuation to any larger set.

Remark 1   This result is related to a construction of Goursat, which shows that for any domain in , there exists a function which has 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 of Proposition .

Remark 2   The function

is defined also outside the unit circle. The subharmonic function has a Riesz measure supported on the unit circle which is dense pure point.

Remark 3   The requirement is by no means necessary. For example, any nonconstant step function is not even continuous, but by Szego's theorem (see [14]) for power series with finitely many distinct coefficients which do not eventually repeat periodically, can not be analytically extended beyond the unit disk.

Remark 4   The condition that all Fourier coefficients are nonzero may be relaxed to the assumption that the set of , where 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.

Proposition 3   Let 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 .

Proof. Set

and let , and let be Diophantine of any type . Inside the unit circle we have

For any for which , 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 itself.

# Ordinary Dirichlet series

Cahen's formula for the abscissa of convergence of an ordinary Dirichlet series is

if does not converge [4]. 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 Denjoy-Koksma inequality.

The first situation applies to real analytic , where we can invoke the cohomology theory of cocycles over irrational rotations:

Proposition 4   If is real analytic, and is Diophantine, then the series for converges and is analytic for . In other words, the abscissa of convergence is 0.

Proof. Since , is an additive coboundary: the Diophantine property of implies that the real analytic function

solves

Because 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. For , we have

The sum is bounded for because so that . 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:

Lemma 5   If is a trigonometric polynomial of period 1 with and is an arbitrary irrational number, then the abscissa of convergence of is 0.

Proof. If is a trigonometric polynomial, then is a coboundary for every irrational because 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 irrational and all .

Remark 5   A special case is the zeta function

which is can be written as , where is the polylogarithm. Integral representations like

(see i.e. [12]) show the analytic continuation for rsp. . It follows that the function has an analytic continuation to the entire complex plane for all irrational if is a trigonometric polynomial. We will use polylogarithms later.

# The bounded variation case

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 is , where the supremum is taken over all partitions of .

Proposition 6   If is Diophantine of type and if is of bounded variation with , then the series for has an abscissa of convergence .

Proof. The Denjoy-Koksma inequality (see Lemma ) implies that for any , the sum satisfies the estimate , with . Cahen's formula for the abscisse of convergence gives

Remark 6   A weaker result could be obtained directly, without Cahen's formula. Choose such that

are summable. Then

is summable.

To compare: if are independent, identically distributed random variables with mean 0 and finite variance , then grows by the law of iterated logarithm like and 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 produces 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 .

# Analytic continuation

The Lerch transcendent is defined as

For , 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 (see [7]).

Lemma 7   For and , there is an integral representation

For fixed , this is analytic in for . For fixed , it is analytic in for .

Proof. By expanding , the claim is equivalent to

A substitution changes this to

Now use .

The improper integral is analytic in because is for and for and for , we have

Lemma 8   The Lerch transcendent has for fixed and an analytic continuation to the entire -plane. In every bounded region in the complex plane, there is a constant such that and .

Proof. For any bounded region in the complex plane we can find a constant such that

Similarly, we can estimate for any integer . The identity

 (1)

allows us to define for : first define in by the recursion (). Then use the identity () again to define it in the strip , then in the strip , etc.

Remark 7   The Lerch transcendent is often written as a function of three variables:

it satisfies the functional equation

See [13]. Using the functional equation to do the analytic continuation is less obvious.

One of the main results in this paper is the following theorem:

Theorem 9   For all Diophantine and every real analytic periodic function satisfying , the series has an analytic continuation to the entire complex plane.

Proof. The Fourier expansion of evaluated at gives

It produces a polylog expansion of

with . Because is real analytic, there exists such that . Since and is Diophantine, so that and

# A commutation formula

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:

Lemma 10 (Commutation formula)   For and , we have

Proof. 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 periodic function at is the same as evaluating the almost periodic Dirichlet series of the periodic function and evaluating it at . But since is of bounded variation, the Dirichlet series has an analytic continuation to all and for example is defined if is Diophantine of type . The commutation formula allows us to define as , even so is not in .

Remark 8   The commutation formula generalizes. The irrational rotation on 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 . We have and : 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 and for odd integer , the function

is a Bernoulli polynomial.

For , we get

where is the number of divisors of . These sums converge absolutely for . If is a positive odd integer, is a Bernoulli polynomial (e.g. for , we have ). For (and still ), we have

The functions , regarded as periodic functions of , may be related by an identity of Ramanujan [15]. Applying Parseval's theorem to these Fourier series, one can deduce

# Unbounded variation

If fails to be of bounded variation, the previous results do not apply. Still, there can be boundedness for the Dirichlet series if . The example appears in the context of KAM theory and was the starting point of our investigations. The product is the determinant of a truncated diagonal matrix representing the Fourier transform of the Laplacian on .

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:

Proposition 11   If is Diophantine of type , then for ,

and converges for .

Proof. because

which gives . Define . Now

for all and also for general by the classical Denjoy-Koksma inequality. Choose . Then the set . The finite orbit never hits that set and the sum is the same when replacing with the untruncated . We get therefore

The rest of the proof is the same as for the classical Denjoy-Koksma inequality.

The Denjoy-Koksma inequality is treated in [5,1]. Here is the exposition as found in [9].

Lemma 12 (Jitomirskaja's formulation of Denjoy-Koksma)   Assume is Diophantine of type and is of bounded variation and . Then satisfies

If is of constant type and is of bounded variation and , then .

Proof. (See [9], Lemma 12). If is a periodic approximation of , then

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 is in . By the intermediate value theorem, there exists a Riemann sum for which every is in an interval (choosing the point gives an lower and gives an upper bound). If .
Now, if and , then

because .
If is of constant type then is bounded and implies .
If is Diophantine of type , then and which implies and so

The general fact deals with the first term. The second term is estimated as follows: from , we have and .

One knows also if the continued fraction expansion of satisfies eventually. See [2].

# Questions

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 . For

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 , where 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 Dirichlet series

is . This series is of some historical interest since Riemann knew in 1861 (at least according to Weierstrass [10]) that is nowhere differentiable.

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2008-11-09