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Previous set of notes: Notes 1. Next set of notes: Notes 3.

In Exercise 5 (and Lemma 1) of 246A Notes 4 we already observed some links between complex analysis on the disk (or annulus) and Fourier series on the unit circle:

  • (i) Functions {f} that are holomorphic on a disk {\{ |z| < R \}} are expressed by a convergent Fourier series (and also Taylor series) {f(re^{i\theta}) = \sum_{n=0}^\infty r^n a_n e^{in\theta}} for {0 \leq r < R} (so in particular {a_n = \frac{1}{n!} f^{(n)}(0)}), where

    \displaystyle  \limsup_{n \rightarrow +\infty} |a_n|^{1/n} \leq \frac{1}{R}; \ \ \ \ \ (1)

    conversely, every infinite sequence {(a_n)_{n=0}^\infty} of coefficients obeying (1) arises from such a function {f}.
  • (ii) Functions {f} that are holomorphic on an annulus {\{ r_- < |z| < r_+ \}} are expressed by a convergent Fourier series (and also Laurent series) {f(re^{i\theta}) = \sum_{n=-\infty}^\infty r^n a_n e^{in\theta}}, where

    \displaystyle  \limsup_{n \rightarrow +\infty} |a_n|^{1/n} \leq \frac{1}{r_+}; \limsup_{n \rightarrow -\infty} |a_n|^{1/|n|} \leq \frac{1}{r_-}; \ \ \ \ \ (2)

    conversely, every doubly infinite sequence {(a_n)_{n=-\infty}^\infty} of coefficients obeying (2) arises from such a function {f}.
  • (iii) In the situation of (ii), there is a unique decomposition {f = f_1 + f_2} where {f_1} extends holomorphically to {\{ z: |z| < r_+\}}, and {f_2} extends holomorphically to {\{ z: |z| > r_-\}} and goes to zero at infinity, and are given by the formulae

    \displaystyle  f_1(z) = \sum_{n=0}^\infty a_n z^n = \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z}\ dw

    where {\gamma} is any anticlockwise contour in {\{ z: |z| < r_+\}} enclosing {z}, and and

    \displaystyle  f_2(z) = \sum_{n=-\infty}^{-1} a_n z^n = - \frac{1}{2\pi i} \int_\gamma \frac{f(w)}{w-z}\ dw

    where {\gamma} is any anticlockwise contour in {\{ z: |z| > r_-\}} enclosing {0} but not {z}.

This connection lets us interpret various facts about Fourier series through the lens of complex analysis, at least for some special classes of Fourier series. For instance, the Fourier inversion formula {a_n = \frac{1}{2\pi} \int_0^{2\pi} f(e^{i\theta}) e^{-in\theta}\ d\theta} becomes the Cauchy-type formula for the Laurent or Taylor coefficients of {f}, in the event that the coefficients are doubly infinite and obey (2) for some {r_- < 1 < r_+}, or singly infinite and obey (1) for some {R > 1}.

It turns out that there are similar links between complex analysis on a half-plane (or strip) and Fourier integrals on the real line, which we will explore in these notes.

We first fix a normalisation for the Fourier transform. If {f \in L^1({\bf R})} is an absolutely integrable function on the real line, we define its Fourier transform {\hat f: {\bf R} \rightarrow {\bf C}} by the formula

\displaystyle  \hat f(\xi) := \int_{\bf R} f(x) e^{-2\pi i x \xi}\ dx. \ \ \ \ \ (3)

From the dominated convergence theorem {\hat f} will be a bounded continuous function; from the Riemann-Lebesgue lemma it also decays to zero as {\xi \rightarrow \pm \infty}. My choice to place the {2\pi} in the exponent is a personal preference (it is slightly more convenient for some harmonic analysis formulae such as the identities (4), (5), (6) below), though in the complex analysis and PDE literature there are also some slight advantages in omitting this factor. In any event it is not difficult to adapt the discussion in this notes for other choices of normalisation. It is of interest to extend the Fourier transform beyond the {L^1({\bf R})} class into other function spaces, such as {L^2({\bf R})} or the space of tempered distributions, but we will not pursue this direction here; see for instance these lecture notes of mine for a treatment.

Exercise 1 (Fourier transform of Gaussian) If {a} is a coplex number with {\mathrm{Re} a>0} and {f} is the Gaussian function {f(x) := e^{-\pi a x^2}}, show that the Fourier transform {\hat f} is given by the Gaussian {\hat f(\xi) = a^{-1/2} e^{-\pi \xi^2/a}}, where we use the standard branch for {a^{-1/2}}.

The Fourier transform has many remarkable properties. On the one hand, as long as the function {f} is sufficiently “reasonable”, the Fourier transform enjoys a number of very useful identities, such as the Fourier inversion formula

\displaystyle  f(x) = \int_{\bf R} \hat f(\xi) e^{2\pi i x \xi} d\xi, \ \ \ \ \ (4)

the Plancherel identity

\displaystyle  \int_{\bf R} |f(x)|^2\ dx = \int_{\bf R} |\hat f(\xi)|^2\ d\xi, \ \ \ \ \ (5)

and the Poisson summation formula

\displaystyle  \sum_{n \in {\bf Z}} f(n) = \sum_{k \in {\bf Z}} \hat f(k). \ \ \ \ \ (6)

On the other hand, the Fourier transform also intertwines various qualitative properties of a function {f} with “dual” qualitative properties of its Fourier transform {\hat f}; in particular, “decay” properties of {f} tend to be associated with “regularity” properties of {\hat f}, and vice versa. For instance, the Fourier transform of rapidly decreasing functions tend to be smooth. There are complex analysis counterparts of this Fourier dictionary, in which “decay” properties are described in terms of exponentially decaying pointwise bounds, and “regularity” properties are expressed using holomorphicity on various strips, half-planes, or the entire complex plane. The following exercise gives some examples of this:

Exercise 2 (Decay of {f} implies regularity of {\hat f}) Let {f \in L^1({\bf R})} be an absolutely integrable function.
  • (i) If {f} has super-exponential decay in the sense that {f(x) \lesssim_{f,M} e^{-M|x|}} for all {x \in {\bf R}} and {M>0} (that is to say one has {|f(x)| \leq C_{f,M} e^{-M|x|}} for some finite quantity {C_{f,M}} depending only on {f,M}), then {\hat f} extends uniquely to an entire function {\hat f : {\bf C} \rightarrow {\bf C}}. Furthermore, this function continues to be defined by (3).
  • (ii) If {f} is supported on a compact interval {[a,b]} then the entire function {\hat f} from (i) obeys the bounds {\hat f(\xi) \lesssim_f \max( e^{2\pi a \mathrm{Im} \xi}, e^{2\pi b \mathrm{Im} \xi} )} for {\xi \in {\bf C}}. In particular, if {f} is supported in {[-M,M]} then {\hat f(\xi) \lesssim_f e^{2\pi M |\mathrm{Im}(\xi)|}}.
  • (iii) If {f} obeys the bound {f(x) \lesssim_{f,a} e^{-2\pi a|x|}} for all {x \in {\bf R}} and some {a>0}, then {\hat f} extends uniquely to a holomorphic function {\hat f} on the horizontal strip {\{ \xi: |\mathrm{Im} \xi| < a \}}, and obeys the bound {\hat f(\xi) \lesssim_{f,a} \frac{1}{a - |\mathrm{Im}(\xi)|}} in this strip. Furthermore, this function continues to be defined by (3).
  • (iv) If {f} is supported on {[0,+\infty)} (resp. {(-\infty,0]}), then there is a unique continuous extension of {\hat f} to the lower half-plane {\{ \xi: \mathrm{Im} \xi \leq 0\}} (resp. the upper half-plane {\{ \xi: \mathrm{Im} \xi \geq 0 \}} which is holomorphic in the interior of this half-plane, and such that {\hat f(\xi) \rightarrow 0} uniformly as {\mathrm{Im} \xi \rightarrow -\infty} (resp. {\mathrm{Im} \xi \rightarrow +\infty}). Furthermore, this function continues to be defined by (3).
Hint: to establish holomorphicity in each of these cases, use Morera’s theorem and the Fubini-Tonelli theorem. For uniqueness, use analytic continuation, or (for part (iv)) the Cauchy integral formula.

Later in these notes we will give a partial converse to part (ii) of this exercise, known as the Paley-Wiener theorem; there are also partial converses to the other parts of this exercise.

From (3) we observe the following intertwining property between multiplication by an exponential and complex translation: if {\xi_0} is a complex number and {f: {\bf R} \rightarrow {\bf C}} is an absolutely integrable function such that the modulated function {f_{\xi_0}(x) := e^{2\pi i \xi_0 x} f(x)} is also absolutely integrable, then we have the identity

\displaystyle  \widehat{f_{\xi_0}}(\xi) = \hat f(\xi - \xi_0) \ \ \ \ \ (7)

whenever {\xi} is a complex number such that at least one of the two sides of the equation in (7) is well defined. Thus, multiplication of a function by an exponential weight corresponds (formally, at least) to translation of its Fourier transform. By using contour shifting, we will also obtain a dual relationship: under suitable holomorphicity and decay conditions on {f}, translation by a complex shift will correspond to multiplication of the Fourier transform by an exponential weight. It turns out to be possible to exploit this property to derive many Fourier-analytic identities, such as the inversion formula (4) and the Poisson summation formula (6), which we do later in these notes. (The Plancherel theorem can also be established by complex analytic methods, but this requires a little more effort; see Exercise 8.)

The material in these notes is loosely adapted from Chapter 4 of Stein-Shakarchi’s “Complex Analysis”.

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In Notes 2, the Riemann zeta function {\zeta} (and more generally, the Dirichlet {L}-functions {L(\cdot,\chi)}) were extended meromorphically into the region {\{ s: \hbox{Re}(s) > 0 \}} in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are known as the non-trivial zeroes of {\zeta}, and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The {\zeta} function obeys the trivial functional equation

\displaystyle \zeta(\overline{s}) = \overline{\zeta(s)} \ \ \ \ \ (1)

 

for all {s} in its domain of definition. Indeed, as {\zeta(s)} is real-valued when {s} is real, the function {\zeta(s) - \overline{\zeta(\overline{s})}} vanishes on the real line and is also meromorphic, and hence vanishes everywhere. Similarly one has the functional equation

\displaystyle \overline{L(s, \chi)} = L(\overline{s}, \overline{\chi}). \ \ \ \ \ (2)

 

From these equations we see that the zeroes of the zeta function are symmetric across the real axis, and the zeroes of {L(\cdot,\chi)} are the reflection of the zeroes of {L(\cdot,\overline{\chi})} across this axis.

It is a remarkable fact that these functions obey an additional, and more non-trivial, functional equation, this time establishing a symmetry across the critical line {\{ s: \hbox{Re}(s) = \frac{1}{2} \}} rather than the real axis. One consequence of this symmetry is that the zeta function and {L}-functions may be extended meromorphically to the entire complex plane. For the zeta function, the functional equation was discovered by Riemann, and reads as follows:

Theorem 1 (Functional equation for the Riemann zeta function) The Riemann zeta function {\zeta} extends meromorphically to the entire complex plane, with a simple pole at {s=1} and no other poles. Furthermore, one has the functional equation

\displaystyle \zeta(s) = \alpha(s) \zeta(1-s) \ \ \ \ \ (3)

 

or equivalently

\displaystyle \zeta(1-s) = \alpha(1-s) \zeta(s) \ \ \ \ \ (4)

 

for all complex {s} other than {s=0,1}, where {\alpha} is the function

\displaystyle \alpha(s) := 2^s \pi^{s-1} \sin( \frac{\pi s}{2}) \Gamma(1-s). \ \ \ \ \ (5)

 

Here {\cos(z) := \frac{e^z + e^{-z}}{2}}, {\sin(z) := \frac{e^{-z}-e^{-z}}{2i}} are the complex-analytic extensions of the classical trigionometric functions {\cos(x), \sin(x)}, and {\Gamma} is the Gamma function, whose definition and properties we review below the fold.

The functional equation can be placed in a more symmetric form as follows:

Corollary 2 (Functional equation for the Riemann xi function) The Riemann xi function

\displaystyle \xi(s) := \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s) \ \ \ \ \ (6)

 

is analytic on the entire complex plane {{\bf C}} (after removing all removable singularities), and obeys the functional equations

\displaystyle \xi(\overline{s}) = \overline{\xi(s)}

and

\displaystyle \xi(s) = \xi(1-s). \ \ \ \ \ (7)

 

In particular, the zeroes of {\xi} consist precisely of the non-trivial zeroes of {\zeta}, and are symmetric about both the real axis and the critical line. Also, {\xi} is real-valued on the critical line and on the real axis.

Corollary 2 is an easy consequence of Theorem 1 together with the duplication theorem for the Gamma function, and the fact that {\zeta} has no zeroes to the right of the critical strip, and is left as an exercise to the reader (Exercise 19). The functional equation in Theorem 1 has many proofs, but most of them are related in on way or another to the Poisson summation formula

\displaystyle \sum_n f(n) = \sum_m \hat f(2\pi m) \ \ \ \ \ (8)

 

(Theorem 34 from Supplement 2, at least in the case when {f} is twice continuously differentiable and compactly supported), which can be viewed as a Fourier-analytic link between the coarse-scale distribution of the integers and the fine-scale distribution of the integers. Indeed, there is a quick heuristic proof of the functional equation that comes from formally applying the Poisson summation formula to the function {1_{x>0} \frac{1}{x^s}}, and noting that the functions {x \mapsto \frac{1}{x^s}} and {\xi \mapsto \frac{1}{\xi^{1-s}}} are formally Fourier transforms of each other, up to some Gamma function factors, as well as some trigonometric factors arising from the distinction between the real line and the half-line. Such a heuristic proof can indeed be made rigorous, and we do so below the fold, while also providing Riemann’s two classical proofs of the functional equation.

From the functional equation (and the poles of the Gamma function), one can see that {\zeta} has trivial zeroes at the negative even integers {-2,-4,-6,\dots}, in addition to the non-trivial zeroes in the critical strip. More generally, the following table summarises the zeroes and poles of the various special functions appearing in the functional equation, after they have been meromorphically extended to the entire complex plane, and with zeroes classified as “non-trivial” or “trivial” depending on whether they lie in the critical strip or not. (Exponential functions such as {2^{s-1}} or {\pi^{-s}} have no zeroes or poles, and will be ignored in this table; the zeroes and poles of rational functions such as {s(s-1)} are self-evident and will also not be displayed here.)

Function Non-trivial zeroes Trivial zeroes Poles
{\zeta(s)} Yes {-2,-4,-6,\dots} {1}
{\zeta(1-s)} Yes {3,5,\dots} {0}
{\sin(\pi s/2)} No Even integers No
{\cos(\pi s/2)} No Odd integers No
{\sin(\pi s)} No Integers No
{\Gamma(s)} No No {0,-1,-2,\dots}
{\Gamma(s/2)} No No {0,-2,-4,\dots}
{\Gamma(1-s)} No No {1,2,3,\dots}
{\Gamma((1-s)/2)} No No {1,3,5,\dots}
{\xi(s)} Yes No No

Among other things, this table indicates that the Gamma and trigonometric factors in the functional equation are tied to the trivial zeroes and poles of zeta, but have no direct bearing on the distribution of the non-trivial zeroes, which is the most important feature of the zeta function for the purposes of analytic number theory, beyond the fact that they are symmetric about the real axis and critical line. In particular, the Riemann hypothesis is not going to be resolved just from further analysis of the Gamma function!

The zeta function computes the “global” sum {\sum_n \frac{1}{n^s}}, with {n} ranging all the way from {1} to infinity. However, by some Fourier-analytic (or complex-analytic) manipulation, it is possible to use the zeta function to also control more “localised” sums, such as {\sum_n \frac{1}{n^s} \psi(\log n - \log N)} for some {N \gg 1} and some smooth compactly supported function {\psi: {\bf R} \rightarrow {\bf C}}. It turns out that the functional equation (3) for the zeta function localises to this context, giving an approximate functional equation which roughly speaking takes the form

\displaystyle \sum_n \frac{1}{n^s} \psi( \log n - \log N ) \approx \alpha(s) \sum_m \frac{1}{m^{1-s}} \psi( \log M - \log m )

whenever {s=\sigma+it} and {NM = \frac{|t|}{2\pi}}; see Theorem 38 below for a precise formulation of this equation. Unsurprisingly, this form of the functional equation is also very closely related to the Poisson summation formula (8), indeed it is essentially a special case of that formula (or more precisely, of the van der Corput {B}-process). This useful identity relates long smoothed sums of {\frac{1}{n^s}} to short smoothed sums of {\frac{1}{m^{1-s}}} (or vice versa), and can thus be used to shorten exponential sums involving terms such as {\frac{1}{n^s}}, which is useful when obtaining some of the more advanced estimates on the Riemann zeta function.

We will give two other basic uses of the functional equation. The first is to get a good count (as opposed to merely an upper bound) on the density of zeroes in the critical strip, establishing the Riemann-von Mangoldt formula that the number {N(T)} of zeroes of imaginary part between {0} and {T} is {\frac{T}{2\pi} \log \frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T)} for large {T}. The other is to obtain untruncated versions of the explicit formula from Notes 2, giving a remarkable exact formula for sums involving the von Mangoldt function in terms of zeroes of the Riemann zeta function. These results are not strictly necessary for most of the material in the rest of the course, but certainly help to clarify the nature of the Riemann zeta function and its relation to the primes.

In view of the material in previous notes, it should not be surprising that there are analogues of all of the above theory for Dirichlet {L}-functions {L(\cdot,\chi)}. We will restrict attention to primitive characters {\chi}, since the {L}-function for imprimitive characters merely differs from the {L}-function of the associated primitive factor by a finite Euler product; indeed, if {\chi = \chi' \chi_0} for some principal {\chi_0} whose modulus {q_0} is coprime to that of {\chi'}, then

\displaystyle L(s,\chi) = L(s,\chi') \prod_{p|q_0} (1 - \frac{1}{p^s}) \ \ \ \ \ (9)

 

(cf. equation (45) of Notes 2).

The main new feature is that the Poisson summation formula needs to be “twisted” by a Dirichlet character {\chi}, and this boils down to the problem of understanding the finite (additive) Fourier transform of a Dirichlet character. This is achieved by the classical theory of Gauss sums, which we review below the fold. There is one new wrinkle; the value of {\chi(-1) \in \{-1,+1\}} plays a role in the functional equation. More precisely, we have

Theorem 3 (Functional equation for {L}-functions) Let {\chi} be a primitive character of modulus {q} with {q>1}. Then {L(s,\chi)} extends to an entire function on the complex plane, with

\displaystyle L(s,\chi) = \varepsilon(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin(\frac{\pi}{2}(s+\kappa)) \Gamma(1-s) L(1-s,\overline{\chi})

or equivalently

\displaystyle L(1-s,\overline{\chi}) = \varepsilon(\overline{\chi}) 2^{1-s} \pi^{-s} q^{s-1/2} \sin(\frac{\pi}{2}(1-s+\kappa)) \Gamma(s) L(s,\chi)

for all {s}, where {\kappa} is equal to {0} in the even case {\chi(-1)=+1} and {1} in the odd case {\chi(-1)=-1}, and

\displaystyle \varepsilon(\chi) := \frac{\tau(\chi)}{i^\kappa \sqrt{q}} \ \ \ \ \ (10)

 

where {\tau(\chi)} is the Gauss sum

\displaystyle \tau(\chi) := \sum_{n \in {\bf Z}/q{\bf Z}} \chi(n) e(n/q). \ \ \ \ \ (11)

 

and {e(x) := e^{2\pi ix}}, with the convention that the {q}-periodic function {n \mapsto e(n/q)} is also (by abuse of notation) applied to {n} in the cyclic group {{\bf Z}/q{\bf Z}}.

From this functional equation and (2) we see that, as with the Riemann zeta function, the non-trivial zeroes of {L(s,\chi)} (defined as the zeroes within the critical strip {\{ s: 0 < \hbox{Re}(s) < 1 \}} are symmetric around the critical line (and, if {\chi} is real, are also symmetric around the real axis). In addition, {L(s,\chi)} acquires trivial zeroes at the negative even integers and at zero if {\chi(-1)=1}, and at the negative odd integers if {\chi(-1)=-1}. For imprimitive {\chi}, we see from (9) that {L(s,\chi)} also acquires some additional trivial zeroes on the left edge of the critical strip.

There is also a symmetric version of this equation, analogous to Corollary 2:

Corollary 4 Let {\chi,q,\varepsilon(\chi)} be as above, and set

\displaystyle \xi(s,\chi) := (q/\pi)^{(s+\kappa)/2} \Gamma((s+\kappa)/2) L(s,\chi),

then {\xi(\cdot,\chi)} is entire with {\xi(1-s,\overline{\chi}) = \varepsilon(\chi) \xi(s,\chi)}.

For further detail on the functional equation and its implications, I recommend the classic text of Titchmarsh or the text of Davenport.

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