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The Functional Equation 18.785 Number theory I Fall 2019 Lecture #17 11/4/2019 17 The functional equation In the previous lecture we proved that the Riemann zeta function ζ(s) has an Euler product and an analytic continuation to the right half-plane Re(s) > 0. In this lecture we complete the picture by deriving a functional equation that relates the values of ζ(s) to those of ζ(1 − s). This will then also allow us to extend ζ(s) to a meromorphic function on C that is holomorphic except for a simple pole at s = 1. 17.1 Fourier transforms and Poisson summation A key tool we will use to derive the functional equation is the Poisson summation formula, a result from harmonic analysis that we now recall. 1 Definition 17.1. A Schwartz function on R is a complex-valued C function f : R ! C that decays rapidly to zero: for all m; n 2 Z≥0 we have m (n) sup x f (x) < 1; x2R (n) where f denotes the nth derivative of f. The Schwartz space S(R) of all Schwartz functions on R is a (non-unital) C-algebra of infinite dimension. Example 17.2. All compactly supported C1 functions are Schwartz functions, as is the Gaussian function g(x) := e−πx2 . Non-examples include functions that do not tend to zero as x ! ±∞ (such as polynomials), and functions like (1 + x2n)−1 and e−x2 sin(ex2 ) that either do not tend to zero quickly enough, or have derivatives that do not tend to zero as x ! ±∞. p Remark 17.3. For any p 2 R≥1, the Schwartz space S(R) is contained in the space L (R) of functions on f : ! for which the Lebesgue integral R jf(x)jpdx exists. The space R C R p p R p 1=p L (R) is a complete normed C-vector space under the L -norm kfkp := ( jf(x)j dx) , R p and is thus a Banach space. The Schwartz space S(R) is not complete under the L -norm, p but it is dense in L (R) (in the subspace topology). One can equip the Schwartz space with a translation-invariant metric of its own under which it is a complete metric space (and thus a Fr´echet space, since it is also locally convex), but the topology of S(R) will not concern n us here. Similar comments apply to S(R ). It follows immediately from the definition and standard properties of the derivative that the Schwartz space S(R) is closed under differentiation, multiplication by polynomials, and linear change of variable. It is also closed under convolution: for any f; g 2 S(R) the function Z (f ∗ g)(x) := f(y)g(x − y)dy R is also an element of S(R). Convolution is commutative, associative, and bilinear. Definition 17.4. The Fourier transform of a Schwartz function f 2 S(R) is the function Z f^(y) := f(x)e−2πixydx; R Andrew V. Sutherland which is also a Schwartz function [1, Thm. 5.1.3]. We can recover f(x) from f^(y) via the inverse transform Z f(x) = f^(y)e+2πixydy; R see [1, Thm. 5.1.9] for a proof of this fact. The maps f 7! f^ and f^ 7! f are thus inverse linear operators on S(R) (they are also continuous in the metric topology of S(R) and thus homeomorphisms). Remark 17.5. The invertibility of the Fourier transform on the Schwartz space S(R) is 1 a key motivation for its definition. For functions in L (R) (the largest space of functions for which our definition of the Fourier transform makes sense), the Fourier transform of a smooth function decays rapidly to zero, and the Fourier transform of a function that decays rapidly to zero is smooth; this leads one to consider the subspace S(R) of smooth functions 1 that decay rapidly to zero. One can show that S(R) is the largest subspace of L (R) closed under multiplication by polynomials on which the Fourier transform is invertible.1 The Fourier transform changes convolutions into products, and vice versa. We have f[∗ g = f^g^ and fgc = f^∗ g;^ for all f; g 2 S(R) (see Problem Set 8). One can thus view the Fourier transform as an isomorphism of (non-unital) C-algebras that sends (S(R); +; ×) to (S(R); +; ∗). 1 ^ y Lemma 17.6. For all a 2 R>0 and f 2 S(R), we have f\(ax)(y) = a f( a ). Proof. Applying the substitution t = ax yields Z 1 Z 1 y f\(ax)(y) = f(ax)e−2πixydx = f(t)e−2πity=adt = f^ : R a R a a d ^ d\ ^ Lemma 17.7. For f 2 S(R) we have dy f(y) = −2πi xf\(x)(y) and dx f(x)(y) = 2πiyf(y). Proof. Noting that xf 2 S(R), the first identity follows from Z Z d ^ d −2πixy −2πixy dy f(y) = dy f(x)e dx = f(x)(−2πix)e dx = −2πi xf\(x)(y); R R since we may differentiate under the integral via dominated convergence. For the second, we note that limx→±∞ f(x) = 0, so integration by parts yields Z Z d\ 0 −2πixy −2πixy ^ dx f(x)(y) = f (x)e dx = 0 − f(x)(−2πiy)e dx = 2πiyf(y): R R The Fourier transform is compatible with the inner product hf; gi := R f(x)g(x)dx on 2 R L (R) (which contains S(R)). Indeed, we can easily derive Parseval's identity: Z Z Z Z hf; gi = f(x)g(x)dx = f^(y)g(x)e+2πixydxdy = f^(y)g^(y)dy = hf;^ g^i; R R R R which when applied to g = f yields Plancherel's identity: 2 ^ ^ ^ 2 kfk2 = hf; fi = hf; fi = kfk2; R 2 1=2 2 where kfk2 = ( jf(x)j dx) is the L -norm. For number-theoretic applications there is R an analogous result due to Poisson. 1I thank Keith Conrad and Terry Tao for clarifying this point. 18.785 Fall 2019, Lecture #17, Page 2 Theorem 17.8 (Poisson Summation Formula). For all f 2 S(R) we have the identity X X f(n) = f^(n): n2Z n2Z Proof. We first note that both sums are well defined; the rapid decay property of Schwartz functions guarantees absolute convergence. Let F (x) := P f(x + n). Then F is a n2Z periodic C1-function, so it has a Fourier series expansion X 2πinx F (x) = cne ; n2Z with Fourier coefficients Z 1 Z 1 Z −2πint X −2πint −2πint cn = F (t)e dt = f(t + m)e dt = f(t)e dt = f^(n): 0 0 m2Z R We then note that X X X f(n) = F (0) = cn = f^(n): n2Z n2Z n2Z Finally, we note that the Gaussian function e−πx2 is its own Fourier transform. Lemma 17.9. Let g(x) := e−πx2 . Then g^(y) = g(y). Proof. The function g(x) satisfies the first order ordinary differential equation g0 + 2πxg = 0; (1) with initial value g(0) = 1. Multiplying both sides by −i and taking Fourier transforms yields 0 0 0 −i(gb + 2πxgc) = −i(2πixg^ + ig^ ) =g ^ + 2πxg^ = 0; via Lemma 17.7. Sog ^ also satisfies (1), andg ^(0) = R e−πx2 dx = 1, sog ^ = g. R 17.1.1 Jacobi's theta function We now define the theta function2 X 2 Θ(τ) := eπin τ : n2Z The sum is absolutely convergent for im τ > 0 and thus defines a holomorphic function on the upper half plane. It is easy to see that Θ(τ) is periodic modulo 2, that is, Θ(τ + 2) = Θ(τ); but it it also satisfies another functional equation. p Lemma 17.10. For all a 2 R>0 we have Θ(ia) = Θ(i=a)= a. p Proof. Put g(x) := e−πx2 and h(x) := g( ax) = e−πx2a. Lemmas 17.6 and 17.9 imply p p p p p h^(y) = g\( ax)(y) =g ^y= a= a = gy= a= a: Plugging τ = ia into Θ(τ) and applying Poisson summation (Theorem 17.8) yields X 2 X X X p p p Θ(ia) = e−πn a = h(n) = h^(n) = gn= a= a = Θ(i=a)= a: n2Z n2Z n2Z n2Z 2The function Θ(τ) we define here is a special case of one of four parameterized families of theta functions Θi(z : τ) originally defined by Jacobi for i = 0; 1; 2; 3, which play an important role in the theory of elliptic functions and modular forms; in terms of Jacobi's notation, Θ(τ) = Θ3(0; τ). 18.785 Fall 2019, Lecture #17, Page 3 17.1.2 Euler's gamma function You are probably familiar with the gamma function Γ(s), which plays a key role in the functional equation of not only the Riemann zeta function but many of the more general zeta functions and L-series we wish to consider. Here we recall some of its analytic properties. We begin with the definition of Γ(s) as a Mellin transform. Definition 17.11. The Mellin transform of a function f : R>0 ! C is the complex function defined by Z 1 M(f)(s) := f(t)ts−1dt; 0 whenever this integral converges. It is holomorphic on Re s 2 (a; b) for any interval (a; b) in R 1 σ−1 which the integral 0 jf(t)jt dt converges for all σ 2 (a; b).
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