In Notes 2, the Riemann zeta function (and more generally, the Dirichlet -functions ) were extended meromorphically into the region 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 are known as the non-trivial zeroes of , and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).
for all in its domain of definition. Indeed, as is real-valued when is real, the function vanishes on the real line and is also meromorphic, and hence vanishes everywhere. Similarly one has the functional equation
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 rather than the real axis. One consequence of this symmetry is that the zeta function and -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 extends meromorphically to the entire complex plane, with a simple pole at and no other poles. Furthermore, one has the functional equation
Here , are the complex-analytic extensions of the classical trigionometric functions , and 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
In particular, the zeroes of consist precisely of the non-trivial zeroes of , and are symmetric about both the real axis and the critical line. Also, 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 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
(Theorem 34 from Supplement 2, at least in the case when 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 , and noting that the functions and 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 has trivial zeroes at the negative even integers , 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 or have no zeroes or poles, and will be ignored in this table; the zeroes and poles of rational functions such as are self-evident and will also not be displayed here.)
|Function||Non-trivial zeroes||Trivial zeroes||Poles|
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 , with ranging all the way from 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 for some and some smooth compactly supported function . 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
whenever and ; 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 -process). This useful identity relates long smoothed sums of to short smoothed sums of (or vice versa), and can thus be used to shorten exponential sums involving terms such as , 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 of zeroes of imaginary part between and is for large . 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 -functions . We will restrict attention to primitive characters , since the -function for imprimitive characters merely differs from the -function of the associated primitive factor by a finite Euler product; indeed, if for some principal whose modulus is coprime to that of , then
(cf. equation (45) of Notes 2).
The main new feature is that the Poisson summation formula needs to be “twisted” by a Dirichlet character , 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 plays a role in the functional equation. More precisely, we have
where is the Gauss sum
From this functional equation and (2) we see that, as with the Riemann zeta function, the non-trivial zeroes of (defined as the zeroes within the critical strip are symmetric around the critical line (and, if is real, are also symmetric around the real axis). In addition, acquires trivial zeroes at the negative even integers and at zero if , and at the negative odd integers if . For imprimitive , we see from (9) that 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:
then is entire with .
— 1. The Gamma function —
There are many ways to define the Gamma function, but we will use the following classical definition:
It is easy to see that the integrals here are absolutely convergent. One can view as the inner product between the multiplicative character and the additive character with respect to multiplicative Haar measure . As such, the Gamma function often appears as a normalisation factor in integrals that involve both additive and multiplicative characters. For instance, by a simple change of variables we see that
whenever and ; indeed, from a contour shift we see that the above identity also holds for complex with , if we use the standard interpretation of the complex exponential with positive real base. Making the further substitution and performing some additional manipulations, we see that the Gamma function is also related to integrals involving Gaussian functions, in that
which follows from (14) by replacing with .
From Cauchy’s theorem and Fubini’s theorem one easily verifies that has vanishing contour integral on any closed contour in the half-space , and thus by Morera’s theorem is holomorphic on this half-space.
From (12) and an integration by parts we see that
of (16) as a definition to meromorphically extend the domain of definition of leftwards by one unit.
Exercise 6 Show that for any natural number (thus , , etc.), and that has simple poles at and no further singularities. Thus one can view the Gamma function as a (shifted) generalisation of the factorial function.
By repeating the proof of (31), we obtain the conjugation symmetry
for all outside of the poles of . Translating this to the -function (5), we see that is meromorphic, with a pole at , and that
The Gamma function is also closely connected to the beta function:
whenever . (Note that this hypothesis makes the integral on the left-hand side absolutely integrable.)
Proof: From (12) and Fubini’s theorem one has
Making the change of variables for and (and using absolute integrability to justify this change of variables), the right-hand side becomes
and the claim follows another appeal to (12) and Fubini’s theorem.
This gives an important reflection formula:
as meromorphic functions (that is to say, the identity holds outside of the poles of the left or right-hand sides, which occur at the integers). In particular, has no zeroes in the complex plane.
Note that the reflection formula, when written in terms of the -function (5), is simply
after removing any singularities from the left-hand side. In particular, has zeroes at and poles at , with no further poles and zeroes. Note that (19) is consistent with the functional equations (4), (3).
Proof: By unique continuation of meromorphic functions, it suffices to verify this identity in the critical strip . By the beta function identity (and the value ), it thus suffices to show that
for in the critical strip.
If we make the substitution , so that , we have
We extend the function to the complex plane (excluding the origin) by the formula , where is the branch of the complex logarithm whose imaginary part lies in the half-open interval . This agrees with the usual power function at (or infinitesimally above) the positive real axis, but instead converges to infinitesimally below this axis. Thus, if one lets be a contour that loops clockwise around the positive real axis, and stays sufficiently close to this axis, we see (using Cauchy’s theorem to justify the passage from infinitesimal neighbourhoods of the real axis to non-infinitesimal ones, and using the hypothesis to handle the contributions near the origin and infinity) we have
On the other hand, outside of the non-negative real axis, is meromorphic, with a simple pole at of residue , and decays faster than at infinity. From the residue theorem we then have
and the claim then follows by putting the above identities together.
As a quick application of (8), if we set and observe that is clearly positive, we have
and thus (by (14)) we recover the classical Gaussian identity
Next, we give an alternate definition of the Gamma function:
Lemma 9 (Euler form of Gamma function) If is not a pole of (i.e., ), then
Proof: It is easy to verify the second identity, and that the product and limit are convergent. One also easily verifies that the expression obeys (16), so it will suffice to establish the claim when .
But by Lemma 7 and a change of variables we have
From (16) one has and , and the claim follows (recall that is never zero).
Exercise 10 (Weierstrass form of Gamma function) If is not a pole of , show that
Exercise 11 (Digamma function) Define the digamma function to be the logarithmic derivative of the Gamma function. Show that the digamma function is a meromorphic function, with simple poles of residue at the non-positive integers and no other poles, and that
for non-integer .
Exercise 12 Show that .
whenever is not a pole of . (Hint: using the digamma function, show that the logarithmic derivatives of both sides differ by a constant. Then test the formula at two values of to verify that the normalising factor of is correct.)
Exercise 14 (Gauss multiplication theorem) For any natural number , establish the multiplication theorem
whenever is not a pole of .
Exercise 15 (Bohr-Mollerup theorem) Establish the Bohr-Mollerup theorem: the function , which is the Gamma function restricted to the positive reals, is the unique log-convex function on the positive reals with and for all .
Now we turn to the question of asymptotics for . We begin with the corresponding asymptotics for the digamma function . Recall (see Exercise 11 from Notes 1) that one has
for any real and any continuously differentiable functions . This gives
for in a sector of the form for some fixed (that is, makes at least a fixed angle with the negative real axis), where and are the standard branches of the argument and logarithm respectively (with branch cut on the negative real axis). From Exercise 11, we obtain the asymptotic
in this regime. (For the other values of , one can use the reflection formula (21) to obtain an analogous asymptotic.) Actually, it will be convenient to sharpen this approximation a bit, using the following version of the trapezoid rule:
Exercise 16 (Trapezoid rule) Let be distinct integers, and let be a continuously twice differentiable function. Show that
(Hint: first establish the case when .)
From this exercise, we obtain a sharper estimate
for some absolute constant . To find this constant , we apply the reflection formula (Lemma 8) and and conclude that
for . Since (up to multiples of )
we conclude that is equal to up to multiples of ; but as is positive on the positive reals, we can normalise so that , thus we obtain the Stirling approximation
In particular, we have the approximation
Also show that the error in (25) is real-valued when , so that
Exercise 20 Using the trapezoid rule, show that for any in the region with , there exists a unique complex number for which one has the asymptotic
for any natural number , where . Use this to extend the Riemann zeta function meromorphically to the region . Conclude in particular that .
to the trapezoid rule when are integers and is continuously three times differentiable. Then show that for any in the region with , there exists a unique complex number for which one has the asymptotic
for any natural number , where . Use this to extend the Riemann zeta function meromorphically to the region . Conclude in particular that ; this is a rigorous interpretation of the infamous formula
Remark 22 One can continue this procedure to extend meromorphically to the entire complex plane by using the Euler-Maclaurin formula; see this previous blog post. However, we will not pursue this approach to the meromorphic continuation of zeta further here.
— 2. The functional equation —
We now give three different (although not wholly unrelated) proofs of the functional equation, Theorem 1.
The first proof (due to Riemann) relies on a relationship between the Dirichlet series
Given that both of the transforms and are linear and (formally, at least) injective, it is not surprising that there should be some linear relationship between the two. It turns out that we can use the Gamma function to mediate such a relationship:
Lemma 23 (Dirichlet series from power series) Let be an arithmetic function such that as . Then for any complex number with , we have
where is the Taylor series (26), which is absolutely convergent in the unit disk . The integral on the right-hand side is absolutely integrable.
Proof: From (13) we have
for any natural number . Multiplying by , summing, and using Fubini’s theorem, we conclude that
and the claim follows. (By restricting to the case when is real and is non-negative, we can see that all integrals here are absolutely integrable.)
for , which can be compared with (12). Now we recall the contour introduced in the proof of Lemma 8, which goes around the positive real axis in the clockwise direction. As in the proof of that lemma, we see that
for sufficiently close to the real axis (specifically, it has to not wind around any of the zeroes of other than ), where we use the branch as in the proof of Lemma 8. Thus we have
The contour integral is in fact absolutely convergent for any , and from the usual argument involving the Cauchy, Fubini, and Morera theorems we see that this integral depends holomorphically on . Thus, we can use (28) as a definition for the Riemann zeta function that extends it meromorphically to the entire complex plane with no further poles (note that has no zeroes to the left of the critical strip, after removing all singularities).
Now suppose that we are in the region , with not an integer. For any natural number , we shift the contour to the rectangular contour , which starts at , goes leftwards to , then upwards to , then rightwards to . As has simple poles at for each non-zero integer with residue , we see from the residue theorem (and the exponential decay of as goes to infinity to the right) that
If , then one can compute that the integral goes to zero as , and thus
From the choice of branch for , one sees that
Inserting these identities into (28), we obtain (4) after a brief calculation, at least in the region when and is not an integer; the remaining cases then follow from unique continuation of meromorphic functions.
Remark 24 The Poisson summation formula was not explicitly used in the above proof of the functional equation. However, if one inspects the contour integration proof of the Poisson summation formula in Supplement 2, one sees an application of the residue theorem which is quite similar to that in the above argument, and so that formula is still present behind the scenes.
Introducing the theta function
Making the change of variables , this becomes
for in the upper half-plane, using the principal branch of the square root; see Exercise 36 of . In particular, blows up like as . We use this formula to transform the previous integral to an integral just on rather than . First observe that
Next, from (30) (using the hypothesis to ensure absolute convergence) and the change of variables we have
Note from the definition of the theta function that decays exponentially fast as . As such, the integral in the right-hand is absolutely convergent for any , and by the usual Morera theorem argument is in fact holomorphic in . Thus (31) may be used to give a meromorphic extension of to the entire complex plane. The right-hand side of (31) is also manifestly symmetric with respect to the reflection , giving the functional equation in the form (7).
Next, we give a short heuristic proof of the functional equation arising from formally applying the Poisson summation formula (8) to the function
ignoring all infinite divergences. Formally, the Fourier transform is then given by
and the functional equation (3) formally follows after some routine calculation if we discard the divergent term on the right-hand side, converts the sum over negative to a sum over positive by the change of variables , and formally identify and with respectively.
The above heuristic argument may be made rigorous by using suitable regularisations. This is the purpose of the exercise below.
- (i) For any , show that the function defined by is continuous, absolutely integrable, and has Fourier transform
for , using the standard branch of the logarithm to define .
- (ii) Rigorously justify the Poisson summation formula
for any . (In Supplement 2, the Poisson summation formula was only established for continuously twice differentiable, compactly supported functions; is neither of these, but one can still recover the formula in this instance by an approximation argument.)
- (iii) Show that as .
- (iv) Show that
as , for either choice of sign .
- (v) Prove (3) for in the critical strip, and then prove the rest of Theorem 1.
Remark 26 There are many further proofs of the functional equation than the three given above; see for instance the text of Titchmarsh for several further proofs. Most of the proofs can be connected in one form or another to the Poisson summation formula. One important proof worth mentioning is Tate’s adelic proof, discussed in this previous post, which is well suited for generalising the functional equation to many other zeta functions and -functions, but will not be discussed further in this post.
Exercise 27 Use the formula from Exercise 21, together with the functional equation, to show that .
Exercise 28 (Relation between zeta function and Bernoulli numbers) In this exercise we give the classical connection between the zeta function and Bernoulli numbers; this connection is not so relevant for analytic number theory, as it only involves values of the zeta function that are far from the critical strip, but is of interest for some other applications.
- (i) For any complex number with , use the Poisson summation formula (8) to establish the identity
- (ii) For as above and sufficiently small, show that
for any natural number , where the Bernoulli numbers are defined through the Taylor expansion
Thus for instance , , and so forth.
- (iii) Show that
- (iv) Use (28) and the residue theorem (now working inside the contour , rather than outside) to give an alternate proof of (32).
Exercise 29 Show that .
Remark 30 The functional equation is almost certainly not sufficient, by itself, to establish the Riemann hypothesis. For instance, there is a classical example of Davenport and Heilbronn of a finite linear combination of Dirichlet L-functions which obeys a functional equation very similar (though not quite identical) to (4), but which possesses zeroes off of the critical line; see e.g. this article for a recent analysis of the counterexample. Eisenstein series can also be used to construct a “natural” variant of a zeta function that has a Dirichlet series and a functional equation, but has zeroes off the critical line. For a “cheaper” counterexample, take two nearby non-trivial zeroes of on the critical line, and “replace” them with two other nearby complex numbers symmetric around the critical line, but not on the line, by introducing the modified zeta function
This function also obeys the functional equation, and behaves very similarly (though not identically) to the Riemann zeta function in all the regions in which we have a good understanding of this function (in particular, it has similar behaviour to around or around the edges of the critical strip), but clearly has zeroes off of the critical line. Such constructions would be particularly hard to exclude by analytic methods if there happened to be a repeated zero of on the critical line, as one could then make extremely close to both of ; it is conjectured that such a repeated zero does not occur, but we cannot exclude this possibility with current technology, which creates a family of “infinitesimal counterexamples” to the Riemann hypothesis which rules out a large number of potential approaches to this hypothesis.
On the other hand, unlike the Davenport-Heilbronn counterexample, does not arise from a Dirichlet series, and certainly does not have an Euler product. One can show (see Section 2.13 of Titchmarsh) that if one insists on the functional equation (4) on the nose (as opposed to, say, the modified functional equation that the Davenport-Heilbronn example obeys, or the functional equation obeyed by a Dirichlet -function) as well as a Dirichlet series representation, then the only possible functions available are scalar multiples of the Riemann zeta function; this was first observed by Hamburger in 1921. It could well be that the analogue of the Riemann hypothesis is in fact obeyed by any function which obeys a suitable functional equation, together with a Dirichlet series representation (with appropriate size bounds on the coefficients) and an Euler product factorisation; a precise form of this statement is the Riemann hypothesis for the Selberg class. But one would somehow need to make essential use of all three of the above axioms to try to prove the Riemann hypothesis, as we have numerous counterexamples that show that zeroes can be produced off the critical line if one drops one or more of these axioms.
— 3. Approximate and localised forms of the functional equation —
In our construction of the Riemann zeta function in Notes 2, we had the asymptotic
for and in the region . Thus, is the limit of the functions as , locally uniformly for in this region. We have an analogous limit for smoothed sums:
converge locally uniformly to on the region . In the critical strip , show that the second term may be replaced by .
It is of interest to understand the rate of convergence of these approximations to the zeta function. We restrict attention to in the critical strip . The first observation is that the smoothed sums have negligible contribution once is much larger than :
for any .
Proof: By the Poisson summation formula (8), we have
If we write and , then after a change of variables we have
where and . In particular we have
so it suffices by the triangle inequality to show that
for any and any non-zero integer . But by hypothesis on , we see that we have the derivative bounds on the support of the smooth compactly supported function . If one repeatedly writes and integrates by parts to move the derivative off of the phase, one obtains the claim.
This gives us a good approximation to in the critical strip, involving a smoothed sum consisting of terms:
We remark that the asymptotic (34) is also valid (with a somewhat worse error term) for the ordinary partial sums (a classical result of Hardy and Littlewood); see Theorem 4.11 of Titchmarsh. However, it will be slightly more convenient for us here to work exclusively with smoothed sums.
From (34) and the triangle inequality, we have the crude bound
for any and ; we leave the details to the interested reader (and we will reprove the convexity bound shortly). Further improvements to (38) for the zeta function and other -functions are known as subconvexity bounds and have many applications in analytic number theory, though we will only discuss the simplest subconvexity bounds in this course.
Exercise 33 describes the zeta function in terms of smoothed sums of . In the converse direction, one can use Fourier inversion to express smoothed sums of in terms of the zeta function:
for all , where
is the Fourier transform of .
we thus rewrite the left-hand side of (39) as the contour integral
The function has a pole at with residue , which by Exercise 28 of Supplement 2 is of size for any . By another appeal to that exercise, together with (35), we see that goes to zero as uniformly when is bounded. By the residue theorem, we can thus shift the contour integral to
and the claim follows by performing the substitution .
Exercise 35 Establish (39) directly from the Fourier inversion formula, without invoking contour integration methods.
Among other things, this lemma shows that growth bounds in the Riemann zeta function are equivalent to growth bounds on smooth exponential sums of :
- (i) One has as .
- (ii) One has the bound
whenever , , and is a compactly supported smooth function.
for any , where is as in (5).
This approximate functional equation can also be established directly from the Poisson summation formula using the method of stationary phase; see Chapter 4 of Titchmarsh. The error term of can be improved further by using better growth bounds on (or by further Taylor expansion of ), but the error term given here is adequate for applications. Note that the true functional equation (3) is formally the case of (40) if one ignores the error term.
Proof: By (39), the left-hand side is
up to negligible errors. Using the rapid decrease of (Exercise 28 of Supplement 2) and (35), we may restrict to the range , up to negligible error. Applying the functional equation (3), we rewrite this as
For , we see from Exercise 17 that
and thus from the fundamental theorem of calculus we have
or equivalently (using )
We can thus write the left-hand side of (40) up to acceptable errors as
for any , and from the functional equation we have
Using Theorem 38, we may split the difference:
for all .
One can also obtain a version of this equation using partial sums instead of smoothed sums, but with slightly worse error terms, known as the Riemann-Siegel formula; see e.g. Theorem 4.13 of Titchmarsh. Setting , we see that we may now approximate by smoothed sums consisting of about terms, improving upon the sum with terms appearing in Exercise 33. Using the triangle inequality, this gives a slight improvement to (38), namely that
whenever and . The equation (41) is particularly useful for getting reasonably good bounds on ; we will see an example of this in subsequent notes.
— 4. Further applications of the functional equation —
One basic application of the functional equation is to improve the control on zeroes of the Riemann zeta function, beyond what was obtained in Notes 2.
From Exercise 37, we now have the crude bounds
and in particular
whenever and . The Jensen formula argument from Proposition 16 of Notes 2 is no longer restricted to the region , and shows that there are zeroes of the Riemann zeta function in any disk of the form . Similarly, Proposition 19 of Notes 2 extends to give the formula
whenever with . Corollary 20 of Notes 2 also extends to show that
We can say more about the zeroes. For any , let denote the number of zeroes of in the rectangle . (If there were zeroes of on the interval , they should each count for towards , but it turns out (as can be computationally verified) that there are no such zeroes.) Equivalently, is the number of zeroes in . We have the following asymptotic for , conjectured by Riemann and established by von Mangoldt:
By the residue theorem (or the argument principle), the number of zeroes in this rectangle is equal to times the contour integral of anticlockwise around the boundary of the rectangle. By (44), the contribution of the upper and lower edges of this contour are ; from the functional equation (Corollary 2) we see that the contribution of the left and right edges of the contour are the same, and from conjugation symmetry we see that the contribution of the upper half of the right edge is the complex conjugate of that of the lower half. Putting all this together, we see that it suffices to show that
or equivalently (after removing the integral from to , which is ), that
for . Since is given by a Dirichlet series that is uniformly bounded on , we have
and the claim follows.
Exercise 41 Establish the more precise formula
whenever and the line avoids all zeroes of , where , and the logarithm is extended leftwards from the region , thus
Theorem 40 then asserts that for all . It is in fact conjectured that as , but this problem has resisted solution for over a century (although it is known that this bound would follow from powerful hypotheses such as the Lindelöf hypothesis).
Remark 42 In principle, can be numerically computed exactly for any , as long as the line has no zeroes of , by evaluating the contour integral of to sufficiently high accuracy. Similarly, one can also obtain a numerical lower bound for the number of zeroes of on the critical line by finding sign changes for the function , which is real-valued on the critical line. If the zeroes are all simple and on the critical line, then this (in principle) allows one to numerically verify the Riemann hypothesis up to height . In practice, faster methods for numerically verifying the Riemann hypothesis (e.g. based on the Riemann-Siegel formula) are available.
We can now obtain global explicit formulae for the log-derivatives of and . Since has zeroes only at the non-trivial zeroes of , and no poles, one heuristically expects a relationship of the form
Unfortunately, the right-hand side is divergent; but we can normalise it by considering the sum
Exercise 43 Show that the sum converges locally uniformly to a meromorphic function away from the non-trivial zeroes of , with an entire function (after removing all singularities). Also establish the bounds
for all in the complex plane. (Hint: You will need to use the Riemann-von Mangoldt formula.)
for all in the strip ; from (22) and the boundedness of in the region one sees that this bound also holds with , and from the functional equation we see that it also holds for . In particular, with as in the preceding exercise, we see that the entire function is bounded by on the entire complex plane. By the generalised Cauchy integral formula (Exercise 9 of Supplement 2) applied to a disk of radius we conclude that has derivative for any , and by sending we conclude that this function is constant; thus we have the representation
The exact values of are not terribly important for applications, but can be computed explicitly:
Exercise 44 By inspecting both sides of the above equations as , show that , and hence .
Exercise 45 (Riemann-von Mangoldt explicit formula) For any non-integer , show that
This is an exact counterpart of the truncated explicit formula (Theorem 21 of Notes 2), although in many applications the truncated formula is a little bit more convenient to use; the untruncated formula supplies all of the “lowest order terms”, but these terms are destined to be absorbed into error terms in most applications anyway.
We similarly have a global smoothed explicit formula, refining Exercise 22 from Notes 2:
with the sums being absolutely convergent. Conclude that
whenever is a smooth function, supported in .
The following variant of the smoothed explicit formula is particularly useful for studying the behaviour of zeroes of the zeta function on the critical line .
Exercise 47 (Riemann-Weil explicit formula) Let be a smooth compactly supported function. Show that
where the non-trivial zeroes of the zeta function are parameterised as , and , with the sum over being absolutely convergent. (Hint: first reduce to the case when (and hence ) is an even function, by eliminating the case when is odd. Then, integrate the meromorphic function around a rectangle with vertices , (say) for some large and apply the residue theorem and the contour integration. It is also possible to derive this formula from Exercise 46 by starting with the case when is supported on the positive real axis, then the negative axis, then for supported in an infinitesimally small neighbourhood of the origin.)
— 5. The functional equation for Dirichlet -functions —
Now we turn to the functional equation for Dirichlet -functions , Theorem 3. Henceforth is a primitive character of modulus . To obtain the functional equation for , we will need a twisted version of the Poisson summation formula (8). The key to performing the twist is the following expression for the additive Fourier coefficients of in the group :
Proof: The formula (46) is trivial when , and by making the substitution we see that it is also true when is coprime to . Now suppose that shares a common factor with , then the right-hand side of (46) vanishes. Writing , we see that is periodic with period , and in particular is invariant with respect to multiplication by any invertible element of whose projection to is one; thus the left-hand side of (46) is invariant under multiplication by . We are thus done unless for all invertible that projects down to one on . But then factors as the product of a character of modulus and a principal character, contradicting the hypothesis that is primitive.
The Gauss sum , being an inner product between a multiplicative character and an additive character , is analogous to the Gamma function , which is also an inner product between a multiplicative character and an additive character . This analogy can be deepened by working in Tate’s adelic formalism, but we will not do so here. But we will give one further demonstration of the analogy between Gauss sums and Gamma functions:
Exercise 49 (Jacobi sum identity) Let be Dirichlet characters modulo a prime , such that are all non-principal. By computing the sum in two different ways, establish the Jacobi sum identity
This should be compared with the beta function identity, Lemma 7.
This determines the magnitude of ; in particular, the quantity defined in (10) has magnitude one, and
Now suppose that is a twice continuously differentiable, compactly supported function. We can expand the sum using (47) (and identifying with as
applying the Poisson summation formula (8) to the modulated function , we conclude that
Remark 50 One can view both the ordinary Poisson summation formula (8) and its twisted analogue (51) as special cases of the adelic Poisson summation formula; see this previous blog post. However, we will not explicitly adopt the adelic viewpoint here.
We now adapt the proofs of the functional equation for to prove Theorem 3, using (51) as a replacement for (8). One can use any of the three proofs for this purpose, but I found it easiest to work with the third proof. We first work heuristically. As before, we formally apply (51) with , ignoring all infinite divergences. Again, the Fourier transform is given by
and so (51) formally yields
The term is formally absent since . If one converts the sum over negative to a sum over positive by the change of variables , and formally identifies and with and , one formally obtains Theorem 3 after some routine calculation.
Exercise 51 Make the above argument rigorous by adapting the argument in Exercise 25.
The following exercise develops the analogue of Riemann’s second proof of the functional equation, which is the proof of the functional equation for Dirichlet -functions that is found in most textbooks.
Exercise 52 Let be a primitive character of conductor , and let be such that . Define the theta-type function
in the half-plane . The purpose of the factor is to make the summand even in , rather than odd (as the theta-function would be trivial if the summand were odd).
One can also adapt the first proof of Riemann to the -function setting, as was done in this paper of Berndt:
Exercise 53 Let be a primitive character of conductor .
The next exercise extends the approximate functional equations for the zeta function to Dirichlet -functions for primitive characters of some conductor . As may be expected from Notes 2, the role of in the error terms will be replaced with .
- (i) Show that
for any , if is a smooth function such that vanishes for and equals for for some constant , and for some sufficiently large depending on .
- (ii) Show that
for any smooth, compactly supported function and any .
- (iii) Suppose that , and let be such that . Let be a smooth compactly supported function. Then
for any and any smooth, compactly supported , where
- (iv) With as in (iii), and as in (i), show that
There are useful variants of the approximate functional equation for -functions that are valid in the low-lying regime , but we will not detail them here.
Exercise 55 Let be a primitive character of modulus , and let . Let denote the number of zeroes of in the rectangle (note that we are now including the lower half-plane as well as the upper half-plane, as the zeroes of need not be symmetric around the real axis when is complex). Show that
Exercise 56 (Riemann-von Mangoldt explicit formula for -functions) Let be a non-integer, and let be a primitive Dirichlet character of conductor .
- (i) If , show that
- (ii) If , show that
for some quantity depending only on .
Exercise 57 Let be the function from Exercise 67 of Notes 2. Let be a non-principal Dirichlet character of conductor . Use the twisted Poisson formula to show that
for any , where the sum is in the conditionally convergent sense. (You may need to smoothly truncate the function before applying the twisted Poisson formula.) Use this and the argument from the previous exercise to establish the bound .
— 6. Appendix: Gauss sum for real primitive characters —
The material here is not needed elsewhere in this course, or even in this set of notes, but I am including it because it is a very pretty piece of mathematics; it is also the very first hint of a much deeper connection between automorphic forms and Galois theory known as the Langlands program, which I will not be able to discuss further here.
which strongly suggests that , but one has to prevent vanishing of (or equivalently, ). (A similar argument using (52) would also give if one could somehow prevent vanishing of .) While it has been conjectured (by Chowla) that is never zero (and should in fact be positive), this is still not proven unconditionally; see this preprint of Fiorilli for recent work in this direction. (Indeed, understanding the vanishing of -functions at the central point is a very deep problem, as attested to by the difficulty of the Birch and Swinnerton-Dyer conjecture.) Nevertheless we still have the following result:
This result is surprisingly tricky to prove. The exercises below give one such proof, essentially due to Dirichlet. First we reduce to quadratic characters modulo a prime:
Exercise 59 (Classification of real primitive characters)
- (i) Let be coprime natural numbers. Show that if are real primitive characters of conductors respectively, then is a real primitive character of conductor , and that .
- (ii) Conversely, if are coprime natural numbers and is a real primitive character of conductor , show that there exist unique real primitive characters of conductors respectively such that . (Hint: use the Chinese remainder theorem to identify with and .
- (iii) If is an odd prime and is a natural number, show that an element of is a quadratic residue if and only if it is a quadratic residue after reduction to . Conclude that there are no real primitive characters of conductor if , and the only real primitive character of conductor is the quadratic character .
- (iv) If , show that an element of is a quadratic residue if and only if it is a quadratic residue after reduction to . Conclude that there are no real primitive characters of conductor if , and show that there is one such character of conductor , two characters of conductor , and no characters of conductor . Verify that for each of these characters.
In view of this exercise and the fundamental theorem of arithmetic, we see that to verify Theorem 58, it suffices to do so when for some odd prime . This can be achieved using the functional equation (30) for the theta function defined in (29):
Exercise 60 (Landsberg-Schaar relation and its consequences) Let be an odd prime, and let be the quadratic character to modulus .
- (i) Show that . (Hint: rewrite both sides in terms of the sum of over quadratic residues or non-residues .)
- (ii) For any natural numbers , establish the Landsberg-Schaar relation
by using the functional equation (30) with and sending .
- (iii) By using the case of the Landsberg-Schaar relation, show that is equal to when and when , and that .
- (iv) By applying the Landsberg-Schaar relation with an odd prime distinct from , establish the law of quadratic reciprocity
Exercise 61 Define a fundamental discriminant to be an integer that is the discriminant of a quadratic number field; recall from Supplement 1 that such take the form if is a squarefree integer with , or if is a squarefree integer with . (Supplement 1 focused primarily on the negative discriminant case when , but the above statement also holds for positive discriminant.) Show that if is a fundamental discriminant, then is a primitive real character, where is the Kronecker symbol. Conversely, show that all primitive real characters arise in this fashion.