The Riemann zeta function is defined in the region by the absolutely convergent series
For , the series on the right-hand side of (1) is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the function can be extended to this region (with a pole at ) by analytic continuation. For instance, it can be shown that after analytic continuation, one has , , and , and more generally
Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous. But as it stands, the formulae look “wrong” for several reasons. Most obviously, the summands on the left are all positive, but the right-hand sides can be zero or negative. A little more subtly, the identities do not appear to be consistent with each other. For instance, if one adds (4) to (5), one obtains
whereas if one subtracts from (5) one obtains instead
However, it is possible to interpret (4), (5), (6) by purely real-variable methods, without recourse to complex analysis methods such as analytic continuation, thus giving an “elementary” interpretation of these sums that only requires undergraduate calculus; we will later also explain how this interpretation deals with the apparent inconsistencies pointed out above.
To see this, let us first consider a convergent sum such as (2). The classical interpretation of this formula is the assertion that the partial sums
converge to as , or in other words that
where denotes a quantity that goes to zero as . Actually, by using the integral test estimate
we have the sharper result
Thus we can view as the leading coefficient of the asymptotic expansion of the partial sums of .
for , which has a vague resemblance to (7), but again the connection is not particularly clear.
The problem here is the discrete nature of the partial sum
which (if is viewed as a real number) has jump discontinuities at each positive integer value of . These discontinuities yield various artefacts when trying to approximate this sum by a polynomial in . (These artefacts also occur in (2), but happen in that case to be obscured in the error term ; but for the divergent sums (4), (5), (6), (7), they are large enough to cause real trouble.)
However, these issues can be resolved by replacing the abruptly truncated partial sums with smoothed sums , where is a cutoff function, or more precisely a compactly supported bounded function that equals at . The case when is the indicator function then corresponds to the traditional partial sums, with all the attendant discretisation artefacts; but if one chooses a smoother cutoff, then these artefacts begin to disappear (or at least become lower order), and the true asymptotic expansion becomes more manifest.
Note that smoothing does not affect the asymptotic value of sums that were already absolutely convergent, thanks to the dominated convergence theorem. For instance, we have
whenever is a cutoff function (since pointwise as and is uniformly bounded). If is equal to on a neighbourhood of the origin, then the integral test argument then recovers the decay rate:
However, smoothing can greatly improve the convergence properties of a divergent sum. The simplest example is Grandi’s series
The partial sums
oscillate between and , and so this series is not conditionally convergent (and certainly not absolutely convergent). However, if one performs analytic continuation on the series
and sets , one obtains a formal value of for this series. This value can also be obtained by smooth summation. Indeed, for any cutoff function , we can regroup
If is twice continuously differentiable (i.e. ), then from Taylor expansion we see that the summand has size , and also (from the compact support of ) is only non-zero when . This leads to the asymptotic
and so we recover the value of as the leading term of the asymptotic expansion.
Exercise 1 Show that if is merely once continuously differentiable (i.e. ), then we have a similar asymptotic, but with an error term of instead of . This is an instance of a more general principle that smoother cutoffs lead to better error terms, though the improvement sometimes stops after some degree of regularity.
Remark 1 The most famous instance of smoothed summation is Cesáro summation, which corresponds to the cutoff function . Unsurprisingly, when Cesáro summation is applied to Grandi’s series, one again recovers the value of .
If we now revisit the divergent series (4), (5), (6), (7) with smooth summation in mind, we finally begin to see the origin of the right-hand sides. Indeed, for any fixed smooth cutoff function , we will shortly show that
(which is also essentially the Mellin transform of ). Thus we see that the values (4), (5), (6), (7) obtained by analytic continuation are nothing more than the constant terms of the asymptotic expansion of the smoothed partial sums. This is not a coincidence; we will explain the equivalence of these two interpretations of such sums (in the model case when the analytic continuation has only finitely many poles and does not grow too fast at infinity) below the fold.
This interpretation clears up the apparent inconsistencies alluded to earlier. For instance, the sum consists only of non-negative terms, as does its smoothed partial sums (if is non-negative). Comparing this with (12), we see that this forces the highest-order term to be non-negative (as indeed it is), but does not prohibit the lower-order constant term from being negative (which of course it is).
while if we subtract from (12) we obtain
These two asymptotics are not inconsistent with each other; indeed, if we shift the index of summation in (17), we can write
and so we now see that the discrepancy between the two sums in (8), (9) come from the shifting of the cutoff , which is invisible in the formal expressions in (8), (9) but become manifestly present in the smoothed sum formulation.
— 1. Smoothed asymptotics —
For sake of argument we shall assume that the smooth cutoff is supported in the interval (the general case is similar, and can also be deduced from this case by redefining the parameter). Thus the sum is now only non-trivial in the range .
for any , . In particular we have
eliminating , we conclude that
Summing in , we conclude the trapezoidal rule
We apply this with , which has a norm of from the chain rule, and conclude that
The same argument does not quite work with (12); one would like to now set , but the norm is now too large ( instead of ). To get around this we have to refine the trapezoidal rule by performing the more precise Taylor expansion
where . Now we have
We cannot simultaneously eliminate both and . However, using the additional Taylor expansion
and thus on summing in , and assuming that vanishes to second order at , one has (by telescoping series)
We apply this with . After a few applications of the chain rule and product rule, we see that ; also, , , and . This gives (12).
The proof of (13) is similar. With a fourth order Taylor expansion, the above arguments give
Here we have a minor miracle (equivalent to the vanishing of the third Bernoulli number ) that the term is automatically eliminated when we eliminate the term, yielding
With , the left-hand side is , the first two terms on the right-hand side vanish, and the norm is , giving (13).
The first few values of can then be computed:
From (19) we see that
for any polynomial (with being the -fold derivative of ); indeed, (19) is precisely this identity with , and the general case then follows by linearity.
As (20) holds for all polynomials, it also holds for all formal power series (if we ignore convergence issues). If we then replace by the formal power series
we conclude the formal power series (in ) identity
leading to the familiar generating function
for the Bernoulli numbers.
If we apply (20) with equal to the antiderivative of another polynomial , we conclude that
which we rearrange as the identity
which can be viewed as a precise version of the trapezoidal rule in the polynomial case. Note that if has degree , the only the summands with can be non-vanishing.
Now let be a smooth function. We have a Taylor expansion
for and some polynomial of degree at most ; also
for and . We conclude that
Summing the telescoping series, and assuming that vanishes to a sufficiently high order at , we conclude the Euler-Maclaurin formula
We apply this with . The left-hand side is . All the terms in the sum vanish except for the term, which is . Finally, from many applications of the product rule and chain rule (or by viewing where is the smooth function ) we see that , and the claim (14) follows.
Exercise 3 Use (21) to derive Faulhaber’s formula (10). Note how the presence of boundary terms at cause the right-hand side of (10) to be quite different from the right-hand side of (14); thus we see how non-smooth partial summation creates artefacts that can completely obscure the smoothed asymptotics.
— 2. Connection with analytic continuation —
Now we connect the interpretation of divergent series as the constant term of smoothed partial sum asymptotics, with the more traditional interpretation via analytic continuation. For sake of concreteness we shall just discuss the situation with the Riemann zeta function series , though the connection extends to far more general series than just this one.
In the previous section, we have computed asymptotics for the partial sums
when is a negative integer. A key point (which was somewhat glossed over in the above analysis) was that the function was smooth, even at the origin; this was implicitly used to bound various norms in the error terms.
Now suppose that is a complex number with , which is not necessarily a negative integer. Then becomes singular at the origin, and the above asymptotic analysis is not directly applicable. However, if one instead considers the telescoped partial sum
with equal to near the origin, then by applying (22) to the function (which vanishes near the origin, and is now smooth everywhere), we soon obtain the asymptotic
for some complex number which is basically the sum of the various terms appearing in (23). By modifying the above arguments, it is not difficult to extend this asymptotic to other numbers than powers of two, and to show that is independent of the choice of cutoff .
From (24) we have
For each fixed , it is not hard to see that the expression is complex analytic in . Also, by a closer inspection of the error terms in the Euler-Maclaurin formula analysis, it is not difficult to show that for in any compact region of , these expressions converge uniformly as . Applying Morera’s theorem, we conclude that our definition of is complex analytic in the region .
We still have to connect this definition with the traditional definition (1) of the zeta function on the other half of the complex plane. To do this, we observe that
for large enough. Thus we have
for . The point of doing this is that this definition also makes sense in the region (due to the absolute convergence of the sum and integral . By using the trapezoidal rule, one also sees that this definition makes sense in the region , with locally uniform convergence there also. So we in fact have a globally complex analytic definition of , and thus a meromorphic definition of on the complex plane. Note also that this definition gives the asymptotic
near , where is Euler’s constant.
We have thus seen that asymptotics on smoothed partial sums of gives rise to the familiar meromorphic properties of the Riemann zeta function . It turns out that by combining the tools of Fourier analysis and complex analysis, one can reverse this procedure and deduce the asymptotics of from the meromorphic properties of the zeta function.
The function is in the Schwartz class. By the Fourier inversion formula, it has a Fourier representation
and so (26) can be rewritten as
Now we have
integrating by parts (which is justified when is large enough) we have
We can thus write (26) as a contour integral
Note that is compactly supported away from zero, which makes an entire function of , which is uniformly bounded whenever is bounded. Furthermore, from repeated integration by parts we see that is rapidly decreasing as , uniformly for in a compact set. Meanwhile, standard estimates show that is of polynomial growth in for in a compact set. Finally, the meromorphic function has a simple pole at (with residue ) and at (with residue ). Applying the residue theorem, we can write (26) as
for any . Using the various bounds on and , we see that the integral is . From integration by parts we have and
and thus we have
The above argument reveals that the simple pole of at is directly connected to the term in the asymptotics of the smoothed partial sums. More generally, if a Dirichlet series
has a meromorphic continuation to the entire complex plane, and does not grow too fast at infinity, then one (heuristically at least) has the asymptotic
where ranges over the poles of , and are the residues at those poles. For instance, one has the famous explicit formula
where is the von Mangoldt function, are the non-trivial zeroes of the Riemann zeta function (counting multiplicity, if any), and is an error term (basically arising from the trivial zeroes of zeta); this ultimately reflects the fact that the Dirichlet series
has a simple pole at (with residue ) and simple poles at every zero of the zeta function with residue (weighted again by multiplicity, though it is not believed that multiple zeroes actually exist).
The link between poles of the zeta function (and its relatives) and asymptotics of (smoothed) partial sums of arithmetical functions can be used to compare elementary methods in analytic number theory with complex methods. Roughly speaking, elementary methods are based on leading term asymptotics of partial sums of arithmetical functions, and are mostly based on exploiting the simple pole of at (and the lack of a simple zero of Dirichlet -functions at ); in contrast, complex methods also take full advantage of the zeroes of and Dirichlet -functions (or the lack thereof) in the entire complex plane, as well as the functional equation (which, in terms of smoothed partial sums, manifests itself through the Poisson summation formula). Indeed, using the above correspondences it is not hard to see that the prime number theorem (for instance) is equivalent to the lack of zeroes of the Riemann zeta function on the line .
With this dictionary between elementary methods and complex methods, the Dirichlet hyperbola method in elementary analytic number theory corresponds to analysing the behaviour of poles and residues when multiplying together two Dirichlet series. For instance, by using the formula (11) and the hyperbola method, together with the asymptotic
which can be obtained from the trapezoidal rule and the definition of , one can obtain the asymptotic
where is the divisor function (and in fact one can improve the bound substantially by being more careful); this corresponds to the fact that the Dirichlet series
has a double pole at with expansion
and no other poles, which of course follows by multiplying (25) with itself.
Remark 3 In the literature, elementary methods in analytic number theorem often use sharply truncated sums rather than smoothed sums. However, as indicated earlier, the error terms tend to be slightly better when working with smoothed sums (although not much gain is obtained in this manner when dealing with sums of functions that are sensitive to the primes, such as , as the terms arising from the zeroes of the zeta function tend to dominate any saving in this regard).