A major topic of interest of analytic number theory is the asymptotic behaviour of the Riemann zeta function in the critical strip in the limit . For the purposes of this set of notes, it is a little simpler technically to work with the log-magnitude of the zeta function. (In principle, one can reconstruct a branch of , and hence itself, from using the Cauchy-Riemann equations, or tools such as the Borel-Carathéodory theorem, see Exercise 40 of Supplement 2.)
One has the classical estimate
(See e.g. Exercise 37 from Supplement 3.) In view of this, let us define the normalised log-magnitudes for any by the formula
informally, this is a normalised window into near . One can rephrase several assertions about the zeta function in terms of the asymptotic behaviour of . For instance:
- (i) The bound (1) implies that is asymptotically locally bounded from above in the limit , thus for any compact set we have for and sufficiently large. In fact the implied constant in only depends on the projection of to the real axis.
- (ii) For , we have the bounds
which implies that converges locally uniformly as to zero in the region .
- (iii) The functional equation, together with the symmetry , implies that
which by Exercise 17 of Supplement 3 shows that
as , locally uniformly in . In particular, when combined with the previous item, we see that converges locally uniformly as to in the region .
- (iv) From Jensen’s formula (Theorem 16 of Supplement 2) we see that is a subharmonic function, and thus is subharmonic as well. In particular we have the mean value inequality
for any disk , where the integral is with respect to area measure. From this and (ii) we conclude that
for any disk with and sufficiently large ; combining this with (i) we conclude that is asymptotically locally bounded in in the limit , thus for any compact set we have for sufficiently large .
From (v) and the usual Arzela-Ascoli diagonalisation argument, we see that the are asymptotically compact in the topology of distributions: given any sequence tending to , one can extract a subsequence such that the converge in the sense of distributions. Let us then define a normalised limit profile of to be a distributional limit of a sequence of ; they are analogous to limiting profiles in PDE, and also to the more recent introduction of “graphons” in the theory of graph limits. Then by taking limits in (i)-(iv) we can say a lot about such normalised limit profiles (up to almost everywhere equivalence, which is an issue we will address shortly):
- (i) is bounded from above in the critical strip .
- (ii) vanishes on .
- (iii) We have the functional equation for all . In particular for .
- (iv) is subharmonic.
Unfortunately, (i)-(iv) fail to characterise completely. For instance, one could have for any convex function of that equals for , for , and obeys the functional equation , and this would be consistent with (i)-(iv). One can also perturb such examples in a region where is strictly convex to create further examples of functions obeying (i)-(iv). Note from subharmonicity that the function is always going to be convex in ; this can be seen as a limiting case of the Hadamard three-lines theorem (Exercise 41 of Supplement 2).
We pause to address one minor technicality. We have defined as a distributional limit, and as such it is a priori only defined up to almost everywhere equivalence. However, due to subharmonicity, there is a unique upper semi-continuous representative of (taking values in ), defined by the formula
for any (note from subharmonicity that the expression in the limit is monotone nonincreasing as , and is also continuous in ). We will now view this upper semi-continuous representative of as the canonical representative of , so that is now defined everywhere, rather than up to almost everywhere equivalence.
By a classical theorem of Riesz, a function is subharmonic if and only if the distribution is a non-negative measure, where is the Laplacian in the coordinates. Jensen’s formula (or Greens’ theorem), when interpreted distributionally, tells us that
away from the real axis, where ranges over the non-trivial zeroes of . Thus, if is a normalised limit profile for that is the distributional limit of , then we have
where is a non-negative measure which is the limit in the vague topology of the measures
Thus is a normalised limit profile of the zeroes of the Riemann zeta function.
Using this machinery, we can recover many classical theorems about the Riemann zeta function by “soft” arguments that do not require extensive calculation. Here are some examples:
Proof: It suffices to show that any limiting profile (arising as the limit of some ) vanishes on the critical line . But if the Riemann hypothesis holds, then the measures are supported on the critical line , so the normalised limit profile is also supported on this line. This implies that is harmonic outside of the critical line. By (ii) and unique continuation for harmonic functions, this implies that vanishes on the half-space (and equals on the complementary half-space, by (iii)), giving the claim.
In fact, we have the following sharper statement:
Theorem 2 (Backlund) The Lindelöf hypothesis is equivalent to the assertion that for any fixed , the number of zeroes in the region is as .
Proof: If the latter claim holds, then for any , the measures assign a mass of to any region of the form as for any fixed and . Thus the normalised limiting profile measure is supported on the critical line, and we can repeat the previous argument.
Conversely, suppose the claim fails, then we can find a sequence and such that assigns a mass of to the region . Extracting a normalised limiting profile, we conclude that the normalised limiting profile measure is non-trivial somewhere to the right of the critical line, so the associated subharmonic function is not harmonic everywhere to the right of the critical line. From the maximum principle and (ii) this implies that has to be positive somewhere on the critical line, but this contradicts the Lindelöf hypothesis. (One has to take a bit of care in the last step since only converges to in the sense of distributions, but it turns out that the subharmonicity of all the functions involved gives enough regularity to justify the argument; we omit the details here.)
Theorem 3 (Littlewood) Assume the Lindelöf hypothesis. Then for any fixed , the number of zeroes in the region is as .
Proof: By the previous arguments, the only possible normalised limiting profile for is . Taking distributional Laplacians, we see that the only possible normalised limiting profile for the zeroes is Lebesgue measure on the critical line. Thus, can only converge to as , and the claim follows.
Even without the Lindelöf hypothesis, we have the following result:
Theorem 4 (Titchmarsh) For any fixed , there are zeroes in the region for sufficiently large .
Among other things, this theorem recovers a classical result of Littlewood that the gaps between the imaginary parts of the zeroes goes to zero, even without assuming unproven conjectures such as the Riemann or Lindelöf hypotheses.
Proof: Suppose for contradiction that this were not the case, then we can find and a sequence such that contains zeroes. Passing to a subsequence to extract a limit profile, we conclude that the normalised limit profile measure assigns no mass to the horizontal strip . Thus the associated subharmonic function is actually harmonic on this strip. But by (ii) and unique continuation this forces to vanish on this strip, contradicting the functional equation (iii).
Exercise 5 Use limiting profiles to obtain the matching upper bound of for the number of zeroes in for sufficiently large .
Remark 6 One can remove the need to take limiting profiles in the above arguments if one can come up with quantitative (or “hard”) substitutes for qualitative (or “soft”) results such as the unique continuation property for harmonic functions. This would also allow one to replace the qualitative decay rates with more quantitative decay rates such as or . Indeed, the classical proofs of the above theorems come with quantitative bounds that are typically of this form (see e.g. the text of Titchmarsh for details).
Exercise 7 Let denote the quantity , where the branch of the argument is taken by using a line segment connecting to (say) , and then to . If we have a sequence producing normalised limit profiles for and the zeroes respectively, show that converges in the sense of distributions to the function , or equivalently
Conclude in particular that if the Lindelöf hypothesis holds, then as .
A little bit more about the normalised limit profiles are known unconditionally, beyond (i)-(iv). For instance, from Exercise 3 of Notes 5 we have as , which implies that any normalised limit profile for is bounded by on the critical line, beating the bound of coming from convexity and (ii), (iii), and then convexity can be used to further bound away from the critical line also. Some further small improvements of this type are known (coming from various methods for estimating exponential sums), though they fall well short of determining completely at our current level of understanding. Of course, given that we believe the Riemann hypothesis (and hence the Lindelöf hypothesis) to be true, the only actual limit profile that should exist is (in fact this assertion is equivalent to the Lindelöf hypothesis, by the arguments above).
Better control on limiting profiles is available if we do not insist on controlling for all values of the height parameter , but only for most such values, thanks to the existence of several mean value theorems for the zeta function, as discussed in Notes 6; we discuss this below the fold.
— 1. Limiting profiles outside of exceptional sets —
In order to avoid an excessive number of extraction of subsequences and discarding of exceptional sets, we now move away from the standard sequential notion of a limit, and instead work with the less popular, but equally valid notion of an ultrafilter limit. Recall that an ultrafilter on a set is a collection of subsets of (which we will call the “-large” sets) which are the sets of full measure with regards to some finitely additive -valued probability measure on (with the power set Boolean algebra ). We call a subset of -small if it is not -large. Given a function into a topological space and a point , we say that converges to along if is -large for every neighbourhood of , and then we call a -limit of .
Exercise 8 Let be a function into a topological space , and let be an ultrafilter on .
- (i) If is compact, show that has at least one -limit.
- (ii) If is Hausdorff, show that has at most one -limit.
- (iii) Conversely, if fails to be compact (resp. Hausdorff), show that there exists a function and an ultrafilter on such that has no -limit (resp. more than one -limit).
In particular, given an ultrafilter on the non-negative reals , which is non-principal in the sense that all compact subsets of are -small,, there exists a unique normalised limiting profile that is the limit of along , and similarly for . Because the distributional topology is second countable, such limiting profiles are also limiting profiles of sequences as in the previous discussion, and so we retain all existing properties of limit profiles such as (i)-(iv). However, in the ultrafilter formalism we can now easily avoid various “small” exceptional sets of , in addition to the compact sets that have already been excluded. For instance, let us call an ultrafilter generic if every Lebesgue measurable subset of of zero upper density (thus has Lebesgue measure as ) is -small. The existence of generic ultrafilters follows easily from Zorn’s lemma. Define a generic limit profile to be a limit profile that arises from a generic ultrafilter; informally, these capture the possible behaviour of the zeta function outside of a set of heights of zero density. To see how these profiles are better than arbitrary limit profiles, we recall from Exercise 2 of Notes 6 that
if the are -separated elements of and are arbitrary complex coefficients. If we set , we can conclude (among other things), that for any constant , one has
for all outside of a set of measure (informally: “square root cancellation occurs generically”). Using this, one can for instance show that
for all outside of a set of measure , which implies that any generic limit profile vanishes on the critical line, and thus must be ; that is to say, the Lindelöf hypothesis is true “generically”.
One can profitably explore the regime between arbitrary non-principal ultrafilters and generic ultrafilters by introducing the intermediate notion of an -generic ultrafilter for any , defined as an ultrafilter with the property that any Lebesgue measurable subset of of “dimension at most ” in the sense that has measure , is -small. One can then interpret many existing mean value theorems on the zeta function (or on other Dirichlet series) as controlling the -generic limit profiles of , or more generally of the log-magnitude of various Dirichlet series (e.g. for various exponents ). For instance, the previous argument shows that
for all outside of a set of measure , which implies that any -generic limit profile is bounded above by on the critical line. One can also recast much of the arguments in Notes 6 in this language (defining limit profiles for various Dirichlet polynomials, and using such profiles and zero-detecting polynomials to establish -generic zero-free regions), although this is mostly just a change of notation and does not seem to yield any major simplifications to these arguments.