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Many modern mathematical proofs are a combination of conceptual arguments and technical calculations. There is something of a tradeoff between the two: one can add more conceptual arguments to try to reduce the technical computations, or vice versa. (Among other things, this leads to a Berkson paradox-like phenomenon in which a negative correlation can be observed between the two aspects of a proof; see this recent Mastodon post of mine for more discussion.)

In a recent article, Heather Macbeth argues that the preferred balance between conceptual and computational arguments is quite different for a computer-assisted proof than it is for a purely human-readable proof. In the latter, there is a strong incentive to minimize the amount of calculation to the point where it can be checked by hand, even if this requires a certain amount of ad hoc rearrangement of cases, unmotivated parameter selection, or otherwise non-conceptual additions to the arguments in order to reduce the calculation. But in the former, once one is willing to outsource any tedious verification or optimization task to a computer, the incentives are reversed: freed from the need to arrange the argument to reduce the amount of calculation, one can now describe an argument by listing the main ingredients and then letting the computer figure out a suitable way to combine them to give the stated result. The two approaches can thus be viewed as complementary ways to describe a result, with neither necessarily being superior to the other.

In this post, I would like to illustrate this computation-outsourced approach with the topic of zero-density theorems for the Riemann zeta function, in which all computer verifiable calculations (as well as other routine but tedious arguments) are performed “off-stage”, with the intent of focusing only on the conceptual inputs to these theorems.

Zero-density theorems concern upper bounds for the quantity for a given and large , which is defined as the number of zeroes of the Riemann zeta function in the rectangle . (There is also an important generalization of this quantity to -functions, but for simplicity we will focus on the classical zeta function case here). Such quantities are important in analytic number theory for many reasons, one of which is through explicit formulae such as the Riemann-von Mangoldt explicit formula

relating the prime numbers to the zeroes of the zeta function (the “music of the primes”). The better bounds one has on , the more control one has on the complicated term on the right-hand side.

Clearly is non-increasing in . The Riemann-von Mangoldt formula, together with the functional equation, gives us the asymptotic

in the case, while the prime number theorem tells us that

The various zero free regions for the zeta function can be viewed as slight improvements to (2); for instance, the classical zero-free region is equivalent to the assertion that vanishes if for some small absolute constant , and the Riemann hypothesis is equivalent to the assertion that for all .

Experience has shown that the most important quantity to control here is the exponent , defined as the least constant for which one has an asymptotic

as . Thus, for instance,

, and is a non-increasing function of , so we obtain the trivial “von Mangoldt” zero density theorem

Of particular interest is the supremal value of , which has to be at least thanks to (3). The density hypothesis asserts that the maximum is in fact exactly , or equivalently that

for all . This is of course implied by the Riemann hypothesis (which clearly implies that for all ), but is a more tractable hypothesis to work with; for instance, the hypothesis is already known to hold for by the work of Bourgain (building upon many previous authors). The quantity directly impacts our understanding of the prime number theorem in short intervals; indeed, it is not difficult using (1) (as well as the Vinogradov-Korobov zero-free region) to establish a short interval prime number theorem

for all if is a fixed exponent, or for almost all if is a fixed exponent. Until recently, the best upper bound for was , thanks to a 1972 result of Huxley; but this was recently lowered to in a breakthrough work of Guth and Maynard.

In between the papers of Huxley and Guth-Maynard are dozens of additional improvements on , though it is only the Guth-Maynard paper that actually lowered the supremum norm . A summary of most of the state of the art before Guth-Maynard may be found in Table 2 of this recent paper of Trudgian and Yang; it is complicated, but it is easy enough to get a computer to illustrate it with a plot:

(For an explanation of what is going on under the assumption of the Lindelöf hypothesis, see below the fold.) This plot represents the combined effort of nearly a dozen papers, each one of which claims one or more components of the depicted piecewise smooth curve, and is written in the “human-readable” style mentioned above, where the argument is arranged to reduce the amount of tedious computation to human-verifiable levels, even if this comes the cost of obscuring the conceptual ideas. (For an animation of how this bound improved over time, see here.) Below the fold, I will try to describe (in sketch form) some of the standard ingredients that go into these papers, in particular the routine reduction of deriving zero density estimates from large value theorems for Dirichlet series. We will not attempt to rewrite the entire literature of zero-density estimates in this fashion, but focus on some illustrative special cases.

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