Effective de la Vallé Poussin style bounds
on the first Chebyshev function

In 1898 Charles Jean de la Vallé Poussin developed an ineffective bound on the first Chebyshev function of the form [1]:

This bound holds for $x$ sufficiently large, $x\geq x_{0}$, and $K$ some unspecified positive constant.

Subsequent work over the last 50 years has developed a large number of related but distinct fully effective bounds of the form [2, 3, 4, 5]:

• •

  For some widely applicable effective bounds of this type see Table I.
  (A straightforward elementary numerical computation is required to determine the numerical coefficients in the Schoenfeld [2] and Trudgian [3] bounds.)

• •

  For some asymptotically more stringent effective bounds of this type, but valid on significantly more restricted regions, see Table II (based on reference [4]), and the extensive tabulations in reference [6].

What I have not yet seen is any attempt to take the effective bounds of Tables I and II and use them to make the original de la Vallé Poussin bound fully effective. Here are two particularly clean fully effective versions of the de la Vallé Poussin bound:

I shall explain how to derive these bounds below.

Note that for any $b>0$, $c>0$, and any $\tilde{c}\in(0,c)$, elementary calculus implies:

The key observation here is that the quantity in braces is explicitly bounded, and achieves a global maximum at $x_{peak}=\exp\left(\left[2b/(c-\tilde{c})\right]^{2}\right)$. Consequently we have the following lemma.

First let us consider some widely applicable derived bounds of the de la Vallé Poussin form, as presented in Table III. Note that the selection of a specific value of $\tilde{c}$ is a choice, and the computation of $\tilde{a}$ is then immediate — there is an infinite number of other effective bounds of de la Vallé Poussin form that we could develop. Determining $x_{*}$ then requires computationally checking low values of $x$.

By now relaxing the prefactor $\tilde{a}$, one can increase the range of validity of the bound, (ie, decrease $x_{*}$). In this way, after some computation, one finds

Note that this could in principle have been deduced as early as 1976, some 46 years ago, from the work of Schoenfeld [2].

Similarly

Note that this particular bound could in principle have been deduced as early as 2016, some 6 years ago, from the work of Trudgian [3]. For the other two widely applicable bounds, some preliminary experimental investigations suggest that it might be possible to reduce the numerical prefactors (295, 385) significantly — but doing so would require rather different techniques from the elementary observations made above.

As always one can trade of stringency against range of validity. In this regard let me mention two specific examples

and

In contrast, for some asymptotically stringent bounds, based on the Johnston–Yang results presented in reference [4], consider Table IV. Note that we can again make the exponential factor smaller, (by increasing $\tilde{c}$), at the cost of making the numerical prefactor $\tilde{a}$ larger. For instance one can deduce the ineffective bound

which can be made effective as (for instance):

Many variations on this theme can be developed.

With some hindsight, deriving effective bounds of the de la Vallé Poussin form is, (given various effective results obtained over the last 50 years [2, 3, 4, 5, 6]), seen to be almost trivial. Certainly, (given the effective results reported in [2, 3, 4, 5, 6]), nothing deeper than elementary calculus and some slightly tedious numerical checking was required. On the other hand, conceptually it is very pleasant to see simple explicit and effective bounds of the de la Vallé Poussin form dropping out so nicely.

MV was directly supported by the Marsden Fund, via a grant administered by the Royal Society of New Zealand.