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In 1737, Euler presented a novel proof of the infinitude of the primes, in showing that the harmonic sum of the primes diverges. Explicitly, Euler’s work furnishes the estimate

in which the sum exhausts the rational primes at most . At this point, it becomes quite elementary to derive the two inequalities

Results of this flavor remained essentially unimproved for over a century, until Chebyshev presented the following landmark theorem in 1852:

**Theorem (Chebyshev): ***There exist positive constants such that*

Thus Chebyshev’s Theorem shows that represents the growth rate (up to constants) of ; stated equivalently in Bachmann-Landau notation, we have .* *Yet more is true: the constants in Chebyshev’s proof are therein made effective, and can be taken as

As a corollary to Chebyshev’s Theorem, we have for. By making this implicit bound on precise, Chebyshev was able to prove Bertrand’s Postulate (thereafter known as the Bertrand-Chebyshev Theorem).

In this post, we’ll prove a variant of Chebyshev’s Theorem in great generality, and discuss some historically competitive bounds on the constants and given above. Lastly, we’ll discuss how Chebyshev’s Theorem relates to proposed “elementary” proofs of the PNT.

Let denote the permutation group on *n* letters. Herein, we consider the following question, brought to light by Landau in 1902:

**Question: ***What is the maximal order of an element in ?*

By convention, we shall refer to this maximal order as . Thus, for instance, Lagrange’s theorem gives the elementary bound . We may obtain sharper bounds by noting that is centerless, but tweaks like this will fail to give more than a gain of a multiplicative constant. A far more productive route comes from the following Proposition:

**Proposition:** *Let be a partition of n into positive integers. Then , with equality if and only if for all i.*

*Proof: *Let be a partition of *n* such that is maximized. If for any *i*, we have , and the partition

admits a strictly larger product. Thus, we may assume that for all *i*. Furthermore, the relation implies that in a maximal partition we may have for at most two such *i*. It now follows (by cases) that our maximal product takes the form

Our result follows from the inequalities .

*Note: buried under all of this is the fact that the function attains a global maximum at , and that 3 is the closest integer to e (this also underlies the marginal appearance of 2, as the second closest integer approximation to e). This will become more evident as we continue.*

As a Corollary, we obtain a new upper bound for the function :

**Corollary: ***For all n, we have , with equality if and only if .*

*Proof: *We note that is the maximum value of , as varies over the partitions of *n*. But , with equality if and only if (1) for all *i *and (2) the are pairwise coprime. In particular, equality holds above if and only if (achieved with the partition ).

*Note: this result is stronger than the commonly-cited bound , which holds for all n.*

The bound arising from our Proposition is nevertheless weak in method (in that we have only used the fact that divides ), and to strengthen it significantly requires the Prime number theorem (PNT). In fact, Landau’s classical result

(which we will derive after the fold) is equivalent to the PNT by means of elementary methods.