Consider the Euler product
 |
(1)
|
where 
is the Riemann zeta function and 
is the 
th prime. 
, but taking the finite product up to 
, premultiplying by a factor 
, and letting 
gives
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the Euler-Mascheroni constant (Havil
2003, p. 173). This amazing result is known as the Mertens theorem.
At least for 
,
the sequence of finite products approaches 
strictly from above (Rosser and Schoenfeld 1962). However,
it is highly likely that the finite product is less than its limiting value for infinitely
many values of 
,
which is usually the case for any such inequality due to the presence of zeros of
on the critical
line ![R[s]=1/2](/images/equations/MertensTheorem/Inline19.svg)
.
An example is Littlewood's famous proof that the sense of the inequality 
, where 
is the prime counting
function and 
is the logarithmic integral, reverses infinitely
often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend
[this] result to show that [the Mertens inequality] fails for large 
; we have not investigated the matter," a full proof of
the reversal of the inequality for terms in the Mertens theorem does not seem to
appear anywhere in the published literature.
A closely related result is obtained by noting that
 |
(4)
|
Considering the variation of (3) with the 
sign changed to a 
sign and the 
moved from the denominator to the numerator then gives
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
The sequence of finite products approaches its limiting value strictly from below for the same range as for the Mertens theorem, since this inequality from below is a consequence of the Mertens inequality from above.
Edwards (2001, pp. 5-6) remarks, "For the first 30 years after Riemann's [1859] paper was published, there was virtually no progress in the field [of prime number asymptotics]," adding as a footnote, "A major exception to this statement was Mertens's Theorem of 1874...." (The celebrated prime number theorem was not proved until 1896.)
