Mertens' theorems

Under the set of Mertens, named after the mathematician Franz Mertens, is understood in the mathematical subfield of analytic number theory, a number of statements about the asymptotic behavior of series which are formed from reciprocals of the primes.

Definitions

For the formulation of the statements, we first recall the O- notation, with the growth of functions ( with the help of a function, where is a constant), can be compared. One writes, if there is a constant and such that, for all of them. The functions appearing in the formulas of Mertens is the natural logarithm, with the designated Mangoldt function and the Chebyshev function

It runs through all primes and all natural numbers; is summed only over those, respectively, and, for each of the specified under the summation sign is fulfilled. From this brief notation use is made in the now presented formulas.

The formulas of Mertens

For the above functions the following relations hold.

There is a constant such that

Here denotes the Euler - Mascheroni constant, and it is

Comments

The original formulas Mertens are ( 2), ( 4) and (5). Mertens calls less precise versions of (4) and (5) " Strange formulas " of Legendre. That the number of the reciprocal values ​​of all primes diverging ( as ) has been known Euler. Formula ( 4) describes precisely how quickly diverges to infinity series. The last formula is a consequence thereof, as shown in the mentioned textbook by Hardy and Wright. The formulas were proved for the first time in 1874 by Franz Mertens. The formula (4) was detected by Chebyshev, but his proof used the Legendre - Gauss guess who was shown in 1896, and thereafter became known as the Prime Number Theorem. Mertens, however, did not use unproven ( in 1874 ) conjecture. His proof is noteworthy for two reasons. Mertens had the idea to first prove ( 2), ( 4) relatively easily follows. Second, we now know that the formula (4) is equivalent to "almost" the prime number theorem: in fact, this is equivalent to