prime harmonic series
square-free part \PMlinkescapephraseset of primes \PMlinkescapephraseharmonic series
The prime11 denotes the set of primes. harmonic series (also known as series of reciprocals of primes) is the infinite sum . The following result was originally proved by Euler (using the Euler product of the Riemann Zeta function) but the following extremely elegant proof is due to Paul Erdős .
The series diverges.
Assume that this series is convergent. If so, then, for a certain , we have:
where is the prime. Now, we define , the number of integers less than divisible only by the first primes. Any of these numbers can be expressed as (i.e. a square multiplied by a square-free number). There are ways to chose the square-free part and clearly , so . Now, the number of integers divisible by less than is , so the number of integers less than divisible by primes bigger than (which we shall denote by ) is bounded above as follows:
The series is in some ways similar to the Harmonic series (http://planetmath.org/HarmonicSeries) . In fact, it is well known that , where is Euler’s constant, and this series obeys the similar asymptotic relation , where and is sometimes called the Mertens constant. Its divergence, however, is extremely slow: for example, taking as the biggest currently known prime, the Mersenne prime , we get (while which is enormous considering ’s also slow divergence).
- 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3 edition (2004), Springer-Verlag, 5–6.
- 2 P. Erdős: Über die Reihe , Mathematica, Zutphen B 7 (1938).
|Title||prime harmonic series|
|Date of creation||2013-03-22 15:07:16|
|Last modified on||2013-03-22 15:07:16|
|Last modified by||Cosmin (8605)|
|Synonym||series of reciprocals of primes|