prime harmonic series

The prime ¹ 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 Erdos [2].

Theorem 1   The series diverges.

Proof. Assume that this series is convergent. If so, then, for a certain , we have:$$ \sum_{i > k}\frac{1}{p_i} < \frac{1}{2},$$ 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:$$\Lambda_k(n) \leq \sum_{i>k}\left\lfloor\frac{n}{p_i}\right\rfloor \leq \sum_{i>k}\frac{n}{p_i} < \frac{n}{2}$$ However, by their definitions, for all and so it is sufficient to find an such that for a contradiction and, using the previous bound for , which is , we see that works.

The series is in some ways similar to the Harmonic series . 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).

Bibliography

1

M. AIGNER & G. M. ZIEGLER: Proofs from THE BOOK, 3 edition (2004), Springer-Verlag, 5-6.

2

P. ERDOS: Über die Reihe , Mathematica, Zutphen B 7 (1938).

Footnotes

... prime¹

denotes the set of primes.