# prime harmonic series

The prime11$\mathbb{P}$ denotes the set of primes. harmonic series (also known as series of reciprocals of primes) is the infinite sum $\sum_{p\in\mathbb{P}}\frac{1}{p}$. 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 [2].

###### Theorem.

The series $\sum_{p\in\mathbb{P}}\frac{1}{p}$ diverges.

###### Proof.

Assume that this series is convergent. If so, then, for a certain $k\in\mathbb{N}$, we have:

 $\sum_{i>k}\frac{1}{p_{i}}<\frac{1}{2},$

where $p_{i}$ is the $i^{\mathrm{th}}$ prime. Now, we define $\lambda_{k}(n):=\#\{m\leq n:p_{i}\!\mid\!m\Rightarrow i\leq k\}$, the number of integers less than $n$ divisible only by the first $k$ primes. Any of these numbers can be expressed as $m=p_{1}^{\omega_{1}}\cdots p_{k}^{\omega_{k}}\cdot s^{2},\;\omega_{i}\in\{0,1\}$ (i.e. a square multiplied by a square-free number). There are $2^{k}$ ways to chose the square-free part and clearly $s\leq\sqrt{n}$, so $\lambda_{k}(n)\leq 2^{k}\sqrt{n}$. Now, the number of integers divisible by $p_{i}$ less than $n$ is $\bigl{\lfloor}\frac{n}{p_{i}}\bigr{\rfloor}$, so the number of integers less than $n$ divisible by primes bigger than $p_{k}$ (which we shall denote by $\Lambda_{k}(n)$) 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, $\lambda_{k}(n)+\Lambda_{k}(n)=n$ for all $n\in\mathbb{N}$ and so it is sufficient to find an $n$ such that $\lambda_{k}(n)\leq\frac{n}{2}$ for a contradiction and, using the previous bound for $\lambda_{k}(n)$, which is $\lambda_{k}(n)\leq 2^{k}\sqrt{n}$, we see that $n=2^{2k+2}$ works. ∎

The series is in some ways similar to the Harmonic series (http://planetmath.org/HarmonicSeries) $H_{n}:=\sum_{k=1}^{n}\frac{1}{n}$. In fact, it is well known that $H_{n}=\log n+\gamma+o(1)$, where $\gamma=0.577215664905\dots$ is Euler’s constant, and this series obeys the similar asymptotic relation $\sum_{k=1}^{n}\frac{1}{p_{k}}=\log\log n+C+o(1)$, where $C=0.26149721\dots$ and is sometimes called the Mertens constant. Its divergence, however, is extremely slow: for example, taking $n$ as the biggest currently known prime, the $42^{\mathrm{nd}}$ Mersenne prime $M_{25964951}$, we get $\log\log(2^{25964951}-1)+C\approx 16.9672$ (while $H_{M_{25964951}}\approx 1.79975\cdot 10^{17}$ which is enormous considering $H_{n}$’s also slow divergence).

## References

• 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3${}^{\mathrm{rd}}$ edition (2004), Springer-Verlag, 5–6.
• 2 P. Erdős: Über die Reihe $\sum\frac{1}{p}$, Mathematica, Zutphen B 7 (1938).
 Title prime harmonic series Canonical name PrimeHarmonicSeries Date of creation 2013-03-22 15:07:16 Last modified on 2013-03-22 15:07:16 Owner Cosmin (8605) Last modified by Cosmin (8605) Numerical id 17 Author Cosmin (8605) Entry type Theorem Classification msc 40A05 Classification msc 11A41 Synonym series of reciprocals of primes Related topic Prime Related topic HarmonicSeries Related topic HarmonicNumber Related topic Series Related topic EulersConstant Related topic HarmonicSeriesOfPrimes