Divergence of the sum of the reciprocals of the primes

The sum of the reciprocals of all prime numbers diverges, that is:
This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rdcenturyBC result that there are infinitely many prime numbers.
There is a variety of proofs of Euler's result, including a lower bound for the partial sums stating that
for all natural numbers n. The iterated natural logarithm indicates that the divergence might be very slow, which is indeed the case, see Meissel–Mertens constant.
Contents
The harmonic series
First, we describe how Euler originally discovered the result. He was considering the harmonic series
He had already used the following "product formula" to show the existence of infinitely many primes.
(Here, the product is taken over all primes p; in the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes, unless noted otherwise.)
Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. (Multiply out the right side as you would like to do.) Of course, the above "equation" is not necessary because the harmonic series is known (by other means) to diverge. This type of formal manipulation was common at the time, when mathematicians were still experimenting with the new tools of calculus.^{[citation needed]}
Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series. (In modern language, we now say that the existence of infinitely many primes is reflected by the fact that the Riemann zeta function has a simple pole at s = 1.)
First proof
Euler took the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the Taylor series expansion for ln(1 − x) as well as the sum of a geometric series:
for a fixed constant C < 1. Since the sum of the reciprocals of the first n positive integers is asymptotic to ln(n), (i.e. their ratio approaches one as n approaches infinity), Euler then concluded
It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to ln(ln(n)) as n approaches infinity. It turns out this is indeed the case; Euler had reached a correct result by questionable means.
Simpler version of the above proof
Since
we get e^{x} > 1 + x and x > ln(1 + x).
So
Hence diverges. But 1/(p_{i} − 1) < 1/p_{i−1} where p_{i} is the i^{th} prime. Hence diverges.
Second proof
The following proof by contradiction is due to Paul Erdős.
Let p_{i} denote the i^{th} prime number. Assume that the sum of the reciprocals of the primes converges, i.e.
Then there exists a positive integer k such that
For a positive integer x let M_{x} denote the set of those n in {1, 2, . . ., x} which are not divisible by any prime greater than p_{k}. We will now derive an upper and a lower estimate for the number M_{x} of elements in M_{x}. For large x, these bounds will turn out to be contradictory.
Upper estimate: Every n in M_{x} can be written as n = r m^{2} with positive integers m and r, where r is squarefree. Since only the k primes p_{1} . . ., p_{k} can show up (with exponent 1) in the prime factorization of r, there are at most 2^{k} different possibilities for r. Furthermore, there are at most √x possible values for m. This gives us the upper estimate
Lower estimate: The remaining x − M_{x} numbers in the set difference {1, 2, . . ., x} \ M_{x} are all divisible by a prime greater than p_{k}. Let N_{i,x} denote the set of those n in {1, 2, . . ., x} which are divisible by the i^{th} prime p_{i}. Then
Since the number of integers in N_{i,x} is at most x/p_{i} (actually zero for p_{i} > x), we get
Using (1), this implies
Contradiction: For every integer x ≥ 2^{2k + 2}, the estimates (2) and (3) cannot hold simultaneously.
Third proof
Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as ln(ln(n)). The proof is an adaptation of the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes.
The proof rests upon the following four inequalities:
 Every positive integer i can be uniquely expressed as the product of a squarefree integer and a square. This gives the inequality

 where for every i between 1 and n the (expanded) product contains to the squarefree part of i and the sum contains to the square part of i (see fundamental theorem of arithmetic).
 The upper estimate for the natural logarithm
 The lower estimate 1 + x < exp(x) for the exponential function, which holds for all x > 0.
 Let n ≥ 2. The upper bound (using a telescoping sum) for the partial sums (convergence is all we really need)
Combining all these inequalities, we see that
Dividing through by 5/3 and taking the natural logarithm of both sides gives
as desired. ∎
Using
(see Basel problem), the above constant ln (5/3) = 0.51082... can be improved to ln(π^{2}/6) = 0.4977...; in fact it turns out that
where M = 0.261497... is the Meissel–Mertens constant (somewhat analogous to the much more famous Euler–Mascheroni constant).
Fourth proof
From Dusart's inequality (see PNT), we get
Then
by the integral test for convergence. This shows that the series on the left diverges.
See also
 Euclid's theorem that there are infinitely many primes
 Small set (combinatorics)
 Brun's theorem
References
 William Dunham (1999). Euler The Master of Us All. MAA. pp. 61–79. ISBN 0883853280.
External links
Categories: Theorems in number theory
 Mathematical series
 Prime numbers
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