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Proof that the sum of the reciprocals of the primes diverges

In the third century BC, Euclid proved the existence of infinitely many prime numbers. In the 18th century, Leonhard Euler proved a stronger statement: the sum of the reciprocals of all prime numbers diverges. Here, we present a number of proofs of this result.

The harmonic series

First, we describe how Euler originally discovered the result. He was considering the harmonic series

sum_{n=1}^infty frac{1}{n} =
1 + frac{1}{2} + frac{1}{3} + frac{1}{4} + cdots

He had already used the following "product formula" to show the existence of infinitely many primes.

sum_{n=1}^infty frac{1}{n} = prod_{p} frac{1}{1-p^{-1}}
=prod_{p} left(1+frac{1}{p}+frac{1}{p^2}+cdots right).

(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.

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:

begin{align} ln left(sum_{n=1}^infty frac{1}{n}right) & {} = ln left(prod_{p} frac{1}{1-p^{-1}}right)

sum_{p} ln left(frac{1}{1-p^{-1}}right)

sum_{p} - ln(1-p^{-1}) & {} = sum_{p} left(frac{1}{p} + frac{1}{2p^2} + frac{1}{3p^3} + cdots right) = left(sum_{p}frac{1}{p} right) + sum_{p} frac{1}{p^2} left(frac{1}{2} + frac{1}{3p} + frac{1}{4p^2} + cdots right) & {} < left(sum_{p}frac{1}{p} right) + sum_{p} frac{1}{p^2} left(1 + frac{1}{p} + frac{1}{p^2} + cdots right) = left(sum_{p} frac{1}{p} right) + left(sum_{p} frac{1}{p(p-1)} right) & {} = left(sum_{p} frac{1}{p} right) + C end{align}

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

frac{1}{2} + frac{1}{3} + frac{1}{5} + frac{1}{7} + frac{1}{11} + cdots = ln ln (+ infty).

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.

Second proof

The following proof by contradiction is due to Paul Erdős.

Let pi denote the i-th prime number. Assume that the sum of the reciprocals of the primes converges, i.e.

sum_{i=1}^infty {1over p_i} < infty,.

Then there exists a positive integer k such that

sum_{i=k+1}^infty {1over p_i} < {1over 2}.qquad(1)

For a positive integer x let Mx denote the set of those n in {1, 2, . . ., x} which are not divisible by any prime greater than pk. We will now derive an upper and a lower estimate for the number |Mx| of elements in Mx. For large x, these bounds will turn out to be contradictory.

Upper estimate: Every n in Mx can be written as n = r m2 with positive integers m and r, where r is square-free. Since only the k primes p1 . . ., pk can show up (with exponent 1) in the prime factorization of r, there are at most 2k different possibilities for r. Furthermore, there are at most √x possible values for m. This gives us the upper estimate

|M_x| le 2^ksqrt{x},.qquad(2)

Lower estimate: The remaining x − |Mx| numbers in the set difference {1, 2, . . ., x} Mx are all divisible by a prime greater than pk. Let Ni,x denote the set of those n in {1, 2, . . ., x} which are divisible by i-th prime pi. Then

{1,2,ldots,x}setminus M_xsubsetbigcup_{i=k+1}^infty N_{i,x};.

Since the number of integers in Ni,x is at most x/pi (actually zero for pi > x), we get

x-|M_x| le sum_{i=k+1}^infty |N_{i,x}|< sum_{i=k+1}^infty {xover p_i},.

Using (1), this implies

{xover 2} < |M_x|,.qquad(3)

Contradiction: For every integer x ≥ 22k + 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 square-free integer and a square. This gives the inequality

sum_{i=1}^n{frac{1}{i}} le prod_{p le n}{biggl(1+frac{1}{p}biggr)}sum_{k=1}^n{frac{1}{k^2}},,
where for every i between 1 and n the (expanded) product contains to the square-free part of i and the sum contains to the square part of i (see fundamental theorem of arithmetic).

ln(n+1) = int_1^{n+1}frac{dx}x

sum_{i

1}^nunderbrace{int_i^{i+1}frac{dx}x}_{{}<1/i}

  • The lower estimate 1 + x < exp(x) for the exponential function, which holds for all x > 0.
  • The upper bound (using a telescoping sum) for the partial sums (convergence is all we really need)

sum_{k=1}^n{frac{1}{k^2}} le1+sum_{k=2}^nunderbrace{biggl(frac1{k-1}-frac1kbiggr)}_{ge1/k^2}=2-frac1nle2.

Combining all these inequalities, we see that

ln(n+1) < sum_{i=1}^nfrac{1}{i} le prod_{p le n}{biggl(1+frac{1}{p}biggr)}sum_{k=1}^n{frac{1}{k^2}} < 2prod_{p le n}{expbiggl(frac{1}{p}biggr)} = 2expbiggl(sum_{p le n}{frac{1}{p}}biggr). Dividing through by 2 and taking the natural logarithm of both sides gives
ln ln(n+1) - ln2 < sum_{p le n}{frac{1}{p}}
as desired. 

Using

sum_{k=1}^infty{frac{1}{k^2}}=frac{pi^2}6 (see Basel problem), the above constant ln 2 = 0.6931... can be improved to ln(π2/6) = 0.4977...; in fact it turns out that
lim_{n to infty } biggl(
sum_{p leq n} frac{1}{p} - ln(ln(n)) biggr)= M, 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

p_n < n ln n + n ln ln n quadmbox{for } n ge 6.

Then

sum_{n=1}^infty frac1{ p_n} ge sum_{n=6}^infty frac1{ p_n} ge sum_{n=6}^infty frac1{ n ln n + n ln ln n} ge sum_{n=6}^infty frac1{2n ln n} = infty by the integral test for convergence. This shows that the series on the left diverges.

See also

References

External links

  • Chris K. Caldwell: "There are infinitely many primes, but, how big of an infinity?", http://www.utm.edu/research/primes/infinity.shtml

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