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 3rd-century-BC result that there are infinitely many prime numbers. There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that for all natural numbers. The double natural logarithm indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant.
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 the set of all primes. Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. 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.
Proofs
Euler's proof
Euler considered 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 as well as the sum of a converging series: for a fixed constant. Then he invoked the relation which he explained, for instance in a later 1748 work, by setting in the Taylor series expansion This allowed him to conclude that It is almost certain that Euler meant that the sum of the reciprocals of the primes less than is asymptotic to as approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. Thus Euler obtained a correct result by questionable means.
Erdős's proof by upper and lower estimates
The following proof by contradiction is due to Paul Erdős. Let denote the th prime number. Assume that the sum of the reciprocals of the primes converges Then there exists a smallest positive integer such that For a positiveinteger, let denote the set of those in which are not divisible by any prime greater than . We will now derive an upper and a lower estimate for, the number of elements in. For large , these bounds will turn out to be contradictory. Upper estimate: Lower estimate: This produces a contradiction: when, the estimates and cannot both hold, because.
Proof that the series exhibits log-log growth
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. The proof is an adaptation of the product expansion idea of Euler. In the following, a sum or product taken over always represents a sum or product taken over a specified set of primes. The proof rests upon the following four inequalities:
Every positive integer can be uniquely expressed as the product of a square-free integer and a square. This gives the inequality
Combining all these inequalities, we see that Dividing through by and taking the natural logarithm of both sides gives as desired. ∎ Using , the above constant can be improved to ; in fact it turns out that where is the Meissel–Mertens constant.
Suppose for contradiction the sum converged. Then, there exists such that. Call this sum. Now consider the convergent geometric series. This geometric series contains the sum of reciprocals of all numbers whose prime factorization contain only primes in the set. Consider the subseries. This is a subseries because is not divisible by any. However, by the Limit comparison test, this subseries diverges by comparing it to the harmonic series. Indeed,. Thus, we have found a divergent subseries of the original convergent series, and since all terms are positive, this gives the contradiction. We may conclude diverges.
Partial sums
While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof is by induction: The first partial sum is, which has the form. If the th partial sum has the form, then the st sum is as the st prime is odd; since this sum also has an form, this partial sum cannot be an integer, and the induction continues. Another proof rewrites the expression for the sum of the first reciprocals of primes in terms of the least common denominator, which is the product of all these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime does divide the denominator. Thus the expression is irreducible and is non-integer.
