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 n. The double natural logarithm indicates that the divergence might be very slow, which is indeed the case, see Meissel–Mertens constant.
Read more about Divergence Of The Sum Of The Reciprocals Of The Primes: The Harmonic Series
Famous quotes containing the word sum:
“The real risks for any artist are taken ... in pushing the work to the limits of what is possible, in the attempt to increase the sum of what it is possible to think. Books become good when they go to this edge and risk falling over it—when they endanger the artist by reason of what he has, or has not, artistically dared.”
—Salman Rushdie (b. 1947)