Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

euler and infinity 63primes is infinite. From the equation∞∑ 1n = ∏ (1 − 1 ) −1,s p s n=1 pwhere p runs over all prime numbers and s>1, he goes boldly to the limit for s → 1,writing the meaningless equation∞∑ 1n = ∏ (1 − 1 ) −1.pn=1 pThe left-hand side is the so-called harmonic series, which diverges to ∞. If therewere only finitely many primes, the right-hand side of the equation would be a finitenumber, a contradiction that proves what we want. Euler goes even further, writing theformula∑ 1= log log ∞,ppwhich is meaningless by itself. On the other hand, Euler’s reasoning is simple,rooted on solid ground, and it is not difficult for the modern mathematician to reconstructit, transforming the meaningless statement into an interesting result. It goes asfollows.Let us think of infinity as the infinity that arises from counting. Thus, 1 + 1 + 1 +···=∞. A way to approach this is via the geometric series,1 + x + x 2 + x 3 +···= 11 − x ,then letting x → 1. Thus, in a sense we can write, as Euler does, the formula1 + 1 + 1 +···= 1 0 =∞.If we take the logarithm of the geometric series, we have the series( )x + x22 + x313 +···=log ,1 − xshowing that the harmonic series, namely, the sum of the reciprocal of the naturalintegers, diverges to infinity, but much more slowly than the sum 1 + 1 + 1 +···.Wecan express this fact by writing the meaningless equation∞∑n=11n = log 1 = log ∞.0Finally, taking the logarithm of the (meaningless) expression relating the harmonicseries to the product over primes, we easily complete the basic thoughts behind Euler’sstatement that∑ 1= log log ∞.pp