Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely less than the harmonic series $1 +\frac{1}{2}+\frac{1}{3}+...$. Moreover, the first sum is almost the logarithm of the second sum. Euler’s proof of this again on adding and subtracting series that do not converge, and so does not meet modern standards of rigor. Unfortunately, Euler’s proof can’t really be “repaired”. http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2029%20infinitely%20many%20primes.pdf

Is it really true that this proof can't be repaired with modern tools? If it can be repaired, however, what has to be done?