Divergence of the sum of the reciprocals of the primes

The theorem that summing the reciprocals of all prime numbers gives an unbounded total, although the primes become increasingly sparse.

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Euler product formula: An identity expressing the Riemann zeta function as an infinite product over primes for complex inputs with real part greater than one. Taking logarithms of the product near exponent one isolates the reciprocal-prime sum.
Leonhard Euler: An eighteenth-century mathematician whose work shaped number theory, analysis, mechanics, and many other fields. Euler first established the divergence using a product over primes and the harmonic series.
Prime number theorem: The theorem that the number of primes up to x is asymptotic to x divided by the natural logarithm of x. It gives a precise distribution law that implies the reciprocal-prime sum diverges at logarithmic-logarithmic scale.
Sum of the reciprocals of the squares of the primes: The infinite sum of 1/p² over prime numbers p; it converges because it is bounded by the sum over all integers. Raising the denominator’s exponent from one to two changes divergence into convergence.
Harmonic series: The infinite series whose terms are the reciprocals of the positive integers; its partial sums grow without bound. Euler’s comparison shows that a bounded prime-reciprocal sum would contradict this series’ divergence.
Euler's proof of the infinitude of primes: A proof that assumes finitely many primes and uses their product to construct a number with a new prime divisor. This closely related argument reflects Euler’s method of extracting consequences from prime products.
Mertens' theorems: Three asymptotic results in number theory concerning primes, including the growth of sums of reciprocal primes. Mertens’ estimate identifies the leading growth as the logarithm of the logarithm of the cutoff.
Sum of reciprocals of the twin primes: The sum of reciprocals of primes p for which p+2 is also prime; whether it converges is unknown. A much thinner prime subset has an unresolved reciprocal sum, unlike the sum over all primes.
Riemann zeta function: The analytic function initially defined by summing reciprocal powers of positive integers where that series converges. Its growth as the exponent approaches one supplies the analytic input in Euler’s proof.
History of analytic number theory: The development of methods using analysis to study integers, primes, and their distribution. The reciprocal-prime theorem became an early example of analysis revealing the distribution of primes.