Editing Harmonic series (mathematics) (section)

===Counting primes and divisors===
{{main|Divergence of the sum of the reciprocals of the primes}}
In 1737, [[Leonhard Euler]] observed that, as a [[formal sum]], the harmonic series is equal to an [[Euler product]] in which each term comes from a [[prime number]]: <math display=block>\sum_{i=1}^{\infty}\frac{1}{i}=\prod_{p\in\mathbb{P}}\left(1+\frac1p+\frac1{p^2}+\cdots\right)=\prod_{p\in\mathbb{P}} \frac{1}{1-1/p},</math> where <math>\mathbb{P}</math> denotes the set of prime numbers. The left equality comes from applying the [[distributive law]] to the product and recognizing the resulting terms as the [[prime factorization]]s of the terms in the harmonic series, and the right equality uses the standard formula for a [[geometric series]]. The product is divergent, just like the sum, but if it converged one could take logarithms and obtain <math display=block>\ln \prod_{p\in\mathbb{P}} \frac{1}{1-1/p}=\sum_{p\in\mathbb{P}}\ln\frac{1}{1-1/p}=\sum_{p\in\mathbb{P}}\left(\frac1p+\frac1{2p^2}+\frac1{3p^3}+\cdots\right)=\sum_{p\in\mathbb{P}}\frac1p+K.</math>
Here, each logarithm is replaced by its [[Taylor series]], and the constant <math>K</math> on the right is the evaluation of the convergent series of terms with exponent greater than one. It follows from these manipulations that the sum of reciprocals of primes, on the right hand of this equality, must diverge, for if it converged these steps could be reversed to show that the harmonic series also converges, which it does not. An immediate corollary is that [[Euclid's theorem|there are infinitely many prime numbers]], because a finite sum cannot diverge.{{r|euler}} Although Euler's work is not considered adequately rigorous by the standards of modern mathematics, it can be made rigorous by taking more care with limits and error bounds.{{r|rubsal}} Euler's conclusion that the partial sums of reciprocals of primes grow as a [[double logarithm]] of the number of terms has been confirmed by later mathematicians as one of [[Mertens' theorems]],{{r|pollack}} and can be seen as a precursor to the [[prime number theorem]].{{r|rubsal}}

Another problem in [[number theory]] closely related to the harmonic series concerns the average number of [[divisor]]s of the numbers in a range from 1 to <math>n</math>, formalized as the [[Average order of an arithmetic function|average order]] of the [[divisor function]],
<math display=block>\frac1n\sum_{i=1}^n\left\lfloor\frac{n}i\right\rfloor\le\frac1n\sum_{i=1}^n\frac{n}i=H_n.</math>
The operation of rounding each term in the harmonic series to the next smaller integer multiple of <math>\tfrac1n</math> causes this average to differ from the harmonic numbers by a small constant, and [[Peter Gustav Lejeune Dirichlet]] showed more precisely that the average number of divisors is <math>\ln n+2\gamma-1+O(1/\sqrt{n})</math> (expressed in [[big O notation]]). Bounding the final error term more precisely remains an open problem, known as [[Dirichlet's divisor problem]].{{r|tsang}}