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First, we describe how Euler originally discovered the result. He was considering the harmonic series
He had already used the following " product formula" to show the existence of infinitely many primes.
(Here, the product is taken over all primes p; in the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes, unless noted otherwise.)
Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. (Multiply out the right hand side as you would like to do.) Of course, the above "equation" is nonsense, because the harmonic series is known (by other means) to diverge. This type of formal manipulation was common at the time, when mathematicians were still experimenting with the new tools of calculus.
Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series. (In modern language, we now say that the existence of infinitely many primes is reflected by the fact that the Riemann zeta function has a simple pole at s = 1.)
Euler took the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the logarithm of each side, then he used the Taylor series expansion for ln(1 − x) as well as the sum of a geometric series:
for some fixed constant C. Since the sum of the reciprocals of the first n positive integers is asymptotic to ln(n), (i.e. their ratio approaches one as n approaches infinity), Euler then concluded
This "equation" appears bizarre to modern eyes; however, it is almost certain that Euler intended it to be interpreted as saying that the sum of the reciprocals of the primes less than n is asymptotic to ln(ln(n)) as n approaches infinity. It turns out this is indeed the case; Euler had reached a correct result by questionable means.
The above proof can be slightly altered to meet the demands of present-day rigour. [to come...]
A proof by contradiction follows.
Assume that the sum of the reciprocals of the primes converges:
Define as the ith prime number. We have:
There exists a positiveIn common usage positive is sometimes used in affirmation, as a synonym for "yes" or to express "certainty". In mathematics, a number is called positive if it is bigger than zero. See negative and non-negative numbers. In functional analysis, a bounded li integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st i such that:
Define N(x) as the number of positive integers n not exceeding x and not divisible by a prime other than the first i ones. Let us write this n as with k square-freeIn mathematics, a square-free integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 32. The small square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26 (which can be done with any integer). Since there are only i primes which could divide k, there are at most choices for k. Together with the fact that there are at most possible values for m, this gives us:
The number of positive integers not exceeding x and divisible by a prime other than the first i ones is equal to x - N(x).
Since the number of integers not exceeding x and divisible by p is at most x/p, we get:
or:
But this is impossible for all x larger than (or equal to) .
Q.E.D.For other meanings of the abbreviation "QED", see QED. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, "that which was to be demonstrated"). This is a translation of the Greek oper edei deixai which was used by many early mathem| Politics | Business | Finance |
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