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The sum of the series is approximately equal to 1.644934. [3] The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be / and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he ...
The sum of the reciprocals of the powerful numbers is close to 1.9436 . [4] The reciprocals of the factorials sum to the transcendental number e (one of two constants called "Euler's number"). The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π 2 / 6 , or ζ(2) where ζ is the Riemann zeta ...
It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log n as n approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. [3] Thus Euler obtained a correct result by questionable means.
[2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6] Additional proofs were published in the 17th century by Pietro Mengoli [2] [7] and by Jacob Bernoulli.
7.3 Relation to the Riemann zeta function. ... the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] ... 4: 25 /12 ~2.08333 5: ...
They do not have a finite sum, as Leonhard Euler proved in 1737. Like rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of these decimal representations of reciprocals of primes. [1]
By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms n −s where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product.
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by ():= = (),.It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n.