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The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
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 ...
While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof [6] is by induction: The first partial sum is 1 / 2 , which has the form odd / even . If the n th partial sum (for n ≥ 1) has the form odd / even , then the (n + 1) st sum is
By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0, (2) H x = H x−1 + 1/x for all complex numbers x except the non-positive integers, and (3) lim m→+∞ (H m+x − H m) = 0 for all complex values x.
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.
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]
This result was first published in Euler's 1737 ... Since the sum of the reciprocal of every power of 2 ... where the denominators consist of all positive integers ...
The sum of the reciprocals of all Eisenstein integers excluding 0 raised to the fourth power is 0: [6] {} = = so / is a root of j-invariant. In general G k ( e 2 π i 3 ) = 0 {\displaystyle G_{k}\left(e^{\frac {2\pi i}{3}}\right)=0} if and only if k ≢ 0 ( mod 6 ) {\displaystyle k\not \equiv 0{\pmod {6}}} .