Riemann zeta function
The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.
Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to
Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes Πp p/p − 1) implies that there are infinitely many primes.
[5] Since the logarithm of p/p − 1 is approximately 1/p, the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite.
The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime.
The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place.
where the η-series is convergent (albeit non-absolutely) in the larger half-plane s > 0 (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine[7][8]).
They are trivial in the sense that their existence is relatively easy to prove, for example, from sin πs/2 being 0 in the functional equation.
The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line.
[14][15] Hardy and J. E. Littlewood formulated two conjectures on the density and distance between the zeros of ζ (1/2 + it) on intervals of large positive real numbers.
for The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.
Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = 1/2.
This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts (Ramanujan summation) such as string theory.
The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime?
There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.
[30] More recent work has included effective versions of Voronin's theorem[31] and extending it to Dirichlet L-functions.
The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on xs − 1; that context gives rise to a series expansion in terms of the falling factorial.
A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + 2πi/ln 2n for some integer n, was conjectured by Konrad Knopp in 1926 [44] and proven by Helmut Hasse in 1930[45] (cf.
Euler summation): The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.
[45] Research by Iaroslav Blagouchine[47][44] has found that a similar, equivalent series was published by Joseph Ser in 1926.
An elegant and very short proof of this representation of the zeta function, based on Carlson's theorem, was presented by Philippe Flajolet in 2006.
as then the Riemann hypothesis is equivalent to Peter Borwein developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.
[57] The function ζ can be represented, for Re(s) > 1, by the infinite series where k ∈ {−1, 0}, Wk is the kth branch of the Lambert W-function, and B(μ)n, ≥2 is an incomplete poly-Bernoulli number.
A classical algorithm, in use prior to about 1930, proceeds by applying the Euler-Maclaurin formula to obtain, for n and m positive integers, where, letting
In one notable example, the Riemann zeta function shows up explicitly in one method of calculating the Casimir effect.
[62] In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series.
[64] The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.
Another interesting series that relates to the natural logarithm of the lemniscate constant is the following There are yet more formulas in the article Harmonic number.
These include the Hurwitz zeta function (the convergent series representation was given by Helmut Hasse in 1930,[45] cf.
The Clausen function Cls(θ) can be chosen as the real or imaginary part of Lis(eiθ).
