In mathematics, the Airy zeta function, studied by Crandall (1996), is a function analogous to the Riemann zeta function and related to the zeros of the Airy function.
The Airy function
is positive for positive x, but oscillates for negative values of x. The Airy zeros are the values at which , ordered by increasing magnitude: .
The Airy zeta function is the function defined from this sequence of zeros by the series
This series converges when the real part of s is greater than 3/2, and may be extended by analytic continuation to other values of s.
Like the Riemann zeta function, whose value is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2:
where is the gamma function, a continuous variant of the factorial. Similar evaluations are also possible for larger integer values of s.
It is conjectured that the analytic continuation of the Airy zeta function evaluates at 1 to
In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
In the physical sciences, the Airy functionAi(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, … by
In number theory, the Mertens function is defined for all positive integers n as
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0:
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as
In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
In mathematics, the Dirichlet beta function is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function is
In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.
In mathematics, Ramanujan's master theorem is a technique that provides an analytic expression for the Mellin transform of an analytic function.
In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies and the potential .
