The zeta function of a mathematical operator O {\displaystyle {\mathcal {O}}} is a function defined as ζ O ( s ) = tr O − s {\displaystyle \zeta _{\mathcal...

zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum...

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent...

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...

mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers...

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,...

The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function ζ ( s ) = ∏ p ∈ P 1 1 − p − s {\displaystyle...

mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate...

{\displaystyle k^{s}\zeta (s)=\sum _{n=1}^{k}\zeta \left(s,{\frac {n}{k}}\right),} where ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. This is a...

{\displaystyle \zeta (\star )} is the Riemann zeta function. The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the...

sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding...

function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or...

that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann...

Bernoulli polynomials. This operator also has a continuous spectrum consisting of the Hurwitz zeta function. The transfer operator of the Gauss map h ( x )...

the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical...

partition function is the Riemann zeta function. This idea underlies Alain Connes's attempted proof of the Riemann hypothesis. The Möbius function is multiplicative...

Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the...

mathematically rigorous definition is via the zeta function of the operator, ζ S ( a ) = tr S − a , {\displaystyle \zeta _{S}(a)=\operatorname {tr} \,S^{-a}\,...

Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle. Let f be a function defined...

Borel) is the operator B : z − 1 C [ [ z − 1 ] ] → C [ [ ζ ] ] {\displaystyle {\mathcal {B}}:z^{-1}\mathbb {C} [[z^{-1}]]\to \mathbb {C} [[\zeta ]]} defined...

The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced...

and in analytic number theory. It is closely related to the Epstein zeta function. There are many generalizations associated to more complicated groups...

realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application...

Los Zetas (pronounced [los ˈsetas], Spanish for "The Zs") was a Mexican criminal syndicate, known as one of the most dangerous of Mexico's drug cartels...

the zeta function, and real-variable analytic continuation, retrieved January 30, 2014. Lepowsky, J. (1999). "Vertex operator algebras and the zeta function"...

{\bigl [}\zeta '(-1,z)-\zeta '(-1){\bigr ]}} where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and ζ...

Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is 1 − ( sin ⁡ ( π u ) π u...

{\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z).} Holomorphic functions of several complex variables satisfy an identity theorem...

symbols for these operators are, up to a sign, 1 2 ( 1 ± i ζ ‖ ζ ‖ ) . {\displaystyle {\frac {1}{2}}\left(1\pm i{\frac {\zeta }{\|\zeta \|}}\right).} These...

the function f is given and g is unknown. Here, L stands for a linear differential operator. For example, one might take L to be an elliptic operator, such...