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If n = 1 and a and b are both 0 or 1/2, then the functions θ a,b (τ,z) are the four Jacobi theta functions, and the functions θ a,b (τ,0) are the classical Jacobi theta constants. The theta constant θ 1/2,1/2 (τ,0) is identically zero, but the other three can be nonzero.
The real part of every nontrivial zero of the Riemann zeta function is 1/2. The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1 / 2 . A proof or disproof of this would have far-reaching implications in number theory, especially for the distribution of prime ...
There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has a positive ...
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as = ((+)) for real values of t.Here the argument is chosen in such a way that a continuous function is obtained and () = holds, i.e., in the same way that the principal branch of the log-gamma function is defined.
In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they generalize classical Eisenstein series. They are named after Henri Poincaré.
In set theory, (pronounced like the letter theta) is the least nonzero ordinal such that there is no surjection from the reals onto . Θ {\displaystyle \varTheta } has been studied in connection with strong partition cardinals and the axiom of determinacy . [ 1 ]
In mathematics, the theta operator is a differential operator defined by [1] [2] θ = z d d z . {\displaystyle \theta =z{d \over dz}.} This is sometimes also called the homogeneity operator , because its eigenfunctions are the monomials in z :
It is also known as Lovász theta function and is commonly denoted by (), using a script form of the Greek letter theta to contrast with the upright theta used for Shannon capacity. This quantity was first introduced by László Lovász in his 1979 paper On the Shannon Capacity of a Graph .