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The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function. Closeup of the summatory Liouville function L(n) in the region where the Pólya conjecture fails to hold. Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10 9. The green spike shows the function itself ...
In mathematics, the Hilbert–Pólya conjecture states 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 hypothesis , by means of spectral theory .
According to the De Finetti's theorem, there must be a unique prior distribution such that the joint distribution of observing the sequence is a Bayesian mixture of the Bernoulli probabilities. It can be shown that this prior distribution is a beta distribution with parameters β ( ⋅ ; α , γ ) {\displaystyle \beta \left(\cdot ;\,\alpha ...
[4]: 23–24 The specific topics treated bear witness to the special interests of Pólya (Descartes' rule of signs, Pólya's enumeration theorem), Szegö (polynomials, trigonometric polynomials, and his own work in orthogonal polynomials) and sometimes both (the zeros of polynomials and analytic functions, complex analysis in general).
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture [5] Salomon Bochner and Marcel Riesz: 236 Bombieri–Lang conjecture: diophantine geometry: Enrico Bombieri and Serge Lang: 181 Borel conjecture: geometric topology: Armand Borel: 981 Bost conjecture: geometric topology: Jean-Benoît Bost: 65 Brennan ...
The zeta function plays an important role in studying dynamical systems. Note that this is the same general type of zeta function as the Riemann zeta function; however, in this case, the corresponding kernel is not known. The existence of such a kernel is known as the Hilbert–Pólya conjecture.
The Polya enumeration theorem translates the recursive structure of rooted ternary trees into a functional equation for the generating function F(t) of rooted ternary trees by number of nodes. This is achieved by "coloring" the three children with rooted ternary trees, weighted by node number, so that the color generating function is given by f ...