Riemann's Explicit Formula and the Prime Number Theorem

This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued...

Orthogonality relations of q-Meixner polynomials: with the use of spectral analysis

Orthogonality relations of q-Meixner polynomials, polynomials in terms of basic hypergeometric series, will be proved by using spectral measures and a difference operator.

Properties of the Racahpolynomials with regard to the Lie algebrarepresentation of sl(2;C)

The Racah polynomial Rn(λ(x)) is a polynomial of degree n and is variable in λ(x). In this thesis two properties of this polynomial will be studied. One is the orthogonal property of the Racah polynomial. And the other is that the Racah polynomial can also be described as a polynomial of degree x and variable over λ(n). The Racah polynomials...

On random tridiagonal matrices and the beta log-gas

In this thesis the beta log-gas probability density function is discussed. It is shown that there is a strong link between this density function and Jacobi matrices. A change of variables exercise shows that the distribution of eigenvalues is exactly like the quadratic beta log-gas. The change of variables gives the normalization constant for...

Spherical and Cherednik-Opdam transforms of Jacobi-type polynomials

The spherical transform maps the orthogonal basis of symmetric Jacobi-type polynomials to an orthogonal basis of (symmetric) Wilson polynomials. The spherical transform is closely related to the Cherednik-Opdam transform, as it is essentially its symmetric version. The symmetric Jacobi-type polynomials can be composed from the non-symmetric...

Binomial formulas for Macdonald polynomials

Symmetric and nonsymmetric Macdonald polynomials associated to root systems are very general families of orthogonal polynomials in multiple variables. Their definition is quite complex, but in certain cases one can define so-called interpolation polynomials that have a surprisingly simple definition and are related to the Macdonald polynomials...

Bewijs van Wigners halve cirkel stelling

Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze...

Statistical analysis of the single-qubit unitarity randomized benchmarking protocol

This thesis provides an overview of the standard randomized benchmarking protocol, the extension to interleaved randomized benchmarking and the related unitarity randomized benchmarking, used to characterize noisy implementations of quantum gates from the Clifford group. It contains a brief overview of the textbook and literature preliminaries...

Brownian Motion and the Airy Function

Deriving the location of the maximum of a Brownian Motion with downward quadratic drift. Proving it is welldefined then finding an algorithmic method to simplify expressions of the moments.

Nulpunten van de Riemann zeta functie

Dit verslag geeft als eerste een overzicht van basis resultaten van de Riemann zeta functie. Vervolgens worden twee meer specifieke resultaten laten zien met betrekking tot de nulpunten van deze Riemann zeta functie.