Sandbox stuff

Contents

1 modular group 1.1 Applications to Number Theory 1.2 Relationship to Lattices 1.3 Relationship to Quadratic Forms 1.4 Congruence Subgroups 2 von Mangoldt function 2.1 See also 3 Scraps 3.1 Georgy Fedoseevich Voronoy

modular group

Applications to Number Theory

[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]

Relationship to Lattices

The lattice Δ_τ is the set of points in the complex plane generated by the values 1 and τ (where τ is not real). So

<math>\Delta_{\tau}=\{m+n\tau : m,n \in Z\}<math>

Clearly τ and -τ generate the same lattice i.e. Δ_τ=Δ_-τ. In addition, τ and τ' generate the same lattice if they are related by a fractional linear transformation that is a member of the modular group.

Relationship to Quadratic Forms

The set of values taken by the positive definite binary quadratic form am²+bmn+cn² is related to the lattice Δ_τ as follows :-

<math>\{ am^2+bmn+cn^2:m,n \in Z \} = \{a|z|^2 : z \in \Delta_\tau \}<math>

where τ is chosen such that Re(τ)=b/2a and |τ|² = c/a.

Congruence Subgroups

[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]

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von Mangoldt function

The von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt.

The von Mangoldt function, conventionally written as Λ(n), is defined as

<math>\Lambda(n) = \begin{cases} \ln p & \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1 \\ 0 & \mbox{otherwise.} \end{cases}<math>

It is an example of an important arithmetic function that is neither multiplicative nor additive.

The von Mangoldt function satisfies the identity

<math>log(n) = \sum_{d|n} \Lambda(n).<math>

The summatory von Mangoldt function, ψ(x), (also known as the Chebyshev function) is defined as

<math>\psi(x) = \sum_{n\le x} \Lambda(n).<math>

von Mangoldt provided a rigorous proof of an explicit formual for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function. This was an important part of the first proof of the prime number theorem.

See also

• Prime counting function

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Scraps

Georgy Fedoseevich Voronoy

Born: 28 April 1868 in Zhuravka, Poltava guberniya, Russia (now Ukraine) Died: 20 Nov 1908 in Warsaw, Poland

Georgy Voronoy studied at the gymnasium in Priluki graduating in 1885. He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics.

After graduating from St Petersburg in 1889, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.

Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences.

Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood.

In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics.

The concept of Voronoi diagrams first appeared in works of Descartes as early as 1644. Descartes used Voronoi-like diagrams to show the disposition of matter in the solar system and its environs.

The first man who studied the Voronoi diagram as a concept was a German mathematician G. L. Dirichlet. He studied the two- and three dimensional case and that is why this concept is also known as Dirichlet tessellation. However it is much better known as a Voronoi diagram because another German mathematician M. G. Voronoi in 1908 studied the concept and defined it for a more general n-dimensional case.

Very soon after it was defined by Voronoi it was developed independently in other areas like meteorology and crystalography. Thiessen developed it in meteorology in 1911 as an aid to computing more accurate estimates of regional rainfall averages. In the field of crystalography German researchers dominated and Niggli in 1927 introduced the term Wirkungsbereich (area of influence) as a reference to a Voronoi diagram.

http://www.comp.lancs.ac.uk/~kristof/research/notes/voronoi/voronoi.pdf