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Journals and Conferences
A graph theoretical analogue of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.
We discuss zeta functions of finite irregular undirected connected graphs (which may be weighted) and apply them to obtain, for example an analog of the prime number theorem for cycles in graphs. We consider 3 types of zeta functions of graphs: vertex, edge, and path. Analogs of the Riemann hypothesis are also introduced.
We present examples of hypergraphs constructed from homogeneous spaces of finite general linear groups. These hypergraphs are constructed using an invariant analogue of a hypervolume and their spectra are analyzed to see if they are Ramanujan in the sense of W.-C.
put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind's time, these conjectures have been phrased in the language of quadratic fields. This is how we will state the conjectures… (More)
2000 iii To m y Mother and Father We could use up two Eternities in learning all that is to be l e arned a b out our own world and the thousands of nations that have arisen and ourished and vanished f r om it. Mathematics alone would occupy me eight million years. |Mark Twain.
It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the rings R of integral algebraic numbers in quadratic complex fields Q √ −D , for D = 19, 43, 67, and 163. In conection with these counterexamples several results were published in an effort… (More)
We investigate Ihara-Selberg zeta functions of Cayley graphs for the Heisenberg group over finite rings Z/p n Z, where p is a prime. In order to do this, we must compute the Galois group of the covering obtained by reducing coordinates in Z/p n+1 Z modulo p n. The Ihara-Selberg zeta functions of the Heisenberg graph mod p n+1 factor as a product of Artin… (More)