This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemann’s unpublished work on the zeta function. By application of standard asymptotic methods for integral estimation, and the use of certain approximations pertaining to special functions, it proves possible to ...

In this article, we give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by Candelas, de la Ossa, and Rodriguez Villegas. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.

1. In t roduc t ion . Several recently invented number-theoretic algorithms are sketched below. They all have the common feature that they rely on bounded precision computations of analytic functions. The main one of these algorithms is a new method of calculating values of the Riemann zeta function at multiple points. This method enables one to verify the truth of the Riemann Hypothesis (RH) f...

One way of characterizing Atle Selberg’s mathematical genius is that he had a “golden touch”. In those domains he thought about in depth, he saw further than generations before him, repeatedly uncovering truths lying below the surface. His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive. The impact o...

We use the mode summation method together with zeta-function regularization to compute the Casimir energy of a dilute dielectric cylinder. The method is very transparent, and sheds light on the reason the resulting energy vanishes.

We present a simple method for evaluation of multiple Euler sums in terms of single and double zeta values. 2000 Mathematics Subject Classification. 11M99, 40B05.

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type ...

Fix a prime number l. In this paper we prove l-adic versions of two related conjectures of Deligne, [4, 8.2, p. 163] and [4, 8.9.5, p. 168], concerning mixed Tate motives over the punctured spectrum of the ring of integers of a number field. We also prove a conjecture [11, p. 300], which Ihara attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fu...

The distribution of many real discrete random variables (e.g., the frequency of words, the population of cities) can be approximated by a zeta distribution, that is known popularly as Zipf’s law, or power law in physics. Here we revisit the relationship between power law distribution of a magnitude and the corresponding power relationship between the magnitude of a certain element and its rank....