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...

Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky’s formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky’s formula by Bass’ method used in the determinant expression for the Ihara zeta function of a graph.

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....

Recently, Guido, Isola and Lapidus [11] defined the Ihara zeta function of a fractal graph, and gave a determinant expression of it. We define the Bartholdi zeta function of a fractal graph, and present its determinant expression.

Fix a prime number l. In this paper we prove a conjecture [16, p. 300], which Ihara attributes to Deligne, about the action of the absolute Galois group on the pro-l completion of the fundamental group of the thrice punctured projective line. It is stated below. Similar techniques are also used to prove part of a conjecture of Goncharov [11, Conj. 2.1], also about the action of the absolute Gal...

Riemann Zeta Function 602 25.2 Definition and Expansions . . . . . . . . 602 25.3 Graphics . . . . . . . . . . . . . . . . . . 603 25.4 Reflection Formulas . . . . . . . . . . . . 603 25.5 Integral Representations . . . . . . . . . 604 25.6 Integer Arguments . . . . . . . . . . . . 605 25.7 Integrals . . . . . . . . . . . . . . . . . . 606 25.8 Sums . . . . . . . . . . . . . . . . . . . 606 25....

Some basic results for Dirichlet series ψ with positive terms via log-convexity properties are pointed out. Applications for Zeta, Lambda and Eta functions are considered. The concavity of the function 1/ψ is explored and, as a main result, it is proved that the function 1/ζ is concave on (1,∞) . As a consequence of this fundamental result it is noted that Zeta at the odd positive integers is b...

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