In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map...

function Ihara zeta function of a graph L-function, a "twisted" zeta function Lefschetz zeta function of a morphism Lerch zeta function, a generalization...

The Ihara zeta function of a graph can be interpreted as an example of the Artin–Mazur zeta function. Lefschetz number Lefschetz zeta-function Artin, Michael;...

supports. The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields. Fixed-point theorems Lefschetz zeta function Holomorphic...

number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s...

Solomon Lefschetz ForMemRS (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental...

number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over...

of the Lefschetz fixed-point theorem in algebraic topology. One application of the Grothendieck trace formula is to express the zeta function of a variety...

algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology, used to express the Hasse–Weil zeta function. Gutzwiller trace formula:...

turns out to equal to its first Chern class.) Lefschetz's original proof proceeded by normal functions, which were introduced by Henri Poincaré. However...

the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already...

function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements. Weil conjectured that such zeta-functions...

Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove two...

path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity)...

mappings acting in such a way that the Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For later history see motive (algebraic...

the strong Tate conjecture, namely that the order of the pole of the zeta function Z(X, t) at t = q−j is equal to the rank of the group of algebraic cycles...

Deninger's papers studies L-functions and their special values. A classical example of an L-function is the Riemann zeta function ζ(s), for which formulas...

groups", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.), The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 363–394...

Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces. An arithmetic cycle of codimension p...

found using fibrations over P 1 {\displaystyle \mathbb {P} ^{1}} such as Lefschetz fibrations. For example, an elliptic K3 surface X {\displaystyle X} is...

accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations...

example, the Riemann hypothesis concerning the zeros of the Riemann zeta function (open as of 2017) can be regarded as being parallel to the Weil conjectures...

quaternion order Hurwitz quaternion Hurwitz scheme Hurwitz surface Hurwitz zeta function Hurwitz's automorphisms theorem Hurwitz's theorem (complex analysis)...

chains, the boundary operator, chain complexes J. W. Alexander, Solomon Lefschetz, Lev Pontryagin, Andrey Kolmogorov, Norman Steenrod, Eduard Čech: the...

Harold M. Edwards for mathematical exposition in his books Riemann's zeta function, Pure and Applied Mathematics, number 58, Academic Press, New York and...

methods of Viggo Brun; achieved major results on zeros of the Riemann zeta function; gave an elementary proof of the prime number theorem (with P. Erdős)...

telescoping sum occurs in the derivation of a probability density function; Lefschetz fixed-point theorem, where a telescoping sum arises in algebraic...

is called the moduli space of formal group laws. Lefschetz 1.  Solomon Lefschetz 2.  The Lefschetz fixed-point theorem says: given a finite simplicial...

457 no. 3 (1995), pp. 484–510, ISSN 0550-3213 [doi] [abs] 24. M Stern, Lefschetz formulae for arithmetic varieties, Inventiones Mathematicae, vol. 115...

points over any finite field, and in multiplicative notation for local zeta-functions. The general idea is that one motive has the same structure in any reasonable...