The Lindelöf hypothesis is a conjecture in number theory, specifically related to the distribution of prime numbers and the Riemann zeta function. Proposed by the Swedish mathematician Ernst Lindelöf in 1908, it posits that the Riemann zeta function \(\zeta(s)\) has a certain bounded behavior for complex numbers \(s\) in the critical strip, where the real part of \(s\) is between 0 and 1.
Li's criterion is a mathematical result that gives conditions for the non-existence of solutions to certain types of differential equations, particularly for higher-order linear differential equations. It is named after the mathematician Li, Chen, and Zhang, who contributed to the understanding of oscillation theory in the context of differential equations. Specifically, in the context of second-order linear differential equations, Li's criterion can relate to the oscillatory behavior of solutions.
The Lerch zeta function, denoted as \(\Phi(z, s, a)\), is a generalization of the Riemann zeta function and is defined for complex numbers.
The Lefschetz zeta function is a mathematical tool used in the field of algebraic topology and dynamical systems to study the properties of continuous maps on topological spaces. It provides a way to encode information about the fixed points of a map and their behavior. Given a continuous map \( f \) from a topological space \( X \) to itself, one can consider the number of fixed points of iterates of this map.
The Langlands–Deligne local constant is a fundamental concept in the theory of automorphic forms and number theory, particularly in the context of the Langlands program. It arises in the study of the local Langlands correspondence, which connects representations of p-adic groups to Galois representations.
The Langlands Program is a vast and influential set of conjectures and theories in the fields of number theory and representation theory, proposed by the mathematician Robert Langlands in the late 1960s. It seeks to establish deep connections between different areas of mathematics, notably between: 1. **Number Theory**: The study of integers and their properties. 2. **Representation Theory**: The study of how algebraic structures, like groups, can be represented through linear transformations of vector spaces.
L-functions are a broad class of complex functions that arise in number theory and are connected to various areas of mathematics, including algebraic geometry, representation theory, and mathematical physics. The concept of an L-function is primarily associated with the study of prime numbers and solutions to polynomial equations, and they encapsulate deep properties of arithmetic objects.
The Igusa zeta function is a mathematical object that arises in number theory and algebraic geometry, particularly in the context of counting points of algebraic varieties over finite fields. It is a generalization of the classical zeta function associated with a variety defined over a finite field. The Igusa zeta function is particularly useful in the study of the solutions of polynomial equations over finite fields.
The Hurwitz zeta function is a generalization of the Riemann zeta function and is defined for complex numbers. It is denoted as \(\zeta(s, a)\), where \(s\) and \(a\) are complex numbers, with \(a > 0\) and typically \(s\) being complex with a real part greater than 1.
The Hilbert–Pólya conjecture is an unproven hypothesis in mathematics that suggests a connection between the zeros of the Riemann zeta function and the eigenvalues of certain self-adjoint operators.
A Hecke character (or Hecke character of the second kind) is a particular type of character associated with algebraic number fields and arithmetic functions. More specifically, these characters arise in the study of modular forms and algebraic K-theory.
The Hasse–Weil zeta function is a mathematical tool used in number theory and algebraic geometry, particularly in the study of algebraic varieties over finite fields and their properties. It generalizes the classical Riemann zeta function and serves as an important object in understanding the distribution of points on algebraic varieties defined over finite fields.
The Hardy–Littlewood zeta-function conjectures refer to a set of conjectures proposed by mathematicians G.H. Hardy and J.E. Littlewood regarding the distribution of prime numbers and, more broadly, the properties of number-theoretic functions.
Hadjicostas's formula is a mathematical formula used in the field of number theory, specifically in relation to the sum of binomial coefficients. It provides a method for calculating the sum of the squares of binomial coefficients.
The Grand Riemann Hypothesis (GRH) is an extension of the famous Riemann Hypothesis (RH), which pertains to the distribution of the non-trivial zeros of the Riemann zeta function \(\zeta(s)\).
The Goss zeta function is a mathematical object that arises in the study of number theory and algebraic geometry, particularly in the context of function fields over finite fields. It is named after the mathematician David Goss, who introduced it while investigating the properties of zeta functions for function fields, similar to how the Riemann zeta function relates to number fields.
The Generalized Riemann Hypothesis (GRH) is a conjecture in number theory that extends the famous Riemann Hypothesis (RH) beyond the critical line of the Riemann zeta function to other Dirichlet L-functions.
The Gan–Gross–Prasad conjecture is a conjecture in the realm of number theory and representation theory, specifically concerning the theory of automorphic forms and nilpotent orbits. Formulated by W. T. Gan, B. Gross, and D. Prasad in the early 2000s, the conjecture relates to the behavior of certain L-functions associated with automorphic representations of groups and has implications for the study of the branching laws of representations.
A functional equation is a relation that defines a function in terms of its value at different points, typically revealing symmetries or properties of the function. In the context of L-functions, these are complex functions arising in number theory and are particularly important in areas such as analytic number theory and the theory of modular forms. ### L-functions L-functions are certain complex functions that encode deep arithmetic properties of numbers.