The Feller–Tornier constant is a constant that arises in the context of probability theory, particularly in relation to random walks and certain types of stochastic processes. It is named after the mathematicians William Feller and Joseph Tornier, who studied the asymptotic behavior of random walks.
The explicit formulas for L-functions typically relate to the values of Dirichlet series associated with characters or other arithmetic objects, and they often connect them to prime numbers through various summation techniques. While there is a variety of specific L-functions, one of the most well-known types of L-functions is associated with Dirichlet characters in number theory.
The Euler product formula is a representation of a function, particularly in number theory, which expresses a function as an infinite product over prime numbers. It is most famously used in relation to the Riemann zeta function, \( \zeta(s) \), for complex numbers \( s \) where the real part is greater than 1.
Equivariant L-functions are a specific class of L-functions that arise in the context of number theory and representation theory, particularly in the study of automorphic forms and motives. The concept of "equivariance" in this context refers to how these functions behave under the action of a certain group, typically a Galois group or a symmetry group associated with the arithmetic structure being studied.
The Eichler–Shimura congruence relations are important results in the field of arithmetic geometry, particularly in the study of modular forms, modular curves, and the arithmetic of elliptic curves. They describe deep relationships between the ranks of certain abelian varieties, specifically abelian varieties that are associated with modular forms.
The Dwork conjecture is a hypothesis in the field of arithmetic algebraic geometry, particularly concerning the interplay between p-adic analysis and the theory of algebraic varieties. It was proposed by the mathematician Bernard Dwork in the context of understanding the zeta function of a family of algebraic varieties over finite fields.
The Dirichlet eta function, denoted as \( \eta(s) \), is a complex function closely related to the Riemann zeta function.
A Dirichlet character is a complex-valued arithmetic function \( \chi: \mathbb{Z} \to \mathbb{C} \) that arises in number theory, particularly in the study of Dirichlet L-functions and Dirichlet's theorem on primes in arithmetic progressions.
The Dirichlet beta function, denoted as \( \beta(s) \), is a special function that generalizes the concept of the Riemann zeta function.
The Dirichlet L-function is a complex function that generalizes the Riemann zeta function and plays a crucial role in number theory, particularly in the study of Dirichlet characters and L-series. It is associated with a Dirichlet character \( \chi \) modulo \( k \), which is a completely multiplicative arithmetic function satisfying certain periodicity and the condition \( \chi(n) = 0 \) for \( n \) not coprime to \( k \).
Dirichlet's theorem on arithmetic progressions states that if \( a \) and \( d \) are two coprime integers (that is, their greatest common divisor \( \gcd(a, d) = 1 \)), then there are infinitely many prime numbers of the form \( a + nd \), where \( n \) is a non-negative integer.
The Dedekind zeta function is an important invariant in algebraic number theory associated with a number field.
The Chowla–Mordell theorem is a result in number theory related to the properties of rational numbers and algebraic equations. Specifically, it deals with the existence of rational points on certain types of algebraic curves.
The Brumer-Stark conjecture is a significant hypothesis in number theory that relates to the structure of abelian extensions of number fields and their class groups. It plays a crucial role in the study of L-functions and their special values, specifically in the context of p-adic L-functions and the behavior of class numbers. The conjecture can be understood in relation to certain aspects of class field theory.
The Birch and Swinnerton-Dyer (BSD) conjecture is a fundamental hypothesis in number theory that relates the number of rational points on an elliptic curve to the behavior of an associated L-function. Specifically, it concerns the properties of elliptic curves defined over the rational numbers \(\mathbb{Q}\).
The Beurling zeta function is a mathematical object related to number theory, specifically in the study of prime numbers. It is named after the Swedish mathematician Arne Magnus Beurling, who introduced it in the 1930s. The Beurling zeta function generalizes the classical Riemann zeta function and is used in the context of "pseudo-primes" or "generalized prime numbers.
The Basel problem is a famous problem in the field of mathematics, specifically in the study of series. It asks for the exact sum of the reciprocals of the squares of the natural numbers. Formally, it is expressed as: \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \] The solution to the Basel problem was famously found by the Swiss mathematician Leonhard Euler in 1734.
The Barnes zeta function is an extension of the classical Riemann zeta function and is defined in the context of number theory and special functions. It is primarily associated with the theory of multiple zeta values and has connections to various areas of mathematics, including algebra, topology, and mathematical physics. The Barnes zeta function, denoted as \( \zeta_B(s, a) \), depends on two parameters: \( s \) and \( a \).