"A Dirichlet series is any series of the form..."
A series of the form ∑(n=1)∞ a_n / n^s, where a_n are constants and s is a complex variable.
Convergence of Series: The concept of convergence plays a crucial role in Dirichlet series, and a thorough understanding of the various convergence tests is necessary to manipulate Dirichlet series effectively.
Dirichlet Functions: The Dirichlet function is defined as a periodic function that takes on the values 0 and 1. These functions are used in Dirichlet series to understand the nature of oscillations.
Fourier Series: The Fourier series is a representation of a function as a sum of sine and cosine functions. It is a crucial tool in the study of Dirichlet series, as it provides a way to represent periodic functions.
Euler's Formula: Euler's formula is a fundamental equation in mathematics that relates the exponential function with the various trigonometric functions. It is used extensively in the study of Dirichlet series.
Gamma Function: The gamma function is a generalization of the factorial function on non-negative integers. It is useful in the study of Dirichlet series as it is related to the Mellin transform.
Mellin Transform: The Mellin transform is a widely used mathematical tool that transforms functions from the time domain to the frequency domain. It is useful in the study of Dirichlet series to analyze their properties.
Modular Forms: Modular forms are defined as functions that are invariant under certain groups of transformations. They have important applications in many areas of mathematics, including the study of Dirichlet series.
Riemann Zeta Function: The Riemann zeta function is defined as an infinite series that converges for certain values of s. It is useful in the study of Dirichlet series, as it is intimately related to the distribution of prime numbers.
Complex Analysis: Complex analysis is the study of functions of complex variables. It plays a major role in the study of Dirichlet series, as many important results concerning these series are obtained using methods from complex analysis.
Analytic Continuation: Analytic continuation is a technique for extending the domain of a function. It is useful in the study of Dirichlet series, as many important results concerning these series are obtained by analytically continuing these functions into the complex plane.
Riemann zeta function: The most famous Dirichlet series is the Riemann zeta function. It is defined as the sum over all positive integers of the reciprocal of the pth power: ζ(s) = 1^−s + 2^−s + 3^−s + · · ·. The Riemann hypothesis remains one of the greatest unsolved problems in mathematics.
Dirichlet L-series: A Dirichlet L-series is a generalization of the Riemann zeta function, where the coefficients are any arithmetic function. It is given by the sum of the terms of the form apn−s, where a is an arithmetic function and s is a complex variable.
Euler product: An Euler product is a type of Dirichlet series given by a product of terms involving primes. It is used to connect many areas of mathematics, including analytic number theory and algebraic geometry.
Mellin transform: A Mellin transform is a type of Dirichlet series that can be used to transform a function of a real variable into a function of a complex variable. It is defined as the integral of the function multiplied by x^(s-1), where s is a complex variable.
Hardy-Littlewood conjecture: This conjecture is a statement about the distribution of prime numbers in arithmetic progressions. It involves a Dirichlet series with a complicated formula for the coefficients.
L-functions: L-functions are Dirichlet series associated with various types of mathematical objects, including elliptic curves, modular forms, and automorphic representations. They have important applications in number theory and related areas.
Dedekind zeta functions: Dedekind zeta functions are Dirichlet series associated with a number field. They play an important role in algebraic number theory and can be used to study the arithmetic properties of the field.
Siegel modular forms: Siegel modular forms are functions of several complex variables that satisfy certain transformation properties. They are related to Dirichlet series with arithmetic coefficients and have applications in the study of automorphic forms.
Multiple Dirichlet series: Multiple Dirichlet series are Dirichlet series with multiple variables. They have applications in the study of special values of L-functions and in the theory of automorphic forms.
"...where s is complex, and a_n is a complex sequence."
"Dirichlet series play a variety of important roles in analytic number theory."
"The most usually seen definition of the Riemann zeta function is a Dirichlet series."
"Dirichlet L-functions are also defined as Dirichlet series."
"It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis."
"The series is named in honor of Peter Gustav Lejeune Dirichlet." Here are some additional questions: