"The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for Re ( s ) > 1, and its analytic continuation elsewhere."
A Dirichlet series with a_1 = 1 and s = 2, which is used to define the Riemann zeta function.
Infinite series: A series is an infinite sum of numbers. An infinite series is said to converge if the sum of the series approaches a finite value.
Convergence tests: A convergence test is a method for determining whether an infinite series converges or diverges.
Comparison tests: The comparison test is a tool for determining whether an infinite series converges or diverges. The test compares the given series to another series with a known convergence status.
Ratio test: The ratio test is a test for determining whether an infinite series converges or diverges based on the limit of the ratio of consecutive terms.
Root test: The root test is a test for determining whether an infinite series converges or diverges based on the limit of the nth root of the absolute value of the terms.
Absolute and conditional convergence: An infinite series is said to be absolutely convergent if the series of the absolute values of the terms converges. A series is said to be conditionally convergent if it converges, but its absolute value series does not.
Zeta series: The zeta function is defined as the infinite series of the reciprocal powers of natural numbers. This series is named after the Greek letter zeta.
Riemann hypothesis: The Riemann hypothesis is a conjecture about the distribution of prime numbers. It is closely related to the zeta function.
Euler-Mascheroni constant: The Euler-Mascheroni constant is a mathematical constant that appears in the zeta function and other mathematical formulas.
Dirichlet series: A Dirichlet series is a series that is related to the zeta function and is used to study the properties of prime numbers.
Riemann Zeta Function: It is a series of the form ζ(z) = ∑n=1∞ 1/n^z, where z is a complex variable.
Hurwitz Zeta Function: It is a generalization of the Riemann Zeta function, given by ζ(s,a) = ∑n=0∞ 1/(n+a)^s, where s is a complex variable and a is a constant.
Epstein Zeta Function: It is a series of the form ζ(a,b,c) = ∑m,n(-1)^(m+n)/(m+a)^2(b+n)^c, where a, b, and c are constants.
Lerch's Zeta Function: It is given by ζ(z, a, s) = ∑n=0∞ 1/(n+a)^s e^(2πin(z-a)), where z is a complex variable and a and s are constants.
Dirichlet Series: It is a series of the form ζ(s) = ∑n=1∞ a(n)/n^s, where a(n) is an arithmetic function and s is a complex variable.
Mahler's Zeta Function: It is a series of the form ζ(z, x) = ∑n=1∞ x^n/f(n)^(z+1), where z is a complex variable, x is a constant, and f(n) is a multiplicative arithmetic function.
Dedekind Zeta Function: It is a series of the form ζ_K(s) = ∑n=1∞ a(n)/n^s, where K is a number field and a(n) is the number of ideals in K of norm n.
Selberg Zeta Function: It is a series of the form ζ(s) = ∏p prime(1-1/p^s)^(-1) exp(sum (∑p^(-ks))/k) , where s is a complex variable and the product is over all primes p.
Barnes Zeta Function: It is a function of the form ζ(z) = Γ(z)/Γ(z+1/2).
Hida's Zeta Functions: They are series of the form ζ(F,s) = ∑(a,n)=1 F(n)/n^s, where s is a complex variable, F is a modular form, and (a,n) is the greatest common divisor of a and n.
Mock Theta Functions: They are functions of the form f(z) = ∑n=0∞ q^(n^2) (1-q)^(n+1), where q is a complex variable.
"The Riemann zeta function plays a pivotal role in analytic number theory..."
"The Riemann zeta function has applications in physics, probability theory, and applied statistics."
"Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century."
"Bernhard Riemann's 1859 article 'On the Number of Primes Less Than a Given Magnitude' extended the Euler definition to a complex variable..."
"...proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers."
"This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics."
"The values of the Riemann zeta function at even positive integers were computed by Euler."
"The first of them, ζ(2), provides a solution to the Basel problem."
"In 1979 Roger Apéry proved the irrationality of ζ(3)."
"The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms."
"Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions, and L-functions, are known." Note: The paragraph provided does not explicitly mention applications of the Riemann zeta function in probability theory and applied statistics, so the quote for question 3 is a general statement indicating its use in those fields rather than a specific quote from the paragraph.