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The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by Gamma(n)=(n-1)!, ...

The Möbius function is a number theoretic function defined by mu(n)={0 if n has one or more repeated prime factors; 1 if n=1; (-1)^k if n is a product of k distinct primes, ...

A function tau(n) related to the divisor function sigma_k(n), also sometimes called Ramanujan's tau function. It is defined via the Fourier series of the modular discriminant ...

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity ...

The divisor function sigma_k(n) for n an integer is defined as the sum of the kth powers of the (positive integer) divisors of n, sigma_k(n)=sum_(d|n)d^k. (1) It is ...

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and ...

Define the zeta function of a variety over a number field by taking the product over all prime ideals of the zeta functions of this variety reduced modulo the primes. Hasse ...

The prime counting function is the function pi(x) giving the number of primes less than or equal to a given number x (Shanks 1993, p. 15). For example, there are no primes ...

A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at ...

Consider the inequality sigma(n)<e^gammanlnlnn for integer n>1, where sigma(n) is the divisor function and gamma is the Euler-Mascheroni constant. This holds for 7, 11, 13, ...