Search Results for ""

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are ...

The so-called explicit formula psi(x)=x-sum_(rho)(x^rho)/rho-ln(2pi)-1/2ln(1-x^(-2)) gives an explicit relation between prime numbers and Riemann zeta function zeros for x>1 ...

rho_(2s)(n)=(pi^s)/(Gamma(s))n^(s-1)sum_(p,q)((S_(p,q))/q)^(2s)e^(2nppii/q), where S_(p,q) is a Gaussian sum, and Gamma(s) is the gamma function.

e is transcendental.

In the American system, 10^(51).

Define E(x;q,a)=psi(x;q,a)-x/(phi(q)), (1) where psi(x;q,a)=sum_(n<=x; n=a (mod q))Lambda(n) (2) (Davenport 1980, p. 121), Lambda(n) is the Mangoldt function, and phi(q) is ...

The largest cube dividing a positive integer n. For n=1, 2, ..., the first few are 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, ... (OEIS A008834).

A generalization of the Gaussian sum. For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states ...

The negative integers ..., -3, -2, -1.

The positive integers 1, 2, 3, ..., equivalent to N.