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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 ...

The integral phi(t,u)=int(e^(piitx^2+2piiux))/(e^(2piix)-1)dx which is related to the Jacobi theta functions, mock theta functions, Riemann zeta function, and Siegel theta ...

The Hadamard product is a representation for the Riemann zeta function zeta(s) as a product over its nontrivial zeros rho, ...

The Gram series is an approximation to the prime counting function given by G(x)=1+sum_(k=1)^infty((lnx)^k)/(kk!zeta(k+1)), (1) where zeta(z) is the Riemann zeta function ...

The Zipf distribution, sometimes referred to as the zeta distribution, is a discrete distribution commonly used in linguistics, insurance, and the modelling of rare events. ...

Let mu(sigma) be the least upper bound of the numbers A such that |zeta(sigma+it)|t^(-A) is bounded as t->infty, where zeta(s) is the Riemann zeta function. Then the Lindelöf ...

A reflection relation is a functional equation relating f(-x) to f(x), or more generally, f(a-x) to f(x). Perhaps the best known example of a reflection formula is the gamma ...

The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue ...

The Montgomery-Odlyzko law (which is a law in the sense of empirical observation instead of through mathematical proof) states that the distribution of the spacing between ...

Apéry's constant is defined by zeta(3)=1.2020569..., (1) (OEIS A002117) where zeta(z) is the Riemann zeta function. Apéry (1979) proved that zeta(3) is irrational, although ...