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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 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 ...

Given a random variable X with continuous and strictly monotonic probability density function f(X), a quantile function Q_f assigns to each probability p attained by f the ...

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function zeta(s) that, roughly speaking, any nonvanishing analytic function can be approximated ...

There are at least two distinct notions of an intensity function related to the theory of point processes. In some literature, the intensity lambda of a point process N is ...

The normal bundle of a submanifold N in M is the vector bundle over N that consists of all pairs (x,v), where x is in N and v is a vector in the vector quotient space ...

Landau (1911) proved that for any fixed x>1, sum_(0<|I[rho]|<=T)x^rho=-T/(2pi)Lambda(x)+O(lnT) as T->infty, where the sum runs over the nontrivial Riemann zeta function zeros ...

Let theta(t) be the Riemann-Siegel function. The unique value g_n such that theta(g_n)=pin (1) where n=0, 1, ... is then known as a Gram point (Edwards 2001, pp. 125-126). An ...

Li's criterion states that the Riemann hypothesis is equivalent to the statement that, for lambda_n=1/((n-1)!)(d^n)/(ds^n)[s^(n-1)lnxi(s)]|_(s=1), (1) where xi(s) is the ...

An anti-analytic function is a function f satisfying the condition (df)/(dz)=0. (1) Using the result ...