A number is called a barrier of a number-theoretic function if, for all , . Neither the totient function nor the divisor function has a barrier.Let be an open set and , then a function is called a barrier for at a point if 1. is continuous, 2. is subharmonic on , 3. , 4. (Krantz 1999, pp. 100-101).
An exponential sum of the form(1)where is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing(2)a notation introduced by Vinogradov, Weyl observed that(3)(4)(5)(6)a process known as Weyl differencing (Montgomery 2001).Weyl was able to use this process to show that if(7)is a real polynomial and at least one of , ..., is irrational, then is uniformly distributed (mod 1).
There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral,(1)where is a Bessel function of the first kind and is a gamma function. It can be derived from Sonine's integral. With , the integral becomes Parseval's integral.In complex analysis, let be a harmonic function on a neighborhood of the closed disk , then for any point in the open disk ,(2)In polar coordinates on ,(3)where and is the Poisson kernel. For a circle,(4)For a sphere,(5)where(6)