Dirichlet’s theorem

The introduction of the r-ordering operator can be understood from the rather elementary Dirichlet s theorem in calculus. Starting with two functions F x) and G y), one forms the integral... 

Dirichlet s theorem on primes in arithmetic progressions Let a and b be relatively prime positive integers. Then the arithmetic progression an + b (for n = 1, 2,...) contains infinitely many primes. 

We begin our discussion of Fourier analysis by considering the representation of a periodic function f t) with a period of 2P, f t + 2P) = f t). If f(f) has a finite number of local extrema and a finite number of times tj e [0,2P] at which it is discontinuous, Dirichlet s theorem states that it may be represented as the Fourier series... 

As follows from Chapter 1, we have formulated an external Dirichlet s boundary value problem, which uniquely defines the attraction field. In this light it is proper to notice the following. In accordance with the theorem of uniqueness its conditions do not require any assumptions about the distribution of density inside of the earth or the mechanism of surface forces between the elementary volumes. In particular, these forces may not satisfy the condition of hydrostatic equilibrium. 

Thus, we have demonstrated that the potential C/g is a solution of Laplace s equation and satisfies the boundary condition at the surface of the ellipsoid of rotation and at infinity. In other words, we have solved the Dirichlet s boundary value problem and, in accordance with the theorem of uniqueness, there is only one function satisfying all these conditions. 

Sometimes, primes in certain congruence classes are needed, e.g., p = 3 mod 4 for Blum integers. For such cases, Dirichlet s prime-number theorem states that in a certain sense, primes are equally distributed over the possible congruence classes Given any modulus v, there are roughly equally many primes congruent mod v for all 6 Zy. If denotes the number of primes in the set 1, that... 

The proof is a rather straightforward application of Dirichlet s prime-number theorem and some o-calculus. 

Blum integers and Williams integers are often used, too. Because of Dirichlet s prime-number theorem, asymptotically about half of all prime numbers are congruent to 3 mod 4, and among these, about half are congruent to 3 mod 8. Hence the factoring assumption made above is a consequence of one for arbitrary numbers with two prime factors. 

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