Details

Snapshot 1: the graph of the Dirichlet -function for ; the corresponding Dirichlet character has only real values, so the zeros of the -function occur in conjugate pairs

Snapshot 2: the graph of ; the corresponding Dirichlet character sometimes has complex values, so the zeros do not occur in conjugate pairs

Introduction:

Let be a positive integer and suppose that and are relatively prime (i.e., and have no common factor greater than 1). Then Dirichlet's theorem says that the arithmetic progression has infinitely many primes. Dirichlet's theorem is proved in [1] by using Dirichlet characters and -functions. The number of integers between 1 and that are relatively prime to is given by , where is Euler's phi, the totient function.

For example, if , then . There are four integers between 1 and 10 that are relatively prime to : , , , and Therefore, each of the arithmetic progressions , , , and contains infinitely many primes, as takes the values .

Dirichlet Characters and -functions:

Given a positive integer , a character function, usually denoted by (the Greek letter chi), is a function defined on the numbers mod such that, for all and , , , and if and only if . There are different characters mod . See reference [2] for more information.

For example, there are four characters mod 5. The third Dirichlet character mod 5 (in Mathematica's numbering system) has the following values at : , , , , and . The second character mod 5 takes complex values: , , , , and . (There are two other characters mod 5: the first is real; the fourth is complex.)

For any modulus , there are at least two characters that take only real values, and usually some that take complex values.

The "Table of Dirichlet Characters" Demonstration (see the related link below) lets you choose the modulus and see the values of all of the different characters mod .

For each character , there is a corresponding Dirichlet -function of the complex variable , defined by

(1) .

In Mathematica notation, the Dirichlet -function for the character mod would be written as

(2) .

These series converge if . With analytic continuation, these functions can be extended to the entire complex plane (see [3]).

What do characters have to do with arithmetic progressions of the form ? This formula from page 252 of reference [1] illustrates the connection. If is a character mod , then , so we can factor out from the sum in equation (1) above to get

.

This shows how arithmetic progressions arise naturally from expressions involving characters and -functions.

Zeros of -Functions:

If a character is real (that is, if all its values are real), then the complex zeros of the corresponding Dirichlet -function occur in conjugate pairs. That is, if is a zero, then so is . For example, we saw above that the third character mod 5 is real. The first two pairs of complex zeros for the corresponding Dirichlet -function are approximately and . These are the four complex zeros whose imaginary parts are closest to 0.

On the other hand, if the character is complex, then the zeros of the corresponding -function are generally not in conjugate pairs. For example, consider the second character mod 5, which takes complex values. The four zeros of whose imaginary parts are closest to 0 are approximately , , , and .

Dirichlet -functions also have real zeros (so-called trivial zeros) at , and at either the negative even integers , or at the negative odd integers .

The Demonstration "A Formula for Primes in Arithmetic Progressions" (see the related link below) uses zeros of Dirichlet -functions to count the primes less than or equal to that are in various arithmetic progressions. Let be an integer. Estimate the primes in an arithmetic progression that are less than or equal to , then estimate the number that are less than or equal to . If (after suitable rounding), these counts differ by 1, then must be prime!

The Generalized Riemann Hypothesis:

The generalized Riemann hypothesis (GRH) is the unproven conjecture that any complex (that is, a so-called nontrivial) zero of whose real part is between 0 and 1, actually has real part equal to 1/2.

The GRH is a generalization of the more-famous Riemann hypothesis (RH), another unproven conjecture, that states that the complex zeros of the Riemann zeta function

all have real part equal to 1/2. For these reasons, the vertical line in the complex plane is called the "critical" line.

The Riemann zeta function is often used to prove theorems about primes, and to estimate the size of , the number of primes less than or equal to .

In a somewhat analogous way, -functions are used to prove theorems, such as Dirichlet's theorem, about primes in arithmetic progressions, and to estimate how many primes there are in the arithmetic progression that are less than or equal to .

The RH and the GRH are among the most important unsolved problems in all of mathematics. They are important, among other reasons, because our current estimates of the number of primes less than or equal to would become much more precise if the RH or the GRH could be proved.

References:

[1] W. J. LeVeque, Topics in Number Theory, vol. 2, Reading, MA: Addison–Wesley, 1961 pp. 81–124.

[2] Dirichlet Characteron Wikipedia.

[3] Dirichlet L-Functionon Wikipedia.

[4] Generalized Riemann Hypothesison Wikipedia.