Does the Dirichlet function converge?

Does the Dirichlet function converge?

1.8 Corollary : For η(s) or for L(s, χ) with χ any non-trivial Dirichlet character χ modulo q, the Dirichlet series converges for σ > 0. We remark that Theorem 1.7 is proved in [4], but the proof is a little less direct than the one given above, and is restricted to the case an ∈ R (for no apparent reason).

How do you prove a Dirichlet is discontinuous?

Let D:R→R denote the Dirichlet function: ∀x∈R:D(x)={c:x∈Qd:x∉Q. where Q denotes the set of rational numbers. Then D is discontinuous at every x∈R.

What kind of discontinuities does Dirichlet function have?

The classification you link to defines an “infinite discontinuity” to mean the case where “one or both of the limits L− and L+ does not exist or is infinite”. That’s precisely what happens with the Dirichlet function everywhere.

Is Thomae’s function continuous?

Thomae Function is Continuous at Irrational Numbers.

Can the Dirichlet function be a derivative?

Since we do not have limits, we also cannot have continuity (even one-sided), that is, the Dirichlet function is not continuous at a single point. Consequently we do not have derivatives, not even one-sided.

How do you solve a Dirichlet problem?

Methods of solution For bounded domains, the Dirichlet problem can be solved using the Perron method, which relies on the maximum principle for subharmonic functions.

Are Dirichlet functions integrable?

The Dirichlet function is Lebesgue-integrable on R and its integral over R is zero because it is zero except on the set of rational numbers which is negligible (for the Lebesgue measure).

Is the Dirichlet function Riemann integrable?

On the other hand, the upper integral of Dirichlet function is b − a, while the lower integral is 0. They don’t match, so that the function is not Riemann integrable.

Why is the Dirichlet function not continuous?

The Dirichlet function is nowhere continuous, since the irrational numbers and the rational numbers are both dense in every interval [ a, b]. On every interval the supremum of f is 1 and the infimum is 0 therefore it is not Riemann integrable.

What is the Dirichlet function in math?

In mathematics, the Dirichlet function is the indicator function 1 ℚ of the set of rational numbers ℚ, i.e. 1 ℚ (x) = 1 if x is a rational number and 1 ℚ (x) = 0 if x is not a rational number (i.e. an irrational number). It is named after the mathematician Peter Gustav Lejeune Dirichlet.

Is the Dirichlet function a Baire class 2 function?

The Dirichlet function is an archetypal example of the Blumberg theorem. for integer j and k. This shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set.