# dirichlet

1. ### Showing the modified Dirichlet function is discontinuous

Show, using the $\epsilon-\delta$ definition of continuity, that the modified Dirichlet function, i.e., $f(x) = x$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational, is discontinuous at all points $c \neq 0$ My attempt: Is the following argument right (using the sequential definition of...
2. ### Dirichlet function

I am trying to prove that f(x)=xD(x) is only continuous at x=0, however I don't know if I can prove it with assuming two different cases for continuity, in the neighbourhood of x=a when x is rational and irrational.. Not sure if it's possible to assume this, would appreciate if I can get...
3. ### Dirichlet characters

Hi! So i have made a table of all the nonprincipal Dirichlet characters mod 16. Is the following table correct? n 1 3 5 7 9 11 13 15 X1(n) 1 1 1 1 1 1 1 1 X2(n) 1 i -i -1 -1 -i i 1 X3(n) 1 -1 -1 1 1 -1 -1 1 X4(n) 1 -i i -1 -1 i -i 1 X5(n) 1 1 -1 -1 1 1 -1 -1 X6(n) 1 i i 1 -1 -i -i...
4. ### Dirichlet characters

Hi all, sorry if this is in the wrong section....?? I am looking to construct a table of values for all the nonprincipal Dirichlet characters mod 16. Can anyone help? Lexi
5. ### Dirichlet function proof

I'm having a hard time understanding this proof. for each point c of R. each open interval of the form (c - \frac{1}{n}, c+ \frac{1}{n}), where n \in N Contains ration x_n and irrational y_n. Ok so far, but then Considering \{x_n\} and \{y_n\}, we have x_n \to c and y_n\to c, by the squeeze...
6. ### PDE problem - Separation of variables - Dirichlet condition

[Solved] PDE problem - Separation of variables - Dirichlet condition Hi all I'm in quite some trouble with this PDE problem, that is part of my assignment for wednesday. First of all I got the PDE problem: PDE: u_t - 2*u_xx = 0 , 0<x<1, t>0 BC: u(0,t) = t, u(1,t)=0 IC...
7. ### example of using dirichlet's test

I need an exmaple of example of using dirichlet's test for assessing if functions series are convergent. please help
8. ### The biharmonic problem with Dirichlet and Neumann conditions concerning unique sol

The function u \in H_0^2(U) (where U\subset \mathbb{R}^n is bounded with smooth boundary \partial U) is a weak solution of the biharmonic equation \Delta(\Delta u)=f \mbox{ in }U and u=\frac{\partial u}{\partial\nu}=0\mbox{ on }\partial U provided \int_U \Delta u\Delta v dx=\int_U f v dx...
9. ### Dirichlet function upper and lower sum

Question: Suppose that f(x)=7 if x is rational and f(x)=3 if x is irrational on the interval [1,3]. Let P be a partition of [1,3]. What is the lower sum: I have that the lower sum is 0. What is the upper sum: I have that the upper sum is 4. What is sup L(f,P) over all partitions: 7 What is...
10. ### Dirichlet Principle

How many pairs of integers (a,b) are necessary to make sure that for two of them, say ({a_1},{b_1}) and ({a_2},{b_2}) it is the case that {a_1} mod 5 = {a_2} mod 5 and {b_1} mod 5 = {b_2} mod 5?
11. ### Approximate functional equation for Dirichlet eta, does any exist?

Referring to Hardy's approximate functional equation for Riemann's Zeta \zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) would anybody know of similar results for the Dirichlet function ...
12. ### Dirichlet Series Example

Hey, Im looking for an example of a Dirichlet series \sum_{n=1}^{\infty}\frac{a_n}{n^s},s \in \mathbb{C}, whose convergence abscissa differs from its absolute convergence abcissa, but the difference is strictly less than 1. Can you help me, and is it clear what Im looking for?
13. ### Coefficients of a Dirichlet series

Let F(s) = \zeta(s)L(s,\chi_1)L(s,\chi_2)L(s,\chi_1\chi_2), for two real primitive characters \chi_1, \chi_2. Multiplication of Euler products of these functions gives us F(s) = \displaystyle \sum_{n=1}^\infty a_n n^{-s} with a_1=1. Davenport (in p.129 of Mult. Numb. Theory) claims that taking...
14. ### Two questions concerning Laplacian (and a Dirichlet Problem)

Hi, got a couple of questions on my latest worksheet (non-assessed) that I'm struggling with. Any help would be appreciated. 1) Find an example of a function u: R^n -> R (where n > 2) that is twice differentiable everywhere, is subharmonic (Laplacian of u is greater than or equal to 0), is...
15. ### Estimating the Dirichlet Kernel

Hi! I need help with the following problem: Let D_n (theta) = sum(k=-N to N) e^ik(theta)= sin ((N+1/2)theta)/sin(theta/2) and define L_n = 1/2Pi integral (from - Pi to Pi) |D_n (theta)| d(theta) prove that L_N is greater than or equal to c log (N) for some constant c>0 Hint: show that...
16. ### dirichlet question

Let D:R-->R be the Dirichlet function defined as: D(x) = 1 when x is rational. D(x) = 0 when x is irrational. Let f:R-->R be a continuous function. Which of the following doesn't not necessarily cause f to a be a constant function? There were 4 options to choose from. The answer was...
17. ### Dirichlet characters

I've been trying to justify the following step in a textbook (Iwaniec-Kowalski) proof, but just can't seem to: \displaystyle \sum_{n\:\equiv\: 0 \mod k} \chi (n) = \sum_{l|k} \mu (l) \sum_{(n,l)=1} \chi (n) where \mu is the Mobius function. It's clearly some application of Mobius inversion...
18. ### Dirichlet Problem-Partial Differential Equations

Let K = \{ (x,y) | -1<x<1 , -1<y<1 \} . Find the unique soloution of dirichlet problem: \Delta u(x,y) =0 , (x,y) \in K , u(x,y) = |x+y| , (x,y) \in \partial K . We need to guess a soloution and not use seperation of variables! Hope you'll be able to help me! Thanks !
19. ### Dirichlet problem for a rectangle

Hello, am new to the dirichlet topic, and am unable to understand how to solve it, neither the approach to solve the problem. could you tell me how to solve this problem, if so it wil be helpful in solving other problems also the problem is \nabla^{2}= 0 0<x<1 0<y<1 u(x,0) = x(x-1)...
20. ### Stick-breaking process vs. flat dirichlet?

This might be a stupid question, but however... I have randomly picked three variables (between 0-1) from an equal distribution. I let them form the "breaking points" on an imagined stick of the length 1. Do the four "pieces" obtained (which together sum up to 1, ofcourse) represent a set...