Proving Dirichelet's Function Converges to 0: A Mathematical Proof

  • Thread starter barksdalemc
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In summary, the conversation discusses the Dirichelet's function and its limit on the interval (0,1). The conversation also includes a proof given by a professor, explaining how to show that the limit of the function is 0. The main point of the conversation is that within any finite distance of a number a, there can only be a finite number of fractions with denominator less than or equal to n. This is important in understanding how to choose a delta value to ensure that f(x) is less than epsilon.
  • #1
barksdalemc
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Homework Statement


Consider the Dirichelet's function defined on (0,1) by

f(x)= 0 is x is irrational
and
f(x)= 1/q if x=p/q

where p and q are positive integers with no common factors. Show that lim f(x) for any x in (0,1) is 0

The Attempt at a Solution



Here is the first line of the proof the professor gave us.

"Let eps>0 and n so large that (1/n)<=eps, where n is a natural number. The only numbers x for which f(x) can > eps are 1/2, 1/3, 2/3, ..., 1/n, ...(n-1)/n. "

I don't understand how the any p/q with n as q can be greater than epsilon is n is chosen so large that 1/n <=eps.
 
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  • #2
barksdalemc said:
I don't understand how the any p/q with n as q can be greater than epsilon is n is chosen so large that 1/n <=eps.
had trouble reading this last sentence

do you mean you don't understand how it's possible that p/n > eps given that 1/n <= eps?
 
Last edited:
  • #3
yes that is exactly where I am confused, but he is saying that f(p/n) > eps. f(p/n) given 1/n.
 
  • #4
i'd list all those numbers to exclude them when I'm choosing delta, then any f(x)<1/n<=eps
 
  • #5
The point is that, within any finite distance of a number a, there can be only a finite number of numerators for a fraction with denominator n: and so only a finite number of fractions with denominator less than or equal to n.
 

1. What is Dirichelet's function?

Dirichelet's function, also known as the Dirichlet eta function, is a mathematical function that is defined as the alternating sum of the reciprocals of positive integers raised to a given power. It is denoted by the Greek letter eta (η) and is commonly used in number theory and analysis.

2. Why is it important to prove that Dirichelet's function converges to 0?

Proving that Dirichelet's function converges to 0 is important because it helps us understand the behavior of the function and its relationship to other mathematical concepts. It also allows us to make more accurate calculations and predictions in various fields, such as number theory, analysis, and physics.

3. What is the significance of proving its convergence to 0?

The significance of proving that Dirichelet's function converges to 0 lies in its application to various mathematical problems and theorems. For example, it is used in the proof of the prime number theorem and the Riemann hypothesis, which are fundamental concepts in number theory. Additionally, understanding its convergence behavior can also lead to new insights and discoveries in mathematics.

4. What are the steps involved in proving that Dirichelet's function converges to 0?

The proof of Dirichelet's function converging to 0 involves using various mathematical techniques, such as algebra, calculus, and analysis. It also requires a deep understanding of number theory and the properties of infinite series. The specific steps involved may vary depending on the approach taken, but generally, it involves manipulating the function and its series representation to show that it approaches 0 as the input value increases.

5. Are there any other methods for proving the convergence of Dirichelet's function to 0?

Yes, there are multiple methods for proving the convergence of Dirichelet's function to 0. Some of the commonly used approaches include using the Euler-Maclaurin summation formula, the Cauchy product, and the Abel summation formula. Each method has its own advantages and may be used depending on the specific problem or theorem being addressed.

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