Number Theory Series Problem

In summary: F(\frac{n}{p_t})But recall that F(n) is defined as \sum_{k=1}^{n}f(\frac{k}{n}). So, we can rewrite the above expression as:µ * F = F(n)-F(n)-...- F(n) = 0This means that µ * F=0, which proves that G=µ * F. I hope this helps to clarify things for you. In summary, we have shown that the Dirichlet product of the Möbius function and the function F is equal to the function G, as required in the problem. Thank you for your attention.
  • #1
Poopsilon
294
1
First note that the operation * denotes the Dirichlet product, and µ denotes the Möbius function.

Ok so here is the problem:

Let f(x) be defined for all rational x in 0≤x≤1 and let [tex]F(n)=\sum_{k=1}^nf(\frac{k}{n})\;\;\;\;\; and \;\;\;\;\;\; G(n)=\sum_{k=1,\;(k,n)=1}^nf(\frac{k}{n}).[/tex]

Prove that G=µ * F.Formally here is what I've got: Set [tex]n=p_{1}^{e_1}...p_t^{e_t}.[/tex] than
[tex]\mu * F = \sum_{d|n}[\;\mu (d)\sum_{k=1}^{\frac{n}{d}}f(\frac{dk}{n})\;][/tex][tex]=\sum_{k=1}^{n}f(\frac{k}{n})-[\;\sum_{k=1}^{\frac{n}{p_1}}f(\frac{p_1k}{n})+...+ \sum_{k=1}^{\frac{n}{p_t}}f(\frac{p_tk}{n})\;]+\sum_{k=1,\;i≠j}^{\frac{n}{p_ip_j}}f(\frac{p_ip_jk}{n})-...+f(\frac{n}{n}).[/tex]

So this is probably a lot to take in but I'm pretty sure it's correct, just remember that the Möbius function is zero whenever d contains multiple copies of a prime in its factorization and thus many terms get zeroed out. Now, after working with some toy examples, the way I see this equaling G is that the initial series, which is just F, gets all its terms satisfying (k,n)>1 canceled out by terms coming from those series inside the brackets. But there are some terms in those series which have doubled up, tripled up, etc. and those terms will be canceled out by the later series(plural) which range over i,j and i,j,l and so on and so forth depending on the size of n. Now! The question is how the hell can I show this rigorously? It's just such a mess that there must be some way of simplifying the situation but I can't see it. Thanks.
 
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  • #2

Thank you for your post. I am a scientist and I would like to respond to your problem. First, I would like to clarify that the Möbius function µ is defined for positive integers, not for rational numbers. So, when we write µ * F, we are actually talking about the Dirichlet product of µ and F, which is defined for positive integers n.

Now, let's take a closer look at the expression you have written for µ * F. As you have correctly noted, the Möbius function µ is zero whenever d contains multiple copies of a prime in its factorization. This means that in the summation, only terms with d=1 will contribute to the final result. Therefore, we can simplify the expression to:

µ * F = \sum_{k=1}^{n}f(\frac{k}{n})-\sum_{k=1}^{\frac{n}{p_1}}f(\frac{p_1k}{n})-...- \sum_{k=1}^{\frac{n}{p_t}}f(\frac{p_tk}{n})

Now, let's focus on the second term in this expression, which is \sum_{k=1}^{\frac{n}{p_1}}f(\frac{p_1k}{n}). Notice that this summation is over all k such that p_1k is a multiple of n, which means that k is a multiple of n/p_1. In other words, k/p_1 is a rational number in the range 0≤x≤1. Therefore, we can rewrite this summation as:

\sum_{k=1}^{\frac{n}{p_1}}f(\frac{p_1k}{n}) = \sum_{k=1}^{\frac{n}{p_1}}f(\frac{k}{\frac{n}{p_1}}) = F(\frac{n}{p_1})

Using the same reasoning, we can rewrite the other terms in a similar way, and we get:

µ * F = \sum_{k=1}^{n}f(\frac{k}{n})-\sum_{k=1}^{n}F(\frac{n}{p_1})-...- \sum_{k=1}^{n}F(\frac{n}{p_t}) = F(n)-F(\frac
 

1. What is Number Theory Series Problem?

Number Theory Series Problem is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves studying patterns, sequences, and operations on numbers to uncover connections and solve problems.

2. What are some common topics in Number Theory Series Problem?

Some common topics in Number Theory Series Problem include prime numbers, divisibility, modular arithmetic, number patterns, and Diophantine equations. These topics are essential in solving various real-world problems and have applications in other fields, such as computer science and cryptography.

3. How is Number Theory Series Problem used in real life?

Number Theory Series Problem has various applications in real life, such as in coding and encryption. It is also used in fields like finance, where it helps in understanding and predicting patterns in the stock market. Number Theory Series Problem also plays a crucial role in creating secure passwords and protecting sensitive information.

4. What are some famous problems in Number Theory Series Problem?

Some famous problems in Number Theory Series Problem include the Goldbach Conjecture, which states that every even number greater than 2 can be expressed as the sum of two prime numbers, and the Twin Prime Conjecture, which proposes that there are infinitely many pairs of prime numbers that differ by 2. Other notable problems include the Collatz Conjecture and the Riemann Hypothesis.

5. Are there any resources for learning more about Number Theory Series Problem?

Yes, there are many resources available for learning more about Number Theory Series Problem. Some popular books on the subject include "Elementary Number Theory" by Kenneth H. Rosen and "An Introduction to the Theory of Numbers" by G.H. Hardy and E.M. Wright. Additionally, there are many online courses and tutorials on platforms like Coursera and Khan Academy, as well as numerous websites and forums dedicated to discussing Number Theory Series Problem.

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