Convergence of sum involving Mobius function (with Riemann Zeta)

In summary, the conversation is about a "heuristic" argument that involves taking the derivative of both sides of a sum and evaluating the limit as s approaches 1. The argument concludes that the sum of a certain expression tends to -1 as s approaches 1, which supports the idea that another sum may also equal -1. The person asking for clarification receives some criticism for not writing clearly and is advised to use shorter sentences.
  • #1
The_Shape
3
0
Hello everyone.

I was hoping someone could clarify this "heuristic" argument I found online. First, what is the analytic function they speak of and is its derivative difficult to compute? Second, does this look like a legit argument? :

If you take the derivative w.r.t s of both sides of sum mu(n)/n^s=1/zeta(s) and have s->1, you get that the sum [mu(n)/n^s]* (-log n) is -zeta'(s)/(zeta(s))^2. Because you probably know that zeta(s)=1/(s-1)+analytic function, zeta'(s)=-1/(s-1)^2+analytic, so just by dividing, zeta'(s)/(zeta(s))^2 is [-1/(s-1)^2 + analytic]/[1/(s-1)^2+terms with 1/(s-1) and analytic], so multiplying by (s-1)^2/(s-1)^2 yields that as s->1, everything is 0 in numerator and denominator except for the leading terms, i.e. [-1+terms tending to zero]/[1+terms tending to zero], so it all cancels to -1. So the sum [mu(n)/n^s]* log(n) tends to -1 as s->1, so that again motivates that the sum mu(n)/n log(n) could well be -1.
 
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  • #2
Piece of advice. Learn to write more clearly. Short sentences for each step would be helpful.
 
  • #3
Uh, ok. Thanks for the advise. I copied this from a website. That's why I posted it here. I was hoping to get some help on it but apparently all I am going to get is snide remarks. I mean, why even reply?
 

1. What is the Mobius function and how is it related to the Riemann Zeta function?

The Mobius function is a number-theoretic function that is defined as follows:
μ(n) = 1 if n is a square-free positive integer with an even number of prime factors
μ(n) = -1 if n is a square-free positive integer with an odd number of prime factors
μ(n) = 0 if n has a squared prime factor
The Riemann Zeta function, denoted as ζ(s), is a mathematical function that is defined for all complex numbers s with real part greater than 1. The Mobius function is related to the Riemann Zeta function through the following formula:
∑μ(n)/n^s = 1/ζ(s)

2. What is the significance of the convergence of sums involving the Mobius function and the Riemann Zeta function?

The convergence of sums involving the Mobius function and the Riemann Zeta function has important implications in number theory and analytic number theory. These functions are closely related to the distribution of prime numbers and the Riemann Hypothesis, which is one of the most famous unsolved problems in mathematics. Understanding their convergence can provide insights into the behavior of prime numbers and potentially lead to a proof of the Riemann Hypothesis.

3. What is the current state of research on the convergence of sums involving the Mobius function and the Riemann Zeta function?

The convergence of sums involving the Mobius function and the Riemann Zeta function is an active area of research in analytic number theory. While some results have been proven, there are still many open questions and conjectures that remain unsolved. Researchers continue to explore different approaches and techniques to study the convergence of these sums and their implications in number theory.

4. Can you provide an example of a sum involving the Mobius function and the Riemann Zeta function?

One example of a sum involving the Mobius function and the Riemann Zeta function is the Mertens function, denoted as M(n), which is defined as the sum of the Mobius function from 1 to n. In other words, M(n) = ∑μ(k) for k = 1 to n. The Mertens function has been extensively studied and is closely related to the Riemann Hypothesis. Its behavior is also connected to the distribution of primes and the prime number theorem.

5. How is the convergence of sums involving the Mobius function and the Riemann Zeta function related to the Riemann Hypothesis?

The Riemann Hypothesis is a conjecture that states that all non-trivial zeros of the Riemann Zeta function lie on the critical line with real part equal to 1/2. One approach to proving the Riemann Hypothesis is through studying the convergence of sums involving the Mobius function and the Riemann Zeta function. In particular, the Mertens function is closely related to the Riemann Hypothesis and its behavior can provide insights into the truth of the conjecture. Therefore, understanding the convergence of these sums is crucial in the study of the Riemann Hypothesis.

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