# Asymptotic expansion for Mertens function

• mhill
In summary, the conjecture is that using the Selberg-Delange method, the next asymptotic equality holds: the sum of the Möbius function up to N is approximately equal to the square root of N times the logarithm of N to the power of a, multiplied by G(c) over the Gamma function of b. The constants a, b, and c are positive and G(c) is equal to G(c) times the Riemann zeta function evaluated at c. It is unlikely that the Mertens function will settle into this asymptotic equality. There is a result that proves it crosses zero an infinite number of times. This conjecture also implies the Riemann hypothesis, as it would mean that the
mhill
My conjecture is that using the Selberg-Delange method the next asymptotic equality holds

$$\sum_{n \le N } \mu (n) \sim (\sqrt n )log^{a}(n) \frac{G(c)}{\Gamma (b)}$$

where a,b and c are positive constants and Gamma stands for (n-1)! , G(c) is just $$G(c) \zeta (c) =1$$

I think it's quite unlikely that the Mertens function settles down into an asymptotic equality, eljose! Hasn't some kind of result been proved to the effect that it crosses zero an infinite number of times?

Further, I think the conjecture implies the Riemann hypothesis, since it would mean that M(n) is bounded by $$o(n^{1/2+\varepsilon})$$ -- right?

## 1. What is the Mertens function?

The Mertens function, denoted by M(x), is defined as the sum of the Möbius function μ(n) from 1 to x. It is named after the German mathematician Franz Mertens and is used in number theory to study the distribution of prime numbers.

## 2. What is an asymptotic expansion?

An asymptotic expansion is a mathematical technique used to approximate a function or sequence as a series of terms, with each term becoming less important as the input value increases. It is particularly useful for analyzing the behavior of functions near certain singular points or asymptotes.

## 3. How is the Mertens function related to asymptotic expansions?

The Mertens function has been extensively studied in number theory, and one of the key results is an asymptotic expansion for the function. This expansion, first derived by Pierre Dusart in 2010, allows for a more precise understanding of the behavior of the Mertens function as x gets very large.

## 4. What is the significance of the asymptotic expansion for the Mertens function?

The asymptotic expansion for the Mertens function has important implications for the distribution of prime numbers. It provides a more accurate estimate of the function's growth rate and helps to better understand the relationship between the Mertens function and the Riemann zeta function.

## 5. Are there any limitations to the asymptotic expansion for the Mertens function?

While the asymptotic expansion for the Mertens function is a significant result in number theory, it is important to note that it is only an approximation and may not hold true for very large values of x. Additionally, the expansion assumes the Riemann hypothesis, which has yet to be proven and remains one of the most important unsolved problems in mathematics.

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