SUMMARY
The discussion centers on the conjecture regarding the asymptotic behavior of the Mertens function using the Selberg-Delange method. The proposed asymptotic equality is expressed as \(\sum_{n \le N } \mu (n) \sim (\sqrt n )\log^{a}(n) \frac{G(c)}{\Gamma (b)}\), where \(a\), \(b\), and \(c\) are positive constants, with \(G(c)\) defined as \(G(c) \zeta (c) = 1\). The conjecture suggests implications for the Riemann hypothesis, indicating that the Mertens function \(M(n)\) is bounded by \(o(n^{1/2+\varepsilon})\). The discussion also touches on the behavior of the Mertens function crossing zero infinitely often.
PREREQUISITES
- Understanding of the Mertens function and its properties
- Familiarity with the Selberg-Delange method
- Knowledge of asymptotic analysis in number theory
- Basic concepts of the Riemann hypothesis
NEXT STEPS
- Research the Selberg-Delange method in detail
- Study the properties and implications of the Mertens function
- Examine the relationship between the Mertens function and the Riemann hypothesis
- Explore asymptotic expansions in analytic number theory
USEFUL FOR
Mathematicians, number theorists, and researchers interested in analytic number theory and the properties of the Mertens function.