The Soft Möbius Function and the Riemann Hypothesis

This chapter extends the Möbius function to the Soft Möbius function. The regular Möbius function, which is related to prime factorization, accepts three values: 1, 0, and − 1. Using Soft logic, we extend the function to the following five values: −1, − 0, 0, + 0, and 1. We prove the basic properties of the Soft Möbius function and discuss the properties of the Soft Mertens function, which is the aggregate sum of the values of the Soft Möbius function. We show that the relationship between the properties of the Mertens function and the Riemann hypothesis is correct also when applied to the relationship between the Soft Mertens function and the Riemann hypothesis.