The Mertens function is a mathematical function denoted as M(n) , which sums the values of the Möbius function \mu(k) for all integers k from 1 to n . The Möbius function is defined based on the prime factorization of integers, taking values of -1, 0, or 1 depending on whether k has an even number of distinct prime factors, an odd number, or is divisible by a square of a prime, respectively. The Mertens function is significant in number theory, particularly in the study of prime numbers and their distribution. It is closely related to the Riemann Hypothesis, a famous unsolved problem in mathematics that concerns the distribution of prime numbers. The behavior of the Mertens function can provide insights into the properties of primes and their density among integers.