06-16-2026, 03:11 AM
Mertens function
BY [WIKIPEDIA]
Summary
The Mertens function is a classical function in number theory defined as the cumulative sum of the Möbius function: $(M(n)=\sum_{k=1}^{n}\mu(k))$. It measures the difference between the number of square-free integers up to (n) with an even number of prime factors and those with an odd number.
Although it changes only by steps of −1, 0, or +1, its behavior is highly irregular and oscillatory, and understanding its growth is closely related to deep problems in analytic number theory, including the distribution of zeros of the Riemann zeta function.
A famous conjecture (the Mertens conjecture) claimed that $(|M(n)|\le \sqrt{n})$ for all (n), but this was disproved in 1985 by Odlyzko and te Riele, though no explicit counterexample is known. The true growth rate of (M(n)) remains unknown, but it is known to be closely connected to the Riemann Hypothesis and is studied using analytic and computational methods.
ARTICLE
BY [WIKIPEDIA]
Summary
The Mertens function is a classical function in number theory defined as the cumulative sum of the Möbius function: $(M(n)=\sum_{k=1}^{n}\mu(k))$. It measures the difference between the number of square-free integers up to (n) with an even number of prime factors and those with an odd number.
Although it changes only by steps of −1, 0, or +1, its behavior is highly irregular and oscillatory, and understanding its growth is closely related to deep problems in analytic number theory, including the distribution of zeros of the Riemann zeta function.
A famous conjecture (the Mertens conjecture) claimed that $(|M(n)|\le \sqrt{n})$ for all (n), but this was disproved in 1985 by Odlyzko and te Riele, though no explicit counterexample is known. The true growth rate of (M(n)) remains unknown, but it is known to be closely connected to the Riemann Hypothesis and is studied using analytic and computational methods.
ARTICLE
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