Is there formula for sum of the Möbius function,
$$M(x)=\sum_{n\leq x}\mu (n)?$$
2026-04-02 05:25:15.1775107515
Is there a formula for Merten's function $M(x)=\sum_{n\leq x}\mu (n)$?
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The summatory function of the Moebius $\mu$-function is called the Mertens function $M(x)$ (which has been denoted by $f(x)$ here). An analytic formula for $M(x)$ is not known, but there are formulas involving the non-trivial zeros of the Riemann $\zeta$-function (sometimes assuming RH), see for example here.