Is there formula for sum of the Möbius function,
$$M(x)=\sum_{n\leq x}\mu (n)?$$
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3$\begingroup$ It's a vague question. What do you mean by 'formula'? It's called Merten's function, and some estimates are possible to derive but it seems it's not what you want. $\endgroup$– Balarka SenMay 30, 2014 at 13:44
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3$\begingroup$ I really don't understand why this question is on hold. If someone is just starting to learn number theory, how would you know what is the precise question to ask about this sum? You wouldn't know that it is called "Merten's function", and you wouldn't know that there is a formula, but that it depends on zeta zeros, and you wouldn't know that often such sums are given estimates but not exact values. Any or all of these things could be explained. $\endgroup$– John MMay 30, 2014 at 17:40
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3$\begingroup$ @John M.: apart from the "unclear what you're asking" issue, the question also is a PSQ (a question that merely states a problem with no additional value-added prose such as where the question was encountered or what has been tried). So the question also fits the "lack of context" closure reason. The two issues are related, of course: the reason that it is unclear what is being asked is that not enough context was included in the question. $\endgroup$– Carl MummertJun 2, 2014 at 1:40
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3$\begingroup$ @Carl Mummert: That would seem to be relevant for a homework-style question, but this seems to be just a question. I would think that people should be able to ask conceptual and basic factual questions without "showing their work." Must we insist that the OP write something like, "I was trying to prove the Riemann Hypothesis earlier today, and I reduced it to a question of the growth of this function. Does anyone have any ideas?" $\endgroup$– John MJun 2, 2014 at 1:50
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4$\begingroup$ @John M.: I agree in principle, and years ago this sort of question was more welcomed. It's always a judgement call to some extent. But, at the same time, it is not particularly hard for the asker to include a sense of why they were looking at the question, what they have tried, or what sort of answer they are looking for - something to add value to the question. I don't think it's unreasonable to ask for more in a question that this one has. For higher-level questions, the user can go to MathOverflow, which welcomes directly stated questions at the graduate and research level. $\endgroup$– Carl MummertJun 2, 2014 at 1:56
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1 Answer
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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.
answered May 30, 2014 at 13:47