Explicit bounds on the Moebius function¶
Collecting references: [Diamond and Erdös, 1980], [Deléglise and Rivat, 1996], [Borwein et al., 2008].
1. Bounds on \(M(D)=\sum\limits_{d\le D}\mu(d)\)¶
The first explicit estimate for \(M(D)\) is due to [von Sterneck, 1898] where the author proved that \(|M(D)|\le \tfrac19 D+8\) for any \(D\ge0\). Here is a first popular estimate.
Theorem ([Mac Leod, 1967])
When \(D\ge 0\), we have \(|M(D)|\le \tfrac1{80} D+5\). When \(D\ge 1119\), we have \(|M(D)|\le D/80\).
We mention at this level the annoted bibliography contained at the end of [Dress, 1983/84].
Theorem ([Costa Pereira, 1989])
When \(D\ge 120\,727\), we have \(|M(D)|\le D/1036\).
Improving on this method, the next result was obtained.
Theorem ([Dress and El Marraki, 1993])
When \(D\ge 617\,973\), we have \(|M(D)|\le D/2360\).
One of the arguments is the next estimate.
Theorem ([Dress, 1993])
When \(33\le D\le 10^{12}\), we have \(|M(D)|\le 0.571\sqrt{D}\).
This has been extended by [Kotnik and van de Lune, 2004] to \(10^{14}\) and recently by Hurst.
Theorem ([Hurst, 2018])
When \(33\le D\le 10^{16}\), we have \(|M(D)|\le 0.571\sqrt{D}\).
Another tool is given in [Cohen and Dress, 1988], where refined explicit estimates for the remainder term of the counting functions of the squarefree numbers in intervals are obtained. The latest best estimate of this shape comes from [Cohen et al., 1996]. This preprint being difficult to get, it has been republished in [Cohen et al., 2007].
Theorem (1996)
When \(D\ge 2\,160\,535\), we have \(|M(D)|\le D/4345\).
These results are used in [Dress, 1999] to study the discrepancy of the Farey series.
Concerning upper bounds that tend to \(0\), [Schoenfeld, 1969] was the pioneer and obtained, among other things, the following estimates.
Theorem (1969)
When \(D>0\), we have \(|M(D)|/D\le 2.9/\log D\).
Those were later improved in:
Theorem ([El Marraki, 1995])
When \(D\ge 685\), we have \(|M(D)|/D\le 0.10917/\log D\).
Ramaré further improved those bounds for larger \(D\).
Theorem ([Ramaré, 2013])
When \(D\ge 1\,100\,000\), we have \(|M(D)|/D\le 0.013/\log D\).
- Some bounds including coprimality conditions were also obtained.
For instance, we have
Theorem ([Ramaré, 2015])
When \(1\le q < D\), we have \(\Bigl|\sum\limits_{\substack{ d\le D, \\ (d,q)=1}}\mu(d)\Bigr|/D\le \frac{q}{\varphi(q)}/(1+\log (D/q))\).
Theorem ([Ramaré, 2015])
For \(1\le q < D\), we have \(\frac{\varphi(q)}{q}\log(D/q)\Bigl|\sum\limits_{\substack{ d\le D, \\ (d,q)=1}}\mu(j)\Bigr|/D \leq \left\{ \begin{array}{cl} 0.997, & D/q > 1, \\ 0.429,& D/q \geq 490, \\ 1/5, & D/q \geq 4536, \\ 0.0918, & D/q \geq 48513. \end{array} \right.\).
The best uniform bound (in \(q\)) of the form above for \(D/q > 1\) were obtained in [de Camargo, 2025].
Theorem ([de Camargo, 2025])
When \(1\le q < D\), we have \(\frac{\varphi(q)}{q}\log(D/q)\Bigl|\sum\limits_{\substack{ d\le D, \\ (d,q)=1}}\mu(d)\Bigr|/D \leq \left\{ \begin{array}{cl} 0.3131, & q = 2,\\ 0.2663, & q = 3, \\ 0.2335, & q = 5 \\ 0.1738, & q = 6, \\ 0.2102, & q \geq 7. \end{array} \right.\). These constants are optimal up to the third decimal place.
2. Bounds on \(m(D)=\sum\limits_{d\le D}\frac{\mu(d)}{d}\)¶
[Mac Leod, 1967] shows that the sum \(m(D)\) takes its minimal value at \(D=13\). A folklore result reads as follows.
Theorem ([Granville and Ramaré, 1996])
When \(D\ge 0\) and for any integer \(r\ge1\), we have \(\Bigl|\sum_{\substack{d\le D,\\(d,r)=1}}\frac{\mu(d)}{d}\Bigr|\le 1\).
In fact, Lemma 1 of [Davenport, 1937] already contains the requisite material. The inequality in the theorem above was rediscovered and generalized in [Tao, 2010] to sums over semi-groups generated by arbitrary sets of prime numbers. Further refinements of the result above were obtained for larger \(D\) as shown below.
Theorem ([Ramaré, 2015])
When \(D\ge 7\), we have \(|\sum_{d\le D}\mu(d)/d|\le 1/10\). We can replace the couple (7, 1/10) by (41, 1/20) or (694, 1/100).
This result has been further extended.
Theorem ([Ramaré, 2013])
When \(D\ge 0\) and for any integer \(r\ge1\) and any real number \(\varepsilon\ge0\), we have \(\Bigl|\sum_{\substack{d\le D,\\(d,r)=1}}\mu(d)/d^{1+\varepsilon}\Bigr|\le 1+\varepsilon\).
Concerning upper bounds that tend to \(0\), here is a first estimate.
Theorem ([El Marraki, 1996])
When \(D\ge33\) we have \(|m(D)|\le 0.2185/\log D\).
When \(D > 1\) we have \(|m(D)|\le 726/(\log D)^2\).
The second bound above was improved:
Theorem ([Bordellès, 2015])
When \(D > 1\) we have \(|m(D)|\le 546/(\log D)^2\).
[Ramaré, 2013] proves several bounds of the shape \(m(D)\ll 1/\log D\). Those results were improved using the tools of [Balazard, 2012], which provide us with a better manner to convert bounds on \(M(D)\) into bounds for \(m(D)\). Here is one result obtained.
Theorem ([Ramaré, 2015])
When \(D\ge 463\,421\) we have \(|m(D)|\le 0.0144/\log D\).
We can, for instance, replace the couple (463 421, 0.0144)by any of (96 955, 1/69), (60 298, 1/65), (1426, 1/20) or (687, 1/12).
In [Ramaré, 2014] and [Ramaré, 2015], the problem of adding coprimality conditions is further addressed. Here is one of the results obtained.
Theorem (2015)
When \(1\le q < D\) we have \(\Bigl|\sum_{\substack{d\le D,\\ (d,q)=1}}\mu(d)/d\Bigr|\le \frac{q}{\varphi(q)}0.78/\log(D/q)\). When \(D/q\ge 24233\), we can replace 0.78 by 17/125.
The estimates above were improved in 2025.
Theorem ([de Camargo, 2025])
When \(1\le q < D\) we have \(\Bigl|\sum_{\substack{d\le D,\\ (d,q)=1}}\mu(d)/d\Bigr|\le \frac{q}{\varphi(q)}0.3055/\log(D/q)\).
3. Bounds on \(\check{m}(D)=\sum\limits_{d\le D}\mu(d)\log(D/d)/d\)¶
The initial investigations on this function go back to [Daublebsky von Sterneck, 1902].
Theorem ([Ramaré, 2015])
When \(3846 \le D\) we have \(|\check{m}(D)-1|\le 0.00257/\log D\). When \(D > 1\), we have \(|\check{m}(D)-1|\le 0.213/\log D\).
This implies in particular that
Theorem (2015)
When \(222 \le D\) we have \(|\check{m}(D)-1|\le 1/1250\). When \(D > 1\), the optimal bound 1 holds.
These bounds are a consequence of the identity:
\(\displaystyle |\check{m}(D)-1|\le \frac{\frac74-\gamma}{D^2}\int_1^D|M(t)|dt+\frac{2}{D}.\) It is also proved that, for any \(D\ge1\), we have
4. Miscellanae¶
Here is an elegant wide ranging estimate, taken from Claim 3.1 of the referenced paper.
Theorem ([Treviño, 2015])
For real \(D\ge1\) we have \(|\sum\limits_{d>D}\mu(d)/d^2|\le 1/D\).
Taking the limit for \(D \rightarrow K+1\) in the inequality above and after replacing \(K+1\) by \(D\), we get \(|\sum\limits_{d\geq D}\mu(d)/d^2|\le 1/D\), which improves upon the trivial upper bound \(1/(D-1)\).
5. The Moebius function and arithmetic progressions¶
The results in this section are scarce.
Theorem ([Bordellès, 2015])
Let \(\chi\) be a non-principal Dirichlet character modulo \(q\ge37\) and let \(k\ge1\) be some integer. Then \(\displaystyle \biggl|\sum_{\substack{n\le x,\\ (n,k)=1}}\frac{\mu(n)\chi(n)}{n}\biggr|\le\frac{k}{\varphi(k)}\frac{2\sqrt{q}\log q}{L(1,\chi)}.\)