Explicit bounds on the Moebius function
Collecting references: , , .
1. Bounds on $M(D)=\sum_{d\le D}\mu(d)$
The first explicit estimate for $M(D)$ is due to where the author proved that $|M(D)|\le \tfrac19 D+8$ for any $D\ge0$. A popular estimate is the one of .
Theorem (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 . shows that
Theorem (1993)
When $D\ge 120\,727$, we have $|M(D)|\le D/1036$.
On elaborating on this method, showed that
Theorem (1993)
When $D\ge 617\,973$, we have $|M(D)|\le D/2360$.
One of the argument is the estimate from
Theorem (1993)
When $33\le D\le 10^{12}$, we have $|M(D)|\le 0.571\sqrt{D}$.
This has been extended by to $10^{14}$ and recently in to $10^{16}$, i.e.
Theorem (2018)
When $33\le D\le 10^{16}$, we have $|M(D)|\le 0.571\sqrt{D}$.
Another tool is 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 . This preprint being difficult to get, it has been republished in .
Theorem (1996)
When $D\ge 2\,160\,535$, we have $|M(D)|\le D/4345$.
These results are used in to study the discrepancy of the Farey series.

Concerning upper bounds that tend to $0$, is the pioneer and shows among other estimates that
Theorem (1969)
When $D>0$, we have $|M(D)|/D\le 2.9/\log D$.
improves that into
Theorem (1995)
When $D\ge 685$, we have $|M(D)|/D\le 0.10917/\log D$.
The latest bound coming from improves that:
Theorem (2012)
When $D\ge 1\,100\,000$, we have $|M(D)|/D\le 0.013/\log D$.
In , bounds including coprimality conditions are proved and here is a typical example.
Theorem (2013)
When $1\le q < D$, we have $\Bigl|\sum_{\substack{ d\le D, \\ (d,q)=1}}\mu(d)\Bigr|/D\le \frac{q}{\varphi(q)}/(1+\log (D/q))$.
2. Bounds on $m(D)=\sum_{d\le D}\mu(d)/d$
shows that the sum $m(D)$ takes its minimal value at $D=13$. A folklore result is generalized in and reads
Theorem (1996)
When $D\ge 0$ and for any integer $r\ge1$, we have $\Bigl|\sum_{\substack{d\le D,\\ (d,r)=1}}\mu(d)/d\Bigr|\le 1$.
In fact, Lemma 1 of already contains the requisite material. The next result is proved in .
Theorem (2013)
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 is further extended in where it is shown that
Theorem (2012)
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 $0$, is the first to get such an estimate.
Theorem (1996)
When $D\ge33$ we have $|m(D)|\le 0.2185/\log D$.
When $D > 1$ we have $|m(D)|\le 726/(\log D)^2$.
This second bound is improved in .
Theorem (2015)
When $D > 1$ we have $|m(D)|\le 546/(\log D)^2$.
proves several bounds of the shape $m(D)\ll 1/\log D$. This is improved in by using . which provides us with a better manner to convert bounds on $M(D)$ into bounds for $m(D)$. Here is one result obtained.
Theorem (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 and , 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.
3. Bounds on $\check{m}(D)=\sum_{d\le D}\mu(d)\log(D/d)/d$
The initial investigations on this function go back to . In it is proved that
Theorem (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: $$ |\check{m}(D)-1|\le \frac{\frac74-\gamma}{D^2}\int_1^D|M(t)|dt+\frac{2}{D}. $$ It is also proved that, for any $D\ge1$, we have $$ 0\le \sum_{\substack{d\le D,\\ (d,q)=1}}\mu(d)\log(D/d)/d \le 1.00303 q/\varphi(q). $$
4. Miscellanae
Here is an elegant wide ranging estimate, taken from Claim 3.1 of .
Theorem (2015)
When $D\ge1$ we have $|\sum_{d>D}\mu(d)/d^2|\le 1/D$.
5. The Moebius function and arithmetic progressions
The results in this section are scarce. We mention a Theorem of .
Theorem (2015)
Let $\chi$  be a non-principal Dirichlet character mmodulo $q\ge37$ and let $k\ge1$ be some integer. Then $$ \biggl|\sum_{\substack{n\le x,\\ (n,k)=1}}\frac{\mu(n)\chi(n)}{n}\biggr| \le\frac{k}{\varphi(k)}\frac{2\sqrt{q}\log q}{L(1,\chi)}. $$
Last updated on February 18th, 2018, by Olivier Ramaré