Explicit bounds on the Moebius function ======================================= .. if-builder:: html .. toctree:: :maxdepth: 2 Collecting references: :cite:`Diamond-Erdos80`, :cite:`Deleglise-Rivat96-2`, :cite:`Borwein-Ferguson-Mossinghoff08`. 1. Bounds on :math:`M(D)=\sum\limits_{d\le D}\mu(d)` ---------------------------------------------------- The first explicit estimate for :math:`M(D)` is due to :cite:`VonSterneck98` where the author proved that :math:`|M(D)|\le \tfrac19 D+8` for any :math:`D\ge0`. Here is a first popular estimate. .. admonition:: Theorem (:cite:`MacLeod69`) :class: thm-tme-emt When :math:`D\ge 0`, we have :math:`|M(D)|\le \tfrac1{80} D+5`. When :math:`D\ge 1119`, we have :math:`|M(D)|\le D/80`. We mention at this level the annoted bibliography contained at the end of :cite:`Dress83`. .. admonition:: Theorem (:cite:`CostaPereira89`) :class: thm-tme-emt When :math:`D\ge 120\,727`, we have :math:`|M(D)|\le D/1036`. Improving on this method, the next result was obtained. .. admonition:: Theorem (:cite:`Dress-ElMarraki93`) :class: thm-tme-emt When :math:`D\ge 617\,973`, we have :math:`|M(D)|\le D/2360`. One of the arguments is the next estimate. .. admonition:: Theorem (:cite:`Dress93`) :class: thm-tme-emt When :math:`33\le D\le 10^{12}`, we have :math:`|M(D)|\le 0.571\sqrt{D}`. This has been extended by :cite:`Kotnik-VanDeLune04` to :math:`10^{14}` and recently by Hurst. .. admonition:: Theorem (:cite:`Hurst18`) :class: thm-tme-emt When :math:`33\le D\le 10^{16}`, we have :math:`|M(D)|\le 0.571\sqrt{D}`. Another tool is given in :cite:`Cohen-Dress88`, 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 :cite:`Cohen-Dress-ElMarraki96`. This preprint being difficult to get, it has been republished in :cite:`Cohen-Dress-ElMarraki07`. .. admonition:: Theorem (1996) :class: thm-tme-emt When :math:`D\ge 2\,160\,535`, we have :math:`|M(D)|\le D/4345`. These results are used in :cite:`Dress99` to study the discrepancy of the Farey series. Concerning upper bounds that tend to :math:`0`, :cite:`Schoenfeld69` was the pioneer and obtained, among other things, the following estimates. .. admonition:: Theorem (1969) :class: thm-tme-emt When :math:`D>0`, we have :math:`|M(D)|/D\le 2.9/\log D`. Those were later improved in: .. admonition:: Theorem (:cite:`ElMarraki95`) :class: thm-tme-emt When :math:`D\ge 685`, we have :math:`|M(D)|/D\le 0.10917/\log D`. Ramaré further improved those bounds for larger :math:`D`. .. admonition:: Theorem (:cite:`Ramare12-2`) :class: thm-tme-emt When :math:`D\ge 1\,100\,000`, we have :math:`|M(D)|/D\le 0.013/\log D`. Some bounds including coprimality conditions were also obtained. For instance, we have .. admonition:: Theorem (:cite:`Ramare12-5`) :class: thm-tme-emt When :math:`1\le q < D`, we have :math:`\Bigl|\sum\limits_{\substack{ d\le D, \\ (d,q)=1}}\mu(d)\Bigr|/D\le \frac{q}{\varphi(q)}/(1+\log (D/q))`. .. admonition:: Theorem (:cite:`Ramare12-5`) :class: thm-tme-emt For :math:`1\le q < D`, we have :math:`\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 :math:`q`) of the form above for :math:`D/q > 1` were obtained in :cite:`Camargo25`. .. admonition:: Theorem (:cite:`Camargo25`) :class: thm-tme-emt When :math:`1\le q < D`, we have :math:`\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 :math:`m(D)=\sum\limits_{d\le D}\frac{\mu(d)}{d}` -------------------------------------------------------------- :cite:`MacLeod69` shows that the sum :math:`m(D)` takes its minimal value at :math:`D=13`. A folklore result reads as follows. .. admonition:: Theorem (:cite:`Granville-Ramare96`) :class: thm-tme-emt When :math:`D\ge 0` and for any integer :math:`r\ge1`, we have :math:`\Bigl|\sum_{\substack{d\le D,\\(d,r)=1}}\frac{\mu(d)}{d}\Bigr|\le 1`. In fact, Lemma 1 of :cite:`Davenport37-1` already contains the requisite material. The inequality in the theorem above was rediscovered and generalized in :cite:`Tao10` to sums over semi-groups generated by arbitrary sets of prime numbers. Further refinements of the result above were obtained for larger :math:`D` as shown below. .. admonition:: Theorem (:cite:`Ramare12-5`) :class: thm-tme-emt When :math:`D\ge 7`, we have :math:`|\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. .. admonition:: Theorem (:cite:`Ramare12-4`) :class: thm-tme-emt When :math:`D\ge 0` and for any integer :math:`r\ge1` and any real number :math:`\varepsilon\ge0`, we have :math:`\Bigl|\sum_{\substack{d\le D,\\(d,r)=1}}\mu(d)/d^{1+\varepsilon}\Bigr|\le 1+\varepsilon`. Concerning upper bounds that tend to :math:`0`, here is a first estimate. .. admonition:: Theorem (:cite:`ElMarraki96`) :class: thm-tme-emt When :math:`D\ge33` we have :math:`|m(D)|\le 0.2185/\log D`. When :math:`D > 1` we have :math:`|m(D)|\le 726/(\log D)^2`. The second bound above was improved: .. admonition:: Theorem (:cite:`Bordelles15`) :class: thm-tme-emt When :math:`D > 1` we have :math:`|m(D)|\le 546/(\log D)^2`. :cite:`Ramare12-2` proves several bounds of the shape :math:`m(D)\ll 1/\log D`. Those results were improved using the tools of :cite:`Balazard12`, which provide us with a better manner to convert bounds on :math:`M(D)` into bounds for :math:`m(D)`. Here is one result obtained. .. admonition:: Theorem (:cite:`Ramare12-5`) :class: thm-tme-emt When :math:`D\ge 463\,421` we have :math:`|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 :cite:`Ramare12-3` and :cite:`Ramare12-5`, the problem of adding coprimality conditions is further addressed. Here is one of the results obtained. .. admonition:: Theorem (2015) :class: thm-tme-emt When :math:`1\le q < D` we have :math:`\Bigl|\sum_{\substack{d\le D,\\ (d,q)=1}}\mu(d)/d\Bigr|\le \frac{q}{\varphi(q)}0.78/\log(D/q)`. When :math:`D/q\ge 24233`, we can replace 0.78 by 17/125. The estimates above were improved in 2025. .. admonition:: Theorem (:cite:`Camargo25`) :class: thm-tme-emt When :math:`1\le q < D` we have :math:`\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 :math:`\check{m}(D)=\sum\limits_{d\le D}\mu(d)\log(D/d)/d` ----------------------------------------------------------------------- The initial investigations on this function go back to :cite:`vonSterneck02`. .. admonition:: Theorem (:cite:`Ramare12-5`) :class: thm-tme-emt When :math:`3846 \le D` we have :math:`|\check{m}(D)-1|\le 0.00257/\log D`. When :math:`D > 1`, we have :math:`|\check{m}(D)-1|\le 0.213/\log D`. This implies in particular that .. admonition:: Theorem (2015) :class: thm-tme-emt When :math:`222 \le D` we have :math:`|\check{m}(D)-1|\le 1/1250`. When :math:`D > 1`, the optimal bound 1 holds. These bounds are a consequence of the identity: :math:`\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 :math:`D\ge1`, we have .. raw:: html
$$ \displaystyle 0\le \sum_{\substack{d\le D,\\ (d,q)=1}}\mu(d)\log(D/d)/d\le 1.00303 q/\varphi(q). $$
4. Miscellanae -------------- Here is an elegant wide ranging estimate, taken from Claim 3.1 of the referenced paper. .. admonition:: Theorem (:cite:`Trevino15`) :class: thm-tme-emt For real :math:`D\ge1` we have :math:`|\sum\limits_{d>D}\mu(d)/d^2|\le 1/D`. Taking the limit for :math:`D \rightarrow K+1` in the inequality above and after replacing :math:`K+1` by :math:`D`, we get :math:`|\sum\limits_{d\geq D}\mu(d)/d^2|\le 1/D`, which improves upon the trivial upper bound :math:`1/(D-1)`. 5. The Moebius function and arithmetic progressions --------------------------------------------------- The results in this section are scarce. .. admonition:: Theorem (:cite:`Bordelles15`) :class: thm-tme-emt Let :math:`\chi` be a non-principal Dirichlet character modulo :math:`q\ge37` and let :math:`k\ge1` be some integer. Then :math:`\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)}.`