EXPLICIT ESTIMATES FOR SUMMATORY

FUNCTIONS LINKED TO THE MÖBIUS μ-FUNCTION

H. Cohen, F. Dress and M. El Marraki

Abstract. Let M(x) be the summatory function of the Möbius function and R(x) be the remainder term for the number of squarefree integers up to x. In this paper, we prove the explicit bounds |M(x)| < x/4345 for x ≥ 2160535 and |R(x)| ≤ 0.02767\sqrt{x} for x ≥ 438653. These bounds are considerably better than preceding bounds of the same type and can be used to improve Schoenfeld type estimates.

1. Introduction

The Möbius function μ is defined by

μ(n) = \{ \begin{matrix}1, & if n = 1, \\ (−1)^{r}, & if n = p_{1}p_{2}...p_{r} (where the p_{i} are distinct primes), \\ 0, & otherwise. \end{matrix}

We also define the summatory functions

M(x) = ∑_{n≤x} μ(n),

and

Q(x) = ∑_{n=1}^{x} |μ(n)|

i.e. Q(x) is the number of squarefree integers up to x.

Finally, we denote by R the remainder term R(x) = Q(x) − \frac{6}{π^{2}} x.

[Möbius 1832] introduced the μ-function to obtain inversion formulas for arithmetic functions (in modern terms, μ is the arithmetic convolution inverse of the constant function 1).

The aim of this paper is to improve substantially on the following explicit upper bounds which were the best to date:

|R(x)| ≤ 0.1333\sqrt{x} for x ≥ 1 664,

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 11N37, 11Y35, 11Y70.

([Cohen and Dress 1988]) and

|M(x)| ≤ \frac{x}{2 360} for x ≥ 617 973

([Dress and El Marraki 1993]). This latter upper bound was itself an improvement on the result |M(x)| ≤ \frac{x}{1 036} for x ≥ 120 727 due to ([Pereira 1989]).

The results are given simultaneously, since they are obtained using convolution formulas where each improvement of an effective estimate of one of the functions gives an improvement for the other one.

It is reasonable to ask what is the point of obtaining bounds of the form |M(x)| ≤ ϵx for “small” ϵ since one knows by the Prime Number Theorem that M(x) = o(x). The answer to this question is in the practical use of the results.

For example, such bounds are used as the starting point for convolutions which give bounds of the form C(α)\frac{x}{(log x)^{α}}, and the size of the constant C(α) is strongly dependent on the size of the constant ϵ. (Note that, contrary to the case of the remainder term for the ψ or θ functions, there does not exist effective upper bounds for |M(x)| better than bounds of the form \frac{x}{(log x)^{α}}.) The first bounds of this form have been given by [Schoenfeld 1960]. Using the inequality |M(x)| < x/4345 given in the present paper, [ElMarraki 1995] has given several bounds of this form, improving by a factor of 15 the corresponding bounds given in [Schoenfeld 1960]. For example, he shows that

|M(x)| < \frac{0.002969x}{(log x)^{1/2}} for x ≥ 142194 .

However, notwithstanding these improvements, the bound |M(x)| < x/4345 is still the best available up to x = 2 · 10^{72}, and so bounds of the form |M(x)| ≤ ϵx are interesting in themselves.

Note also that bounds of the form |M(x)| ≤ ϵx are also essential in the study of the discrepancy of the Farey sequence.

To bound |R(x)|, we use the method introduced by [Cohen et Dress 1988].

To bound |M(x)| we use a method initiated by [von Sterneck 1898], which itself was inspired by Tchebychev’s method for ψ(x) [Tchebychev 1854]. The method that we use here contains improvements introduced by [Mac Leod 1969] and [Costa Pereira 1989] for constructing a function which in some sense is close to the constant function 1 (see Section 5 below), and a method introduced in [Dress 1977] for fine estimates of the complementary part of the bound.

2. Estimates of Q(x) and R(x): The basic formula

Let n be a positive integer which we shall choose later. For j = 1, 2, ..., n, we set X_{j} := \sqrt{\frac{x}{j}}. We have:

Q(x) = ∑_{a^{2}≤x} μ(a)[\frac{x}{a^{2}} ] = ∑_{a≤X_{n}} μ(a)[\frac{x}{a^{2}} ] + ∑_{X_{n}<x≤X_{1}} μ(a)[\frac{x}{a^{2}} ].

SUMMATORY FUNCTIONS OF THE MÖBIUS μ FUNCTION 3

In the second sum, we group terms giving the same value of [\frac{x}{a^{2}} ]. Since [\frac{x}{a^{2}} ] = j if and only if [X_{j+1}] + 1 ≤ a ≤ [X_{j}], we get:

\begin{aligned}Q(x) & = ∑_{a≤X_{n}} μ(a)[\frac{x}{a^{2}} ] + ∑_{j=1}^{n−1} ( ∑_{a=[X_{j+1}]+1}^{[X_{j}]} μ(a))j \\ & = ∑_{a≤X_{n}} μ(a)[\frac{x}{a^{2}} ] + ∑_{j=1}^{n−1} j(M(X_{j}) − M(X_{j+1})), \end{aligned}

hence, finally:

(2.1)  Q(x) = ∑_{a≤X_{n}} μ(a)[\frac{x}{a^{2}} ] + ∑_{j=1}^{n−1} M(X_{j}) − (n − 1)M(X_{n}).

This formula can be used in several ways, with quite different optimal values of n:

– numerical computation of Q(x); if we can store in the computer’s main memory all the values M(y) up to \sqrt{x}, we must the minimize the number of terms in formula (2.1), which gives an n of the order of x^{1/3}, and the running time is also O(x^{1/3});

– obtaining upper bounds for |R(x + y) − R(x)|; depending on the size of y with respect to x, the optimal value of n will either be x^{1/3} or \frac{x}{y}; this is how the following bounds were obtained:

(2.2)  |R(x + y) − R(x)| < 0.7343 \frac{y}{x^{1/3}} + 1.4327 x^{1/3}

et

(2.3)  |R(x + y) − R(x)| < 1.6749 \sqrt{y} + 0.6212 \frac{x}{y},

valid for all x and y ≥ 1, given in [Cohen and Dress 1988]. Contrary to the other bounds, the bounds for |R(x + y) − R(x)| depend very little on the quality of the effective bounds for |M(x)| and cannot be substantially improved.

– bounds for |R(x)|, using a bound for |M(x)| of the form ϵ(x)x, where ϵ is either a constant or a slowly varying function which tends to 0 when x → ∞. We then choose n close to \frac{1}{4π\sqrt{3ϵ(x)}}#\frac{0.046}{\sqrt{ϵ(x)}}, and we then get a bound close to 1.7\sqrt{ϵ(x)}\sqrt{x}. It is this type of

bounds that we study in detail below (under the Riemann hypothesis, we would obtain in this way a bound of the form O(x^{1/4} log x)).

To obtain bounds for |R(x)| in terms of \sqrt{x}, we transform the first term in the fundamental formula (2.1):

∑_{a≤X_{n}} μ(a)[\frac{x}{a^{2}} ] = x ∑_{a≤X_{n}} \frac{μ(a)}{a^{2}} − \frac{1}{2}M(X_{n}) − ∑_{a≤X_{n}} μ(a)(\{\frac{x}{a^{2}} − \frac{1}{2}\}),

4 H. COHEN, F. DRESS AND M. EL MARRAKI

and we also have

\begin{aligned}∑_{a≤X_{n}} \frac{μ(a)}{a^{2}} & = \frac{6}{π^{2}} − ∑_{a>X_{n}} \frac{μ(a)}{a^{2}} = \frac{6}{π^{2}} − ∫_{X_{n}}^{∞} \frac{dM(t)}{t^{2}} \\ & = \frac{6}{π^{2}} + \frac{M(X_{n})}{X_{n}^{2}} − 2 ∫_{X_{n}}^{∞} \frac{dM(t)}{t^{3}}. \end{aligned}

Grouping all the terms, we obtain

\begin{aligned}R(x) & = Q(x) − \frac{6}{π^{2}} x \\ & = −2x ∫_{X_{n}}^{∞} \frac{dM(t)}{t^{3}} + ∑_{j=1}^{n−1} M(X_{j}) + \frac{1}{2}M(X_{n}) − ∑_{a≤X_{n}} μ(a)(\{\frac{x}{a^{2}} \} − \frac{1}{2}). \end{aligned}

We will use this formula in the form |R(x)| ≤ T_{1} + T_{2} + T_{3}, where

T_{1} := 2x ∫_{X_{n}}^{∞} \frac{|M(t)|}{t^{3}} dt,

T_{2} := ∑_{j=1}^{n−1} |M(X_{j})| + \frac{1}{2}|M(X_{n})|,

T_{3} := ∑_{a≤X_{n}} |μ(a)(\{\frac{x}{a^{2}} \} − \frac{1}{2})|.

Initially, we can apply these formulas by using the bounds |M(x)| ≤ \frac{x}{2 360} for x ≥ 617 973 (see introduction).

As mentioned in the introduction, there will be an interaction between all the bounds. We will first give the basic lemma, not with the bound \frac{x}{2 360} but with the parametric bounds |M(x)| < ϵ x for x ≥ x_{1} = x_{1}(ϵ) and |R(x)| < b\sqrt{x} for x ≥ x_{2} = x_{2}(b).

Lemma 1. For X_{n} = \sqrt{\frac{x}{n}} ≥ max(x_{1}, x_{2}), we have T_{1} ≤ 2ϵ \sqrt{n} \sqrt{x} T_{2} ≤ 2ϵ \sqrt{n} \sqrt{x} − 1.46 ϵ \sqrt{x}

T_{3} ≤ \sqrt{(\frac{6\sqrt{x}}{π^{2}\sqrt{n}} + b\frac{x^{1/4}}{n^{1/4}} )(\frac{\sqrt{x}}{18\sqrt{n − 0.02}} + \frac{(12x)^{1/3}}{6} − \frac{2}{3}(n − \frac{1}{2}))}

The proofs of the first two bounds are immediate:

T_{1} = 2x ∫_{X_{n}}^{∞} \frac{|M(t)|}{t^{3}}dt ≤ 2ϵx ∫_{X_{n}}^{∞} \frac{dt}{t^{2}} = 2ϵ \frac{x}{X_{n}} ,

and

\begin{aligned}T_{2} = ∑_{j=1}^{n−1} |M(X_{j})| + \frac{1}{2} |M(X_{n})| & ≤ ϵ(X_{1} + X_{2} + ... + X_{n−1} + \frac{1}{2}X_{n}) \\ & = ϵ\sqrt{x}(1 + \frac{1}{\sqrt{2}} + ... + \frac{1}{\sqrt{n − 1}} + \frac{1}{2\sqrt{n}}) \end{aligned}

and the result follows from the elementary inequality (1 + \frac{1}{\sqrt{2}} + ... + \frac{1}{\sqrt{n−1}} + \frac{1}{2\sqrt{n}}) < 2\sqrt{n} − 1.46.

The proof of the third bound is much more tricky. We first use Schwartz’s inequality:

T_{3} ≤ \sqrt{∑_{a≤X_{n}} μ^{2}(a)} \sqrt{∑_{a≤X_{n}} |μ(a)|(\{\frac{x}{a^{2}} \} − \frac{1}{2})^{2}}.

Let h(a) be the characteristic function of integers which are not divisible by 4 or by 9. We have |μ(a)| ≤ h(a), so if we set:

S(z) := ∑_{a≤z} h(a)(\{\frac{x}{a^{2}} \} − \frac{1}{2})^{2},

we have

T_{3} ≤ \sqrt{Q(X_{n})} \sqrt{S(X_{n})}.

We trivially have

Q(X_{n}) ≤ \frac{6 \sqrt{x}}{π^{2}\sqrt{n}} + b \frac{x^{1/4}}{n^{1/4}} .

To bound S(X_{n}), we set S_{k} := ∑_{X_{k+1}+1≤a≤X_{k}} h(a)(\{\frac{x}{a^{2}} \})^{2} and H(z) := ∑_{a≤z} h(z) = [z] − [\frac{z}{4}] − [\frac{z}{9}] + [\frac{z}{36}]. Thus, we have S(X_{n}) < S_{n} + S_{n+1} + ... + S_{N} + \frac{1}{4}H(X_{N}).

We first note that H(z) = \frac{2}{3}z + ρ(z) with |ρ(z)| ≤ \frac{4}{3}.

An easy but detailed and delicate study of a few functions gives the following inequality:

\begin{aligned}S_{k} = ((\frac{16k}{9} + \frac{4}{3} + \frac{1}{6k} )\sqrt{k} & − (\frac{16k}{9} + \frac{4}{9} + \frac{1}{6(k + 1)})\sqrt{k + 1})\sqrt{x} \\ & + \frac{1}{4}(ρ(\sqrt{\frac{x}{k}}) − ρ(\sqrt{\frac{x}{k + 1}})) + r_{k} with |r_{k}| ≤ \frac{2}{3}. \end{aligned}

For k ≥ 4, the main term can be bounded by

\frac{1}{18}(\frac{1}{\sqrt{k − 0.02}} − \frac{1}{\sqrt{k + 0.98}})\sqrt{x}.

Finally, we obtain

S(X_{n}) ≤ \frac{\sqrt{x}}{18\sqrt{n − 0.02}} + \frac{1}{9}(\sqrt{\frac{x}{N}} + 6N) − \frac{2}{3}(n − \frac{1}{2}).

The best bound is obtained by choosing N = [(\frac{x}{144})^{1/3}], which gives

S(X_{n}) ≤ \frac{\sqrt{x}}{18\sqrt{n − 0.02}} + \frac{(12 x)^{1/3}}{6} − \frac{2}{3}(n − \frac{1}{2})

as claimed.

Note that the bounds for Q(X_{n}) and S(X_{n}) are valid for n ≥ 4 and \sqrt{\frac{x}{n}} ≥ x_{2}.

3. Estimates for small x

The bound for T_{3} given in the above lemma is useful only when x is rather large. Hence it is important to have estimates for “small” values of x.

Proposition 2. We have

|R(x)| ≤ 0.02\sqrt{x} for x ∈ [2 050 244 , 1.6 10^{15}].

We first show that the bounds (2.2) and (2.3) for |R(x + y) − R(x)| imply that, if we know that |R(x)| ≤ 0.001\sqrt{x} and |R(1.0001 x)| ≤ 0.001\sqrt{1.0001 x} then |R(z)| ≤ 0.02\sqrt{z} for all z in the interval [x, 1.0001 x]. We first use (2.2) and (2.3) together to give a single increasing bound f_{x}(y) for |R(x + y) − R(x)|, where

\begin{matrix}f_{x}(y) = 0.7343 \frac{y}{x^{1/3}} + 1.4327 x^{1/3} if y < 1.3007 x^{2/3} \\ f_{x}(y) = 1.6749 \sqrt{y} + 0.6212 \frac{x}{y} if y ≥ 1.3007 x^{2/3}. \end{matrix}

We have f_{x}(0.00005 y) = 0.36715 10^{−4} x^{2/3} +1.4327 x^{1/3} if x < (26014)^{3} = 1.76044 10^{13}, and f_{x}(0.00005 y) = 1.18433 10^{−2}\sqrt{x} + 12414 if x ≥ 1.76044 10^{13}. Hence our claim is valid as soon as f_{x}(0.00005 y) ≤ 0.019 \sqrt{x}, which is true when x ≥ 5.92 10^{11}.

The explicit computations of the necessary values of R(x) are done using formula (2.1) with x = [\frac{x^{1/3}}{2} ].

Thus we have proved that the bound of Proposition 2 is valid on the interval

[5.92 10^{11} , 1.6 10^{15}].

It is a simple (but long) matter to check explicitly that it is also valid on the interval [2 050 244, 5.92 10^{11}], thus proving the proposition.

4. Estimates for Q(x) and R(x): Numerical results Theorem 3. We have

|R(x)| ≤ 0.036438 \sqrt{x} for x ≥ 82 005

Form 82 005 to 2 050 244, we check directly. From 2 050 244 to 1.6 10^{15}, the result follows from the proposition above. Above 1.6 10^{15} we use the lemma of Section 2 with ϵ = \frac{1}{2 360}, x_{1} = 617 973, b = 0.1333, x_{2} = 1 664. One easily checks that the optimal value of n is n = 109. Thus, for x ≥ 1.6 10^{15} we obtain

\begin{matrix}Q(X_{n}) ≤ \sqrt{\frac{x}{109}}(\frac{6}{π^{2}} + b\frac{109^{1/4}}{(1.6 10^{15})^{1/4}} ) \\ S(X_{n}) ≤ \sqrt{\frac{x}{109}}(\frac{\sqrt{109}}{18\sqrt{107.98}} + \frac{12^{1/3} 109^{1/2}}{6(1.6 10^{15})^{1/6}} ) \end{matrix}

hence

T_{3} ≤ \sqrt{Q(X_{n})S(X_{n})} = 0.019361 \sqrt{x}.

Since

T_{1} + T_{2} ≤ 0.847458 10^{−3} (2\sqrt{109} − 0.73)\sqrt{x} = 0.017077 \sqrt{x},

Theorem 3 follows.

5. Estimates for M(x): description of the method

Let E be the function E(x) = ∑_{n≤x} μ(n)[\frac{x}{n}], which is constant and equal to 1, and which trivially satisfies the condition

∑_{n≤x} μ(n)(1 − E(\frac{x}{n})) = M(x) − ∑_{n≤x} μ(n).

We consider an approximation F (x) = ∑_{i=1}^{m} c_{i} [\frac{x}{a_{i}} ] of this function. We have

∑_{n≤x} μ(n)(1 − F (\frac{x}{n})) = M(x) − ∑_{a_{i}≤x} c_{i} .

If ∑_{i=1}^{m} \frac{c_{i}}{a_{i}} = 0 and if F is a “good” approximation of E, we can thus obtain good bounds for |M(x)|. We refer to [El Marraki 1991] for a detailed description of this method.

Here, we use the same method with a more efficient function F and better bounds for R(x) (which are used in the final estimates).

We use the following function

F (x) = ∑_{i=1}^{m} c_{i} [\frac{x}{a_{i}} ] + c_{m+1}[\frac{x}{a_{m+1}} ] + c_{m+2}[\frac{x}{a_{m+2}} ]

with the following four conditions:

(i) F (x) = 1 for 1 ≤ x ≤ k_{0},

(ii) the function G(x) := |1 − F (x)| is close to 1 for k_{0} ≤ x ≤ 2 k_{0},
(iii) I(F) := \frac{6}{π^{2}} ∫ \frac{|1−F (u)|}{u^{2}}du is “small”,

(iv) the function F is periodic, i.e. ∑_{i=1}^{m+2} \frac{c_{i}}{a_{i}} = 0 (this last condition is absolutely necessary).

We build the function F in the following way. For condition (i), we choose an integer k_{0}, we set m_{1} = Q(k_{0} − 1), and we define the first m_{1} terms of F by setting (a_{i}) to be the increasing sequence of squarefree integers less than or equal to k_{0} and c_{i} = μ(a_{i}). We then add corrective terms of the form c_{i}[\frac{x}{a_{i}} ] so that condition (ii) is satisfied. We then compute the integral I(f), and we add other terms so that condition (iii) is satisfied. Finally, we add two terms with not necessary integral a_{i} so that the periodicity condition (iv) is satisfied.

We have written a program which, given k_{0}, automatically computes a number of possible functions, which only differ by the number of corrective terms. Heuristic reasons and experimental observation show that efficient values of k_{0} are obtained when both M(k_{0}) and m(k_{0}) = ∑_{n≤k_{0}} \frac{μ(n)}{n} are small. The upper bound |M(x)| ≤ \frac{x}{2 360} (see Introduction) was obtained using a function where k_{0} = 26 442, for which M(k_{0}) = −1 and m(k_{0}) = 1.59 10^{−5}.

Here we will use a function where k_{0} = 100 882, for which M(k_{0}) = 1 and m(k_{0}) = 4.34 10^{−7}. This function F, given by our program, is built with m_{1} = 61 334 and m = 63 951 as follows:

F (x) = ∑_{i=1}^{61 334} c_{i}[\frac{x}{a_{i}} ] + ∑_{i=61 335}^{63 951} c_{i}[\frac{x}{a_{i}} ] + [\frac{x}{1 436 009.0}] − [\frac{x}{47 666 734 237 381.39}]

We obtain an upper bound max(G) of G by splitting G = |1 − F | into an affine part (which vanishes) and a periodic part:

\begin{aligned}|1 − F (x)| & = |1 + \frac{1}{2} ∑_{i=1}^{m+2} c_{i} − x ∑_{i=1}^{m+2} \frac{c_{i}}{a_{i}} + ∑_{i=1}^{m+2} c_{i}(\{\frac{x}{a_{i}} \} − \frac{1}{2})| \\ & ≤ |1 + \frac{1}{2} ∑_{i=1}^{m+2} c_{i}| + 0 + | ∑_{i=1}^{m_{1}} c_{i}(\{\frac{x}{a_{i}} \} − \frac{1}{2})| + | ∑_{i=m_{1}+1}^{m+2} c_{i}(\{\frac{x}{a_{i}} \} − \frac{1}{2})|. \end{aligned}

The term |1 + \frac{1}{2} ∑_{i=1}^{m+2} c_{i}| is equal to 2.

We bound the term | ∑_{i=1}^{m_{1}} c_{i}(\{\frac{x}{a_{i}} \} − \frac{1}{2})| by using a technique due to Costa Pereira and explained in detail in Lemma 5.1 of [El Marraki 1991]. The result is that this term is bounded by 14 891. Finally, the term | ∑_{i=m_{1}+1}^{m+2} c_{i}(\{\frac{x}{a_{i}} \} − \frac{1}{2})| can be trivially bounded by \frac{1}{2} ∑_{i=m_{1}+1}^{m+2} |c_{i}|, which gives 7634.5.

The properties of the function F can be summarized in the following table:

\begin{matrix}k_{0} & m_{1} & m & max(G) \\ 100 882 & 61 334 & 63 951 & 22 527.5 \end{matrix}

6. Estimates for M(x): Numerical results To bound |M(x)| we use the same method as in [Dress et El Marraki 1993]. It can be

summarized in the following lemma:

Lemma 4. If x ≥ τ and τ ≥ max(a_{i}, (N + 1)x(b)), we have

|M(x)| ≤ (\frac{6}{π^{2}} u + \frac{b v}{\sqrt{τ}} + \frac{w}{τ} )x,

where, if we denote by N the upper bound for direct computations,

\begin{aligned}u(N) & = ∑_{n=1}^{N} \frac{1}{n}(G(n) − G(n − 1)) − \frac{1}{N + 1} G(N), \\ v(N) & = ∑_{n=1}^{N} \frac{1}{\sqrt{n}}|G(n) − G(n − 1)| − \frac{1}{\sqrt{N + 1}} G(N) \end{aligned}

u = u(N) + \frac{1}{N + 1} max(G), v = v(N) + \frac{1}{\sqrt{N + 1}} max(G), w = | ∑_{i=1}^{m+2} c_{i}|.

Recall also the following two results.

(6.1)  |M(x)| ≤ 0.570591 \sqrt{x} for 33 ≤ x ≤ 10^{12}

[Dress 1993]

(6.2)  |M(x)| ≤ \frac{x}{4 257} for 10^{12} ≤ x ≤ 2.590 10^{15}

[Dress and El Marraki 1993].

Theorem 5. We have

|M(x)| ≤ \frac{x}{4 257} for x ≥ 2 159 561

To prove this, we use the function F given above (for which w = 8), the value b = 0.036438 given in Theorem 3, and τ = 2.590 10^{15}. We split the interval [2 159 561 ; +∞[ in four consecutive sub-intervals as follows.

(i) For x ∈ [2 159 561 ; 5 900 070]: we make a direct check on the computer. (ii) For x ∈ [5 900 070 ; 10^{12}]: the bound (6.1) implies |M(x)| ≤ \frac{x}{4257}. (iii) For x ∈ [10^{12} ; 2.590 10^{15}]: this is (6.2). (iv) Finally, for x ∈ [2.590 10^{15}, +∞]: the bound follows from the use of the function

F (x) as explained in Lemma 4.

The first table below gives the basic results for using Lemma 4 up to N = 1.62 10^{9}. The total running time on a Sparcstation 10 was approximately 6 months. We decided to stop there since we would only very slightly improve the final result by going any further.

\begin{matrix}\frac{N}{10^{6}} & u(N) & v(N) \\ 60 & 0.00033225264406 & 6993.26247 \\ 120 & 0.00033278045122 & 9798.92803 \\ 180 & 0.00033297524343 & 11924.63733 \\ 240 & 0.00033307188379 & 13701.60869 \\ 300 & 0.00033313444950 & 15258.91664 \\ 360 & 0.00033317548479 & 16659.86951 \\ 420 & 0.00033320547881 & 17943.01439 \\ 480 & 0.00033322821023 & 19132.56364 \\ 540 & 0.00033324602713 & 20245.63763 \\ 600 & 0.00033326069524 & 21294.91286 \\ 660 & 0.00033327290186 & 22290.10075 \\ 720 & 0.00033328301510 & 23238.79402 \\ 780 & 0.00033329184416 & 24146.22679 \\ 840 & 0.00033329909164 & 25017.15026 \\ 900 & 0.00033330533451 & 25855.13303 \\ 960 & 0.00033325085140 & 26664.09029 \\ 1020 & 0.00033332552935 & 27446.77662 \\ 1080 & 0.00033332997683 & 28205.29763 \\ 1140 & 0.00033333387044 & 28941.39330 \\ 1200 & 0.00033333739947 & 29656.91071 \\ 1260 & 0.00033334068787 & 30353.50586 \\ 1320 & 0.00033334359056 & 30560.02903 \\ 1380 & 0.00033334623636 & 31222.56618 \\ 1440 & 0.00033334870142 & 31869.60823 \\ 1500 & 0.00033335094814 & 32502.25952 \\ 1560 & 0.00033335300301 & 33121.36884 \\ 1620 & 0.00033335488681 & 33727.85805 \end{matrix}

The second table gives the results obtained after applying Lemma 4, with b = 0.036438 and τ = 2.590 10^{15}. The term w(=8), which only changes the 10-th significant figure, has been completely neglected. The bound is given through the value u = u(N) + \frac{1}{N+1} max(G), v = v(N) + \frac{1}{\sqrt{N+1}} max(G), M = \frac{6}{π^{2}} u + \frac{bv}{\sqrt{τ}} and \frac{1}{M}.

\begin{matrix}\frac{N}{10^{6}} & \frac{6}{π^{2}} u & \frac{bv}{\sqrt{τ}} & M = \frac{6}{π^{2}} u + \frac{bv}{\sqrt{τ}} & \frac{1}{M} \\ 60 & 0.00043023667963 & 0.00000500916169 & 0.00043524584133 & 2297.55 \\ 180 & 0.00027850843979 & 0.00000853907793 & 0.00028704751773 & 3483.74 \\ 300 & 0.00024817171955 & 0.00001092610491 & 0.00025909782447 & 3859.55 \\ 420 & 0.00023517196904 & 0.00001284773765 & 0.00024801970670 & 4031.94 \\ 540 & 0.00022795054659 & 0.00001449629117 & 0.00024244683777 & 4124.62 \\ 660 & 0.00022335574718 & 0.00001596003228 & 0.00023931577946 & 4178.58 \\ 780 & 0.00022017493691 & 0.00001728894260 & 0.00023746387950 & 4211.17 \\ 900 & 0.00021784209912 & 0.00001851245600 & 0.00023635455513 & 4230.93 \\ 1020 & 0.00021606416987 & 0.00001965201822 & 0.00023571618809 & 4242.39 \\ 1140 & 0.00021465591992 & 0.00002072211581 & 0.00023537803574 & 4248.48 \\ 1200 & 0.00021405740402 & 0.00002123440431 & 0.00023529180832 & 4250.04 \\ 1260 & 0.00021351594765 & 0.00002173314560 & 0.00023524909325 & 4250.81 \\ 1320 & 0.00021302366185 & 0.00002188100289 & 0.00023490466474 & 4257.05 \\ 1380 & 0.00021257418079 & 0.00002235536058 & 0.00023492954137 & 4256.60 \end{matrix}

Note that I(F) := \frac{6}{π^{2}} ∫ \frac{|1−F (u)|}{u^{2}}du#\frac{1}{4 930}.

It is trivial to control the accuracy of the computations: we deal with sums ∑_{n} \frac{1}{n}(G(n)− G(n − 1)) and ∑_{n} \frac{1}{\sqrt{n}} |G(n) − G(n − 1)|, where G(n) − G(n − 1) is exact since it is integral. If we sum up to N = 1.62 10^{9} using double precision floating point real numbers in C, which have a relative accuracy of at least 10^{−15}, the accuracy of the final result is at least 10^{−6}, which is more than enough.

This finishes the proof of Theorem 5.

7. Final Numerical Estimates

Theoreme 3 bis. We have

|R(x)| ≤ 0.02767 \sqrt{x} for x ≥ 438 653

We follow again the proof of Theorem 3, but now with ϵ = \frac{1}{4 257}, x(ϵ) = 2 159 561, b = 0.036438, x_{1}(b) = 82 005. The optimal value of n for minimizing the sum T_{1} + T_{2} + T_{3} is now n = 197. Thus, we have for x ≥ 1.6 10^{15}:

\begin{matrix}Q(X_{n}) ≤ \sqrt{\frac{x}{197}}(\frac{6}{π^{2}} + b\frac{197^{1/4}}{(1.6 10^{15})^{1/4}} ) \\ S(X_{n}) ≤ \sqrt{\frac{x}{197}}(\frac{\sqrt{197}}{18\sqrt{196.98}} + \frac{12^{1/3} 197^{1/2}}{6(1.6 10^{15})^{1/6}} ) \end{matrix}

hence

T_{3} ≤ \sqrt{Q(X_{n})S(X_{n})} = 0.01479 \sqrt{x}.

Since

T_{1} + T_{2} ≤ 0.472478 10^{−3} (2\sqrt{197} − 0.73)\sqrt{x} = 0.012845 \sqrt{x},

we obtain the theorem.

Theorem 5 bis. We have

|M(x)| ≤ \frac{x}{4 345} for x ≥ 2 160 535

We follow the proof of Theorem 5. The following table gives the results of applying Lemma 4 with b = 0.02767 and τ = 2.401 10^{15}. As in the preceding table, the bound is given through the value u = u(N) + \frac{1}{N+1} max(G), v = v(N) + \frac{1}{\sqrt{N+1}} max(G), M = \frac{6}{π^{2}} u + \frac{bv}{\sqrt{τ}} and \frac{1}{M}.

\begin{matrix}\frac{N}{10^{6}} & \frac{6}{π^{2}} u & \frac{bv}{\sqrt{τ}} & M = \frac{6}{π^{2}} u + \frac{bv}{\sqrt{τ}} & \frac{1}{M} \\ 60 & 0.00043023667963 & 0.00000395069480 & 0.00043418737443 & 2303.15 \\ 180 & 0.00027850843979 & 0.00000673471787 & 0.00028524315766 & 3505.78 \\ 300 & 0.00024817171955 & 0.00000861735126 & 0.00025678907082 & 3894.25 \\ 420 & 0.00023517196904 & 0.00001013293110 & 0.00024530490015 & 4076.56 \\ 540 & 0.00022795054659 & 0.00001143313505 & 0.00023938368164 & 4177.39 \\ 660 & 0.00022335574718 & 0.00001258757859 & 0.00023594332577 & 4238.31 \\ 780 & 0.00022017493691 & 0.00001363568192 & 0.00023381061883 & 4276.97 \\ 900 & 0.00021784209912 & 0.00001460065937 & 0.00023244275849 & 4302.13 \\ 1020 & 0.00021606416987 & 0.00001549942503 & 0.00023156359490 & 4318.47 \\ 1140 & 0.00021465591992 & 0.00001634340437 & 0.00023099932430 & 4329.02 \\ 1260 & 0.00021351594765 & 0.00001714079730 & 0.00023065674496 & 4335.45 \\ 1380 & 0.00021257418079 & 0.00001763153440 & 0.00023020571519 & 4343.94 \\ 1440 & 0.00021216218064 & 0.00001799690788 & 0.00023015908852 & 4344.82 \\ 1500 & 0.00021178312766 & 0.00001835415541 & 0.00023013728308 & 4345.23 \\ 1560 & 0.00021143322102 & 0.00001870375628 & 0.00023013697731 & 4345.24 \\ 1620 & 0.00021110922195 & 0.00001904623100 & 0.00023015545295 & 4344.89 \end{matrix}

This proves Theorem 5 bis.

It is not really useful to iterate once more the process, since the improvements would be marginal (around 0.9% for |R(x)| and 0.07% for |M(x)|).

References

[Cohen et Dress 1988] H. Cohen et F. Dress, Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteur carré, Publications mathématiques d’Orsay : Colloque de théorie analytique des nombres, Marseille, 1985 (1988) p 73–76.

[Pereira 1989] N. Costa Pereira, Elementary estimates for the Chebyshev function Ψ(x) and the Möbius function M(x), Acta Arithmetica, LII (1989), p. 307–337.

[Dress 1977] F. Dress, Majorations de la fonction sommatoire de la fonction de Mobius, Bull. Soc. Math. Fr., mémoire n^{o} 49–50 (1977), p. 47–52.

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[El Marraki 1995] M. El Marraki, Fonction sommatoire de la fonction de Möbius,

3. Majorations asymptotiques effectives fortes, Journal de Th. des Nombres de Bordeaux, 7 (1995), p. 407–433.

[Mac Leod 1969] R. A. Mac Leod, A new estimate for the sum M(x) = ∑_{n≤x} μ(n), Acta Arithmetica 13 (1967), p. 49–59 (Erratum in Acta Arithmetica 16 (1969), p. 99–100).

[Möbius 1832] A. F. Möbius, Uber eine besondere Art von Untersuchrung des Reihen, J. reine Angew. Math. 9 (1832), p. 105–123.

[Schoenfeld 1960] L. Schoenfeld, An improved estimate for the summatory function of the Möbius function, Acta Arithmetica 15 (1960), p. 221–233.

[von Sterneck 1898] D. von Sterneck, Bemerkung über die Summierung einiger zahlentheoretischen Funktionen, Monatsh. Math. Phys. 9 (1898), p. 43–45.

[Tchebychev 1854] P. L. Tchebychev, Mémoire sur les nombres premiers, J. de Math. 17 (1852), p. 366–390, et Mémoire des savants étrangers, Saint-Pétersbourg (1854).

Laboratoire d’Arithmétique Algorithmique eXpérimentale (A2X), U.M.R. 9936 du C.N.R.S., Université Bordeaux I, 351 Cours de la Libération, F-33505 TALENCE Cedex, FRANCE

E-mail address: cohen@math.u-bordeaux.fr, dress@math.u-bordeaux.fr, marraki@math.u-bordeaux.fr