Averages of non-negative multiplicative functions

Collecting references:
1. General tools
When looking for averages of functions that look like 1 or like the divisor function, Lemma 3.2 of [Ramaré, 1995 †Ramaré, O. 1995
On Snirel'man's constant
Ann. Scu. Norm. Pisa, 21, 645--706. http://math.univ-lille1.fr/ ramare/Maths/Article.pdf.
] offers an efficient easy path. The technique of comparison of two arithmetical function via their Dirichlet series is known as the Convolution method and is for instance decribed at length in [Berment & Ramaré, 2012 †Berment, P., & Ramaré, O. 2012
Ordre moyen d'une fonction arithmétique par la méthode de convolution
Revue de Mathématique Spéciale, 212(1), 1--15.
], and in the course that can be found here.
Theorem (1995)
Let (gn)n1$(g_n)_{n\ge1}$, (hn)n1$(h_n)_{n\ge1}$ and (kn)n1$(k_n)_{n\ge1}$ be three complex sequences. Let H(s)=nhnns$H(s)=\sum_nh_nn^{-s}$, and H¯¯¯(s)=n|hn|ns$\overline{H}(s)=\sum_n|h_n|n^{-s}$. We assume that g=hk$g=h\star k$, that H¯¯¯(s)$\overline{H}(s)$ is convergent for R(s)1/3$\Re(s)\ge-1/3$ and further that there exist four constants A$A$, B$B$, C$C$ and D$D$ such that
ntkn=Alog2t+Blogt+C+O(Dt1/3) for t>0.
Then we have for all t>0$t>0$ :
ntgn=ulog2t+vlogt+w+O(Dt1/3H¯¯¯(1/3))
with u=AH(0)$u=AH(0)$, v=2AH(0)+BH(0)$v=2AH^{\prime}(0)+BH(0)$ and w=AH(0)+BH(0)+CH(0)$w=AH^{\prime\prime}(0)+BH^{\prime}(0)+CH(0)$. We have also
ntngn=Utlogt+Vt+W+O(2.5Dt2/3H¯¯¯(1/3))
with
U=W=AH(0),V=2AH(0)+2AH(0)+BH(0),A(H(0)2H(0)+2H(0))+B(H(0)H(0))+CH(0).
This Lemma says that one derives information from gn$g_n$ from informations on the model kn$k_n$. When this model is kn=1$k_n=1$, the values concerning A$A$, B$B$ and C$C$ are given by the first half of Lemma 3.3 of [Ramaré, 1995 †Ramaré, O. 1995
On Snirel'man's constant
Ann. Scu. Norm. Pisa, 21, 645--706. http://math.univ-lille1.fr/ ramare/Maths/Article.pdf.
]:
Lemma (1995)
For all t>0$t>0$, we have nt1/n=logt+γ+O(0.9105t1/3).$\sum_{n\le t}1/n=\log t+\gamma+\mathcal{O}^*(0.9105 t^{-1/3}).$
When this model is kn=τ(n)$k_n=\tau(n)$, the number of divisors of n$n$, the values concerning A$A$, B$B$ and C$C$ are given by Corollary 2.2 of [Berkane et al., 2012 †Berkane, D., Bordellès, O., & Ramaré, O. 2012
Explicit upper bounds for the remainder term in the divisor problem
Math. of Comp., 81(278), 1025--1051.
]:
Lemma (2011)
We have, for all t>0$t>0$, ntτ(n)/n=12log2t+2γlogt+γ2γ1+O(1.16/t1/3)$\sum_{n\le t}\tau(n)/n =\tfrac12\log^2t+2\gamma\log t +\gamma^2-\gamma_1+ \mathcal{O}^*(1.16/t^{1/3})$ where γ1$\gamma_1$ is the second Laurent-Stieljes constant -- for instance [Kreminski, 2003 †Kreminski, R. 2003
Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants
Math. Comp., 72(243), 1379--1397 (electronic).
] and [Coffey, 2006 †Coffey, M.W. 2006
New results on the Stieltjes constants: asymptotic and exact evaluation
J. Math. Anal. Appl., 317(2), 603--612.
]. In particular, we have γ1=0.0728158454836767248605863758749013191377+O(1040).$\gamma_1= -0.0728158454836767248605863758749013191377 + \mathcal{O}^*(10^{-40}).$
The constants H(0)$H(0)$, H(0)$H'(0)$ and H′′(0)$H''(0)$ are to be computed. In most cases, the Dirichlet series has an Euler product, in which case, (see section 3 of [Ramaré, 1995 †Ramaré, O. 1995
On Snirel'man's constant
Ann. Scu. Norm. Pisa, 21, 645--706. http://math.univ-lille1.fr/ ramare/Maths/Article.pdf.
]) we have
H(0)=p(1+mhpm),$H(0)=\prod_p(1+\sum_mh_{p^m}),$ then H(0)H(0)=pmmhpm1+mhpm(logp),$\frac{H^{\prime}(0)}{H(0)}= \sum_p \frac{\sum_mmh_{p^m}}{1+\sum_mh_{p^m}}(-\log p),$ and also
H(0)H(0)=(H(0)H(0))2+pmm2hpm1+mhpm[mmhpm1+mhpm]2log2p.
When looking for an upper bound, it is common to compare sums to an Euler product, via,
nyf(n)/npy1+1mlogy/logpf(pm)
valid when f$f$ is non-negative and multiplicative. Lemma 4 of [Daboussi & Rivat, 2001 †Daboussi, H., & Rivat, J. 2001
Explicit upper bounds for exponential sums over primes
Math. Comp., 70(233), 431--447.
] extends this. Let z$z$ be a parameter and vz(n)$v_z(n)$ be the characteristic function of those integers that have all their prime factors pz$p\le z$.
Theorem (2000)
Let z2$z\ge2$, f$f$ a multiplicative function with f0$f\ge0$ and S=pzf(p)1+f(p)logp$S=\sum_{p\le z}\frac{f(p)}{1+f(p)}\log p$. We assume that S>0$S>0$ and write K(t)=logt1(1/t)$K(t)=\log t-1-(1/t)$. For any y$y$ such that logyS$\log y\ge S$, we have
n>yvz(n)μ2(n)f(n)pz(1+f(p))exp(logylogzK(logyS))
nyvz(n)μ2(n)f(n)pz(1+f(p)){1exp(logylogzK(logyS))}
and in particular, when logy7S$\log y\ge 7S$, we have
n>yvz(n)μ2(n)f(n)pz(1+f(p))exp(logylogz)
nyvz(n)μ2(n)f(n)pz(1+f(p)){1exp(logylogz)}.
It is sometimes required to compare a function close to 1$1$ (or more generally to the divisor function τk$\tau_k$) to a function close to 1/n$1/n$ or τk(n)/n$\tau_k(n)/n$. Theorem 01 of [Hall & Tenenbaum, 1988 †Hall, R., & Tenenbaum, G. 1988
Divisors
Cambridge Tracts in Mathematics, vol. 90. Cambridge: Cambridge University Press.
] offers a fast way of doing so.
Theorem (1988)
Let f$f$ be a non-negative multiplicative function such that, for some A$A$ and B$B$,
pyf(p)logpAy(y0),pν2f(pν)pνlogpνB.
Then, for x>1$x>1$,
nxf(n)(A+B+1)xlogxnxf(n)n
Theorem 21.1 of [Ramaré, 2009 †Ramaré, O. 2009
Arithmetical aspects of the large sieve inequality
Harish-Chandra Research Institute Lecture Notes, vol. 1. New Delhi: Hindustan Book Agency. With the collaboration of D. S. Ramana.
] offers a fully explicit estimate for the average of a general non-negative multiplicative function, but it is often numerically rather poor. As the result above it relies on the technique developped by [Levin & Fainleib, 1967 †Levin, B.V., & Fainleib, A.S. 1967
Application of some integral equations to problems of number theory
Russian Math. Surveys, 22, 119--204.
].
Theorem (2009)
Let g$g$ be a non-negative multiplicative function. Let A$A$ and κ$\kappa$ be three positive real parameters such that, for any Q1$Q\ge1$, one has
p2,ν1pνQg(pν)log(pν)=κlogQ+O(L)
and p2ν,k1g(pk)g(pν)log(pν)A.$\sum_{p\ge2} \sum_{\nu,k\ge1}g\bigl(p^k\bigr)g\bigl(p^{\nu}\bigr)\log\bigl(p^{\nu}\bigr) \le A.$ Then, when Dexp(2(L+A))$D\ge\exp(2(L+A))$, we have
dDg(d)=C(logD)κ(1+O(B/logD))
where B=2(L+A)(1+2(κ+1)eκ+1)$B=2(L+A)\bigl(1+2(\kappa+1)e^{\kappa+1}\bigr)$ and
C=1Γ(κ+1)p2{(ν0g(pν))(11p)κ}.
2. Estimates of some special functions
[Cohen & Dress, 1988 †Cohen, H., & Dress, F. 1988
Estimations numériques du reste de la fonction sommatoire relative aux entiers sans facteur carré
Prépublications mathématiques d'Orsay : Colloque de théorie analytique des nombres, Marseille, 73--76.
] contains the following Theorem.
Theorem (1988)
Let R(x)=nxμ2(n)6x/π2$R(x)=\sum_{n\le x}\mu^2(n)-6x/\pi^2$. We have |R(x+y)R(x)|1.6749y+0.6212x/y$|R(x+y)-R(x)|\le 1.6749\sqrt{y}+0.6212 x/y$ and |R(x+y)R(x)|0.7343y/x1/3+1.4327x1/3$|R(x+y)-R(x)|\le 0.7343y/x^{1/3}+1.4327 x^{1/3}$ for x,y1$x,y\ge1$.
Elementary estimates for the Chebyshev function ψ(X)$\psi(X)$ and for the Möbius function M(X)$M(X)$
Acta Arith., 52, 307--337.
]. [Cohen et al., 2007 †Cohen, H., Dress, F., & El Marraki, M. 2007
Explicit estimates for summatory functions linked to the Möbius {μ$\mu$}-function
Funct. Approx. Comment. Math., 37(, part 1), 51--63.
] contains:
Theorem (2008)
We have |nxμ2(n)6x/π2|0.02767x$|\sum_{n\le x}\mu^2(n)-6x/\pi^2|\le 0.02767\sqrt{x}$ for x438653$x\ge 438653$. One can replace (0.02767,438653)$(0.02767, 438653)$ by (0.1333,1004)$(0.1333, 1004)$.

Lemma 3.4 of [Ramaré, 2014 †Ramaré, O. 2014
An explicit density estimate for Dirichlet L$L$-series
To appear in Math. Comp., 35pp.
] gives us:
Theorem (2013)
We have logx+0.578nxμ2(n)/nlogx+1.166$\log x+0.578\le \sum_{n\le x}\mu^2(n)/n\le \log x+1.166$ for x1$x\ge1$ When x1000$x\ge1000$, one can also replace the couple (0.578,1.166)$(0.578, 1.166)$ by (1.040,1.048)$(1.040, 1.048)$.
An improved estimate for the summatory function of the Möbius function
Acta Arith., 15, 223--233.
] for an earlier version.
The main result [Berkane et al., 2012 †Berkane, D., Bordellès, O., & Ramaré, O. 2012
Explicit upper bounds for the remainder term in the divisor problem
Math. of Comp., 81(278), 1025--1051.
Theorem (2012)
We define Δ(x)=nxτ(n)x(logx+2γ1)$\Delta(x)=\sum_{n\le x}\tau(n)-x(\log x+2\gamma-1)$. We have
• When x1$x\ge 1$, we have |Δ(x)|0.961x1/2$|\Delta(x)|\le 0.961\, {x^{1/2}}$.
• When x1981$x\ge 1\,981$, we have |Δ(x)|0.482x1/2$|\Delta(x)|\le 0.482\, {x^{1/2}}$.
• When x5560$x\ge 5\,560$, we have |Δ(x)|0.397x1/2$|\Delta(x)|\le 0.397\, {x^{1/2}}$.
• When x5$x\ge 5$, we have |Δ(x)|0.764x1/3logx$|\Delta(x)|\le 0.764\, {x^{1/3}\log x}$.
For evaluation of the average of the divisor function on integers belonging to a fixed residue class modulo 6, see Corollary to Proposition 3.2 of [Deshouillers & Dress, 1988 †Deshouillers, J.-M, & Dress, F. 1988
Sommes de diviseurs et structure multiplicative des entiers
Acta Arith., 49(4), 341--375.
]. For more complicated sums and when x$x$ is large with respect to k$k$, one can use [Mardjanichvili, 1939 †Mardjanichvili, C. 1939
Estimation d'une somme arithmétique
].
Theorem (1939)
Let k$k$ and $\ell$ be two positive integers. We have for any real number x1$x\ge1$
mxτk(m)xk(k!)k1k1(logx+k1)k1.
See [Deshouillers & Dress, 1988 †Deshouillers, J.-M, & Dress, F. 1988
Sommes de diviseurs et structure multiplicative des entiers
Acta Arith., 49(4), 341--375.
] for some upper bounds linked with τ3$\tau_3$.
[Bordellès, 2002 †Bordellès, O. 2002
Explicit upper bounds for the average order of {dn(m)$d_n(m)$} and application to class number
JIPAM. J. Inequal. Pure Appl. Math., 3(3), Article 38, 15 pp. (electronic).
] contains the following bounds, better than the above when x$x$ is small with respect to k$k$.
Theorem (2002)
Let k1$k\ge1$ be a positive integer.
• When x1$x\ge1$ is a real number, we have mxτk(m)x(logx+γ+(1/x))k1$\sum_{m\le x}\tau_k(m)\le x(\log x+\gamma+(1/x))^{k-1}$.
• When x6$x\ge6$ is a real number, we have mxτk(m)2x(logx)k1$\sum_{m\le x}\tau_k(m)\le 2x(\log x)^{k-1}$.
3. Euler products
[Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.
], [Bordellés, 2005 †Bordellés, O. 2005
An explicit Mertens' type inequality for arithmetic progressions
J. Inequal. Pure Appl. Math., 6(3), paper no 67 (10p).
],

Last updated on September 3rd, 2013, by Olivier Ramaré