Various asymptotic properties of multiplicative arithmetic functions, i.e., nonzero func-tionsf :NÑCsatisfying fpmnq “fpmqfpnq, provided thatpm, nq “1, are well known in the literature. It is one of the main objectives of elementary and analytic number theory to deduce asymptotic formulas with sharp error terms for sumsř
nďxfpnq, where fpnqis a special multiplicative function or it is belonging to a certain class of such functions.
For example, the Dirichlet divisor problem consists in finding the infimum of exponents θ such that the formula
ÿ
nďx
τpnq “xlogx` p2γ´1qx`Opxθ`εq, (1.1) holds for every εą0. It is known that 1{4ďθ ď131{416 .
“0.314903. More exactly, the best error term in (1.1) up to date isOpx131{416plogxq26947{8320q, due to Huxley [46].1
More generally, for positive integers a1 ď ¨ ¨ ¨ ď ak consider the generalized divisor functionτpa1, . . . , ak;nq:“ř
da11¨¨¨dakk “n1 and let ∆pa1, . . . , ak;xq stand for the remainder term in the related asymptotic formula, i.e.,
ÿ
nďx
τpa1, . . . , ak;nq “ Hpa1, . . . , ak;xq `∆pa1, . . . , ak;xq,
whereHpa1, . . . , ak;xqis the main term. See, e.g., the book by Kr¨atzel [56, Ch. 6]. In the case a1 “ ¨ ¨ ¨ “ak “1 we have the Piltz divisor functionτkpnq, and let ∆kpxq denote, as usual, the corresponding error term (Piltz divisor problem).
1In a recent preprint of 13 September 2017, Bourgain and Watt [12] proved the better result θ ď 517{1648 .
“0.313713. The same error term is valid for the Gauss circle problem.
The squarefree divisor problem goes back to the work of Mertens (1874). Letτp2qpnq “ 2ωpnq denote the number of squarefree divisors of n. One has
ÿ
nďx
τp2qpnq “ 6 π2x
ˆ
logx`2γ´1´2ζ1p2q ζp2q
˙
`OpRpxqq, (1.2)
with Rpxq !x1{2δpxq, where
δpxq:“expp´cplogxq3{5plog logxq´1{5q, (1.3) c being a positive constant. See Suryanarayana and Siva Rama Prasad [90]. If the Riemann hypothesis (RH) is true, then Rpxq !x4{11`ε, due to Baker [6].
Another example, we quote here is the asymptotic formula ÿ
nďx
φpnq “ 3
π2x2`O`
xplogxq2{3plog logxq4{3˘
, (1.4)
concerning Euler’s functionφpnq, with the best error term known to date, due to Walfisz [120, Satz 1, p. 144]. The formula
ÿ
nďx
1
φpnq “Aplogx`γ´Bq `O`
x´1plogxq2{3˘
, (1.5)
where
A“ ζp2qζp3q
ζp6q “ 315ζp3q
2π4 , B “ÿ
p
logp
p2´p`1, (1.6)
with the weaker error termOpx´1logxq goes back to the work of Landau. See the book by De Koninck and Iv´ıc [26, Th. 1.1]. The error term in (1.5) was obtained by Sita Rama Chandra Rao [81].
Now consider the class W of multiplicative functions f : N Ñ r0,1s. According to a celebrated result of Wirsing [123], iff is in the classW, then the mean value
Mpfq:“ lim
xÑ8
1 x
ÿ
nďx
fpnq exists and
Mpfq “ ź
p
ˆ 1´ 1
p
˙ 8
ÿ
ν“0
fppνq pν ,
with the convention that the product is zero provided that the seriesř
p 1´fppq
p diverges.
Other type of results are concerning the maximal order of certain multiplicative func-tions. For example, the following useful theorem on the maximal order of a class of prime-independent multiplicative functions was proved by Suryanarayana and Sita Rama
Chandra Rao [89]: Let f be a positive function satisfying fpnq “ Opnβq for some fixed β ą0. Let F be the multiplicative function with Fppνq “fpνq for every prime power pν (ν ě1). Then
lim sup
nÑ8
logFpnqlog logn
logn “sup
mě1
logfpmq
m .
This applies to the function Fpnq “ τpnqand gives lim sup
nÑ8
logτpnqlog logn
logn “log 2, (1.7)
which is a well known result. The same formula is true for τpnq replaced by τp2qpnq. If Fpnq “τpeqpnq, the number of exponential divisors of n, then we obtain
lim sup
nÑ8
logτpeqpnqlog logn
logn “ log 2 2 , proved earlier by Erd˝os. See [87, Th. 6.2].
Ramanujan [77] derived pointwise convergent series representations of arithmetic func-tions with respect to the sums cqpnq, now called Ramanujan sums. For example, letσpnq denote the sum of divisors of n. For every fixed n PN,
σpnq
n “ζp2q
8
ÿ
q“1
cqpnq
q2 (1.8)
“ π2 6
ˆ
1` p´1qn
22 `2 cosp2πn{3q
32 ` 2 cospπn{2q 42 ` ¨ ¨ ¨
˙ ,
which shows how the values of σpnq{n fluctuate harmonically about their mean value π2{6.
Delange [30] proved the following general theorem concerning such expansions, called Ramanujan (or Ramanujan-Fourier) expansions of arithmetic functions. Let f : N ÑC be an arithmetic function. Assume that
8
ÿ
n“1
2ωpnq|pµ˚fqpnq|
n ă 8. (1.9)
Then for every nP Nwe have the absolutely convergent Ramanujan expansion fpnq “
8
ÿ
q“1
aqcqpnq, where the coefficientsaq are given by
aq “
8
ÿ
m“1
pµ˚fqpmqq
mq pqPNq.
Delange also pointed out how this result can be formulated for multiplicative functions f. By Wintner’s theorem condition (1.9) ensures that the mean value Mpfq exists and a1 “Mpfq.
A nonzero function f : Nk Ñ C is said to be multiplicative if fpm1n1, . . . , mknkq “ fpm1, . . . , mkqfpn1, . . . , nkq, provided that pm1¨ ¨ ¨mk, n1¨ ¨ ¨nkq “ 1. Therefore, if f is multiplicative, then it is determined by the values fppν1, . . . , pνkq, where p is prime and ν1, . . . , νk PNY t0u. More exactly, fp1, . . . ,1q “ 1 and for anyn1, . . . , nkP N,
fpn1, . . . , nkq “ ź
p
fppνppn1q, . . . , pνppnkqq.
If the case k “ 1 this reduces to the usual multiplicativity. Some simple examples of multiplicative functions of k variables are pn1, . . . , nkq and rn1, . . . , nks. Among other examples of such functions we mention spn1, . . . , nkq and cpn1, . . . , nkq, representing the total number of subgroups and the number of cyclic subgroups, respectively, of the group pZn1 ˆ ¨ ¨ ¨ ˆZnk,`q. Let %r denote the characteristic function of the set of ordered r-tuples pn1, . . . , nrq P Nr such that n1, . . . , nr are pairwise relatively prime. Then %r is a multiplicative function ofr variables and it satisfies
ÿ
d1|n1,...,dr|nr
%rpd1, . . . , drq “τpn1¨ ¨ ¨nrq pn1, . . . , nr PNq. (1.10) A detailed study of multiplicative functions of several variables was carried out by Vai-dyanathaswamy [119] more than eighty-five years ago. However, the paper [119] includes algebraic and arithmetic properties, essentially. Even to the present day, there are only a few asymptotic results in the literature for multiplicative functions of several variables.
My paper [108] is a survey on this topic.
The mean value of a function f :NkÑCis Mpfq:“ lim
x1,...,xkÑ8
1 x1¨ ¨ ¨xk
ÿ
n1ďx1,...,nkďxk
fpn1, . . . , nkq,
provided that this limit exists. As a generalization of Wintner’s theorem (valid in the one variable case), Ushiroya [116, Th. 1] proved the next result: If f is a function of k variables, not necessary multiplicative, such that
8
ÿ
n1,...,nk“1
|pµk˚fqpn1, . . . , nkq|
n1¨ ¨ ¨nk
ă 8, then the mean valueMpfq exists and
Mpfq “
8
ÿ
n1,...,nk“1
pµk˚fqpn1, . . . , nkq n1¨ ¨ ¨nk ,
where˚ denotes the Dirichlet convolution defined by pf ˚gqpn1, . . . , nkq “
ÿ
d1|n1,...,dk|nk
fpd1, . . . , dkqgpn1{d1, . . . , nk{dkq,
and µkpn1, . . . , nkq “µpn1q ¨ ¨ ¨µpnkq is the M¨obius function of k variables (the inverse of the constant 1 function under ˚).
For multiplicative functions the above result was formulated by us [108, Prop. 19] as follows (see Ushiroya [116, Th. 4] for the same result in a slightly different form and for its proof): Letf :Nk ÑCbe a multiplicative function. Assume that
ÿ
p
8
ÿ
ν1,...,νk“0 ν1`¨¨¨`νkě1
|pµk˚fqppν1, . . . , pνkq|
pν1`¨¨¨`νk ă 8.
Then the mean value Mpfq exists and Mpfq “ ź
p
ˆ 1´1
p
˙k 8
ÿ
ν1,...,νk“0
fppν1, . . . , pνkq pν1`¨¨¨`νk .
We are not aware of more general mean value results concerning the several variables case. Asymptotic formulas for sums of typeř
n1,...,nkďxfpn1, . . . , nkq, with certain special functionsf, were derived by Balazard, Naimi, P´etermann [8] and de la Bret`eche [28] using analytic methods. For example, in paper [8] the authors use an effective Perron inversion formula ink variables to prove (by a very complicated process) that
ÿ
n1,...,nkďx
µpn1q ¨ ¨ ¨µpnkq
rn1, . . . , nks “Pkplogxq `Opδpxqq,
wherePkptqis a polynomial in t and δpxq is defined by (1.3). They also prove that Pkptq is identically zero, whenk is odd (in the case k“1 this is equivalent to the prime number theorem).