2.5.1 Gcd-sum function
The gcd-sum function, also called Pillai’s arithmetical function is defined by Ppnq “
n
ÿ
k“1
pk, nq.
The function Ppnq is multiplicative and for everyn PN, Ppnq “ÿ
d|n
d φpn{dq “ÿ
d|n
dτpdqµpn{dq.
Hence the arithmetic mean of p1, nq, . . . ,pn, nq is given by Apnq:“ Ppnq
n “ÿ
d|n
φpdq
d “τpnqź
pν||n
ˆ
1´ν{pν`1q p
˙
, (2.19)
which is ,,close” to τpnq.
Various properties, generalizations and analogs of the functionPpnqwere investigated by several authors. See my survey paper [101] and my subsequent papers [102, 106].
Chidambaraswamy and Sitaramachandrarao [20] proved the following result:
ÿ
nďx
Ppnq “ x2 2ζp2q
ˆ
logx`2γ´ 1
2´ ζ1p2q ζp2q
˙
`Opx1`θ`εq, (2.20)
whereθis the exponent appearing in the Dirichlet divisor problem (1.1). They also proved that
lim sup
nÑ8
logApnqlog logn
logn “log 2, (2.21)
which is well known for the function τpnq instead of Apnq. See (1.7).
Our next result is an asymptotic formula for the quadratic moment of the function Apnq. Letα4 be the exponent appearing in the Piltz divisor problem forτ4pnq. It is known that α4 ď1{2 (result of Hardy and Littlewood) and it is conjectured that α4 “ 3{8, cf.
Titchmarsh [94, Ch. 12].
Theorem 2.5.1 (T´oth [101, Th. 1]). i) For any εą0, ÿ
nďx
Apnq2 “xpC1log3x`C2log2x`C3logx`C4q `Opx1{2`εq, (2.22) where
C1 “ 1 π2
ź
p
ˆ 1` 1
p3 ´ 4 ppp`1q
˙ ,
C2, C3, C4 are constants given in terms of the constants appearing in the asymptotic for-mula (2.1) for ř
nďxτpnq2.
ii) Assume that α4 ă 1{2. Then the error term in (2.22) is Opx1{2δpxqq, where δpxq is defined by (1.3).
iii) If RH is true, then the error term in (2.22) is Opxp2´α4q{p5´4α4qλpxqq, where λpxq:“exppplogxq1{2plog logxq14q.
Remark 2.5.2. LetMpxq “ř
nďxµpnq denote the Mertens function. The error term of iii) comes from the estimate Mpxq ! ?
x λpxq, the best up to now, valid under RH, due to Soundararajan [86].1
Remark 2.5.3. Later, by using the analytic method (properties of the zeta function), Zhang and Zhai [127, Th. 1] established the following asymptotic formula, where k ě 1 is a fixed integer:
ÿ
nďx
Apnqk“xQ2k´1plogxq `Opxβk`εq,
where Q2k´1ptq is a polynomial of degree 2k´1 in t and β2 “ 1{2, β3 “ 5{8, β4 “ 7{9, β5 “31{36, β6 “207{224, βj “1´2´2j{3{50 (j ě7).
Note that in the casek “2 this is the same error as in i) of Theorem 2.5.1.
The next formula was proved by Chen and Zhai [19, Th. 4], sharpening my result [101, Th. 6], which is recovered forN “0:
ÿ
nďx
1 Ppnq “
N
ÿ
j“0
Kj
plogxqj´1{2 `O
ˆ 1 plogxqN`1{2
˙
, (2.23)
valid for every real x ě 2 and every fixed integer N ě 0 where Kj (0 ď j ď N) are computable constants,
K0 “ 2
?π ź
p
ˆ 1´1
p
˙1{2 8
ÿ
ν“0
1 Pppνq.
Referring to my similar formulas [101, Th. 6] concerning the functions Ppeqpnq and Prpnq, to be defined in the next Sections, Chen and Zhai [19, Th. 4] obtained results analogous to (2.23).
1Balazard and Roton showed in their preprint [9] that under RH the slightly better estimateMpxq !
?xexppplogxq1{2plog logxq5{2`εqholds.
2.5.2 Exponential analog of the gcd-sum function
In paper [96] I introduced the function Ppeqpnq “
n
ÿ
j“1 κpjq“κpnq
pj, nqpeq,
representing an analog of Pillai’s function Ppnq “ řn
j“1pj, nq. The function Ppeqpnq, called the exponential gcd-sum function, is also multiplicative and for every prime power pν (ν ě1),
Ppeqppνq “
ν
ÿ
t“1
ppt,νq “ÿ
d|ν
pdφpν{dq,
so herePpeqppq “ p, Ppeqpp2q “ p`p2, Ppeqpp3q “2p`p3, Ppeqpp4q “2p`p2 `p4, etc.
Theorem 2.5.4 (T´oth [96, Th. 3]).
ÿ
nďx
Ppeqpnq “C4x2`Opxplogxq5{3q, (2.24) where the constant C4 is given by
C4 “ 1 2
ź
p
˜ 1`
8
ÿ
a“2
Ppeqppaq ´pPpeqppa´1q p2a
¸ .
Theorem 2.5.5 (T´oth [96, Th. 4]).
lim sup
nÑ8
Ppeqpnq
nlog logn “ 6
π2eγ. (2.25)
It is a simple consequence of (2.25) that the error term in (2.24) is Ωpxlog logxq.
P´etermann [74, Th. 2] proved the stronger result that it is Ω˘pxlog logxq.
2.5.3 A gcd-sum function involving regular integers (mod n)
An integer k is called a regular integer (mod n), if there exists an integerx such that k2x ” k (mod n), i.e., the residue class of k is a regular element (in the sense of J. von Neumann) of the ring Zn of residue classes (mod n). In general, an element k of a ring R is said to be (von Neumann) regular if there is an x P R such that k “ kxk. If every kP R has this property, then R is called a von Neumann regular ring.
Letn ą1 be an integer with prime factorization n“pν11¨ ¨ ¨pνrr. It can be shown that k ě 1 is regular (mod n) if and only if for every i P t1, . . . , ru either pi - k or pνii | k.
These integers occur in the literature also in another context. It is said that an integerk
possesses aweak order(mod n) if there exists an integermě1 such that km`1 ”k (mod n). Then the weak order of k is the smallest m with this property. It turns out that k is regular (modn) if and only ifk possesses a weak order (mod n). See my paper [99].
Let Regn “ tk : 1 ďk ď n, k is regular (modn)u and let %pnq “ # Regn denote the number of regular integersk (modn) such that 1ďk ďn. This function is multiplicative and %ppνq “φppνq `1“ pν ´pν´1`1 for every prime power pν (ν ě1), where φ is the Euler function. The average order of the function%pnqwas considered by Joshi [51]. One has
ÿ
nďx
%pnq “ 1
2Ax2`Rpxq, where
A “ź
p
ˆ
1´ 1
p2pp`1q
˙
“ζp2qź
p
ˆ 1´ 1
p2 ´ 1 p3 ` 1
p4
˙ .
“0.8815
is the so called quadratic class-number constant, and Rpxq “ Opxlog3xq, given in [51]
using elementary arguments. This was improved into Rpxq “ Opxlogxq by Herzog and Smith [41], using analytic methods. Also,Rpxq “Ω˘px?
log logxq, see [41].
In the paper [100] I introduced the function Prpnq:“
ÿ
kPRegn
pk, nq,
which is another analog of the gcd-sum function Ppnq, discussed above. I showed that the function Prpnqis multiplicative and for every n PN,
Prpnq “ ÿ
de“n pd,eq“1
dφpeq (2.26)
“nź
p|n
ˆ 2´1
p
˙
“2ωpnqnź
p|n
ˆ 1´ 1
2p
˙
, (2.27)
which is ,,close” to 2ωpnqn.
Letψpnq “nś
p|np1`1{pq denote the Dedekind function and let αpnq “ ÿ
p|n
logp
p´1, βpnq “ÿ
p|n
logp p2´1. Theorem 2.5.6 (T´oth [100, Th. 2]). We have
ÿ
nďx
Prpnq “ x2
2ζp2qpK1logx`K2q `Opx3{2δpxqq, (2.28)
where the constantsK1 and K2 are given by K1 :“
8
ÿ
n“1
µpnq nψpnq “
ź
p
ˆ
1´ 1
ppp`1q
˙ ,
K2 :“K1 ˆ
2γ´ 1
2´ 2ζ1p2q ζp2q
˙
´
8
ÿ
n“1
µpnqplogn´αpnq `2βpnqq
nψpnq ,
and δpxq is given by (1.3).
If RH is true, then the error term of (2.28) is Opxp7´5θq{p5´4θqηpxqq, where θ is the exponent in the Dirichlet divisor problem (1.1) and
ηpxq:“exp`
Bplogxqplog logxq´1˘ , with a positive constant B.
Theorem 2.5.7 (T´oth [100, Th. 1]). The minimal order of Prpnq is 3n{2 and the maximal order of logpPrpnq{nq is log 2 logn{log logn.
Zhang and Zhai [125] pointed out that the estimate ofř
nďxPrpnqis closely related to the squarefree divisor problem. They showed that under RH the error term in (2.28) is Opx15{11`εq, due to the result of Baker [6], quoted in the Introduction.
De Koninck and K´atai [27] introduced two wide classes of arithmetic functions,R and U, the first of which includes the functionPpnq{n, and the second includes Prpnq{n. They deduced asymptotic formulas forř
nďxRpnq,ř
nďxUpnqandř
pďxRpp´1q,ř
pďxUpp´1q, whereR PR, U PU.
Zhang and Zhai [126] improved the error term forř
nďxUpnqand also deduced a short interval result for ř
xďnďx`yUpnq.