On The Asymptotic Expansion Of A Generalized Smith’s Determinant
Mehdi Hassani
yReceived 6 April 2020
Abstract
In this paper we study the generalized Smith’s determinant s(n) := det [(gcd(i; j))s]16i;j6n, where s6= 0is …xed real. For large values ofnwe obtain asymptotic expansions oflogj s(n)j, and fors >1we obtain Stirling type approximations for s(n). Furthermore, we prove that fors <0the sign of s(n)is independent ofs, and is same as the sign of( 1) n, where ndenotes the number of integersm2[1; n]
having odd number of distinct prime divisors.
1 Introduction
In 1875 Smith [8] considered the determinant of the matrix[aij]16i;j6nwith elements given byaij= gcd(i; j), greatest common divisor ofiandj. He proved that
det [gcd(i; j)]16i;j6n= Yn m=1
'(m);
where'(m)denotes the Euler function ofm, counting the number of positive integers not exceedingmand coprime to m. The above determinant is known as Smith’s determinant. Since Smith’s work this …eld has been studied extensively. For a recent account of the theory of gcd-matrices we refer the reader to [4] and the references given there. Also, see [2, p. 123] for some classical generalizations of Smith’s determinant, including the assertion that iff is an arithmetic function then
det [f(gcd(i; j))]16i;j6n= Yn m=1
X
djm
(d)f m
d ; (1)
where (d)denotes the Möbius function of d, which is1 ifd= 1, is( 1)k ifdis equal to the product of k distinct primes, and is0otherwise. In this paper we are motivated by the asymptotic growth of generalized Smith’s determinant (1) forf(n) =ns. The exponentsis an arbitrary non-zero real. For the cases >0we prove the following result.
Theorem 1 Let s >0 be …xed real and
+
s(n) = det [(gcd(i; j))s]16i;j6n: (2) De…ne the absolute constant s by
s=X
p
1
plog 1 1
ps ; (3)
Mathematics Sub ject Classi…cations: 15A15, 11A25.
yDepartment of Mathematics, University of Zanjan, University Blvd., 45371-38791, Zanjan, Iran
187
wherepruns over all primes. Then, asn! 1,
log +s(n) = 8>
>>
><
>>
>>
:
snlogn+ ( s s)n+O(n1 s) (0< s <1);
nlogn+ ( 1 1)n+12logn+O(log logn) (s= 1);
snlogn+ ( s s)n+s2logn+slogp
2 +O ns11 (1< s62):
Also, for each s >2the following approximation holds log +s(n) =snlogn+ ( s s)n+s
2logn+slogp
2 + X
16j6s2
sB2j
(2j)(2j 1)n2j 1 +O 1 ns 1 ;
whereBi denotes thei-th Bernoulli number.
Stirling approximation forn!asserts thatn! = ne np
2 n(1 +O(1n)). By taking exponent we obtain the following Stirling type approximation for +s(n)for each s >1.
Corollary 1 Let +s(n) be the determinant de…ned by (2). Then, asn! 1,
+ s(n) =
8<
:
n e
sn n s
p(2 n)s 1 +O ns11 (1< s <2);
n e
sn n s
p(2 n)s 1 +O 1n (s>2);
where s is an absolute constant de…ned by
s=Y
p
1 1
ps
1 p
;
andpruns over all primes.
A more sophisticated argument, similar to that used in our paper [3], enables us to consider the case of negative values of exponent.
Theorem 2 Let s >0 be …xed real and
s(n) = det (gcd(i; j)) s 16i;j6n:
Then, for any positive integerr there exist computable constants c1; : : : ; cr such that as n! 1,
log s(n) = ( s+ +E)n+s Xr j=1
cjn
logjn+O n logr+1n ;
where sis de…ned by (3), is Euler’s constant, andE is the constant in Mertens’ approximation given by E= lim
x!1
X
p6x
logp
p logx: (4)
Furthermore, the sign of s(n)is independent of s, and is same as the sign of ( 1) n, where n denotes the number of integersm2[1; n]having odd number of distinct prime divisors.
2 Proof of Theorem 1
Proof. Forf(n) =ns, we conclude from (1) that
+s(n) = Yn m=1
msgs(m) =n!s Yn m=1
gs(m);
wheregs(m) =P
djm (d)d s. Sincegsis multiplicative, we get gs(m) = Y
pakm
gs(pa) = Y
pakm
1 1
ps =Y
pjm
1 1
ps : Thus,
+
s(n) =n!s Yn m=1
Y
pjm
1 1
ps ; and
log +s(n) =slogn! + Xn m=1
X
pjm
log 1 1
ps : (5)
Stirling’s approximation [7, p. 294] forlogn!asserts that given any positive integerr, as n! 1,
logn! =nlogn n+1
2logn+ logp 2 +
Xr j=1
B2j
(2j)(2j 1)n2j 1 +O 1
n2r+1 : (6)
To approximate the double sum in (5), we change the order of summations. Thus, Xn
m=1
X
pjm
log 1 1
ps =X
p6n
log 1 1 ps
X
m6n pjm
1 =X
p6n
log 1 1 ps
n p
=X
p6n
log 1 1 ps
n
p+O(1)
=nX
p6n
1
plog 1 1 ps +O
0
@X
p6n
log 1 1 pk
1 A
= sn+nX
p>n
1
plog 1 1 ps
1
+O 0
@X
p6n
log 1 1 ps
1 A:
Since log(1 t) tas t!0, we get X
p>n
1
plog 1 1 ps
1 X
p>n
1 ps+1
Z 1
n
dx xs+1
1 ns: Also, by using the approximationP
p6n 1
p log lognwe obtain X
p6n
log 1 1 ps
X
p6n
1 ps
8<
: Rn
2 dx xs
1
ns 1 (s6= 1);
log logn (s= 1):
Hence,
Xn m=1
X
pjm
log 1 1
ps = sn+O n1 s (s6= 1)
log logn (s= 1) : (7)
We let r= [s2] in (6). Note that 2[s2] + 1 >s 1. Therefore, by considering (5) and (7) we conclude the proof.
3 Proof of Theorem 2
Proof. Lets >0. We conclude from (1) that
s(n) = Yn m=1
m shs(m) =n! s Yn m=1
hs(m);
wherehs(m) =P
djm (d)ds. Sincehs is multiplicative, we get hs(m) = Y
pakm
hs(pa) = Y
pakm
(1 ps) =Y
pjm
(1 ps)
= ( 1)!(m)Y
pjm
(ps 1) = ( 1)!(m) (m)sY
pjm
1 1
ps ;
where !(m) counts the number of distinct prime factors of m, and (m) denotes the product of distinct prime factors ofm. Thus,
s(n) = ( 1)
Pn m=1!(m)
n! s Yn m=1
(m)
!s Yn m=1
Y
pjm
1 1
ps : (8)
This relation implies that the sign of s(n)depends on the value of Pn
m=1!(m), which is independent of s. Moreover, the sign of s(n)is same as the sign of ( 1) n, where
n= X
16m6n
!(m)is o dd
1:
denoting the number of integersm2[1; n]which have odd number of distinct prime divisors. Furthermore, we conclude from (8) that
log s(n) = slogn! +s Xn m=1
log (m) + Xn m=1
X
pjm
log 1 1 ps : To approximatePn
m=1log (m)we recall the notion ofthe index of composition ofn, which is de…ned by (n) = logn
log (n);
for each integer n > 2. Note that (n) “somehow” measures how much the integer n > 2 is composite!
Fornsquare-free it takes the value (n) = 1, and for integersnhaving square factors in heart, it takes the value (n)>1. De Koninck and Kátai [1] proved that given any positive integerr, there exist computable constantsd1; : : : ; drsuch that
(x) :=X
k6x
1
(k) =x+ Xr j=1
dj
x
logjx+O x
logr+1x : (9)
By using Abel summation we get Xn
k=1
log (k) = Xn k=2
1
(k)logk= (n) logn (2 ) log 2 Z n
2
(t) t dt:
To deal with the last integral, we study the functionsLj(t)de…ned for each integerj >1 by the following anti-derivative
Lj(t) :=
Z dt logjt;
Note thatL1(t)is the logarithmic integral function, which admits the following expansion
L1(t) = li(t) = Xr i=1
(i 1)! t
logit+O t
logr+1t : (10)
Integrating by parts gives
Lj 1(t) =
Z 1
logj 1t (dt) = t
logj 1t+ (j 1) Z dt
logjt: Hence, forj >2 the functionsLj(t)satisfy the recurrence
Lj(t) = 1
j 1Lj 1(t) t
(j 1) logj 1t: By repeated using this recurrence we deduce that
(j 1)! Lj(t) = li(t)
j 1
X
i=1
(i 1)! t logit: Hence, by using the expansion (10), for16j 6r we obtain
Lj(t) = Xr i=j
(i 1)!
(j 1)!
t
logit+O t
logr+1t : (11)
We deduce from the expansion (9) that Z n
2
(t) t dt=
Z n 2
0
@1 + Xr j=1
dj
1
logjt +O 1 logr+1t
1 Adt
=n+ Xr j=1
djLj(n) 0
@2 + Xr j=1
djLj(2) 1
A+O n logr+1n : Withrreplaced byr+ 1in (9), we obtain
(n) logn=nlogn+d1n+ Xr j=1
dj+1 n
logjn+O n logr+1n : Combining the above expansions, we obtain
Xn k=1
log (k) =nlogn+ (d1 1)n+ Xr j=1
dj+1 n
logjn djLj(n) Cr+O n logr+1n ;
where
Cr= 2 + (2 ) log 2 + Xr j=1
djLj(2)
is a constant depending only onr. Thus,Cr=Or(1). Moreover, we deduce from the expansion (11) that Xr
j=1
dj+1
n
logjn djLj(n) = Xr j=1
0
@dj+1 n logjn
Xr i=j
dj
(i 1)!
(j 1)!
n login
1
A+O n logr+1n : Note that
Xr j=1
0
@dj+1 n logjn
Xr i=j
dj(i 1)!
(j 1)!
n login
1 A=
Xr j=1
cj n
logjn+O n logr+1n ; wherecjs are computable constants in terms ofdjs. Thus, lettingc0=d1 1, we obtain
Xn m=1
log (m) =nlogn+c0n+ Xr j=1
cj n
logjn+O n logr+1n : To compute the precise value ofc0 we write
Xn m=1
log (m) = Xn m=1
logY
pjm
p= Xn m=1
X
pjm
logp=X
p6n
n
p logp=nM(n) R(n); (12) where
M(n) :=X
p6n
logp p ; and
R(n) :=X
p6n
n p logp:
It is known due to Landau [5, p. 198] that
M(n) = logn+E+O 1
logn ; (13)
whereE is the constant given by (4). To estimate R(n)we let S(n) =X
p6n
n p ; and
L(n) = X
p 6n
n p : It is known due to Lee [6] that
L(n) = (1 ) n
logn+O n log2n :
We observe that although the summationL(n)has the summationS(n)in heart, but their di¤erence is not too large in comparison the true size ofL(n). More precisely,
L(n) S(n) = X
p 6n
>2
n
p < X
p 6n
>2
1 = X
p6n1
>2
1 = X
26 6loglog 2n
(n1)
X
26 6loglog 2n
n1
logn1 6 n12 logn
X
26 6loglog 2n
pnlogn;
where (t)denotes the number of primespnot exceeding t, and we use the simple estimate (t) logtt in the above argument. Hence,
S(n) = (1 ) n
logn+O n
log2n : (14)
Let$(k)to be1whenk is prime and0 otherwise. By using Abel summation we get
R(n) = Xn k=2
nn k o
$(k) logk=S(n) logn S(2 ) log 2 Z n
2
Fn(t) t dt;
where
Fn(t) =X
p6t
n p :
Since06Fn(t)6 (t) logtt, by using the approximation (14) we deduce that
R(n) = (1 )n+O n logn
Z n 2
O t
logt dt
t = (1 )n+O n logn :
Thus, by substituting (13) and the last approximation in (12) we obtain the truncated approximation Xn
m=1
log (m) =nlogn+ ( +E 1)n+O n logn ;
implying thatc0= +E 1. Hence, given any positive integerr, there exist computable constantsc1; : : : ; cr such that
Xn m=1
log (m) =nlogn+ ( +E 1)n+ Xr j=1
cjn
logjn+O n logr+1n : By using this approximation and the relations (6) and (7) we conclude the proof.
References
[1] J. M. De Koninck and I. Kátai, On the mean value of the index of composition of an integer, Monatsh.
Math., 145(2005), 131–144.
[2] L. E. Dickson, History of the Theory of Numbers, Vol. I, AMS Chelsea Publishing, 2002.
[3] M. Hassani and M. Esfandiari, On the geometric mean of the values of positive multiplicative arithmetical functions, Commun. Math., to appear.
[4] P. Haukkanen, J. Wang and J. Sillanpää, On Smith’s determinant, Linear Algebra Appl., 258(1997), 251–269.
[5] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, AMS Chelsea Publishing, 1974.
[6] J. Lee, The second central moment of additive functions, Proc. Amer. Math. Soc., 114(1992), 887–895.
[7] F. W. J. Olver, Asymptotics and special functions, Academic Press, 1974.
[8] H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. Lond. Math. Soc., 7(1875/76), 208–212.
