Scientia Magna. International Book Series, vol. 12, no. 1

2017·2.8 MB·English

by  Huaning, Liu

SCIENTIA MAGNA

– International Book Series (vol. 12, no. 1) –

Editor

Professor Huaning Liu
School of Mathematics
Northwest University
Xi’an, Shaanxi, China

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i
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ii
Contents
Jing Huang and Ping Song: On the mean value of exponential divisor function 1
Ping Song and Jing Huang: The mean value of P∗(n) over cube-full numbers 7
OpeyemiOmotoyinboandAdesanmiMogbademu: SomeconvexfunctionsforHermite-
Hadamard integral inequalities 14
Niraj Kumar and Garima Manocha: Study of the Jacobson radical for a certain class
of entire Dirichlet series 23
D. Senthilkumar and R. Murugan: Weyl’s theorem for m-quasi N-class A operators
k
32
S. Sekar and G. Kumar: Almost contra generalized α regular-continuous functions in
topological spaces 46
S. Sekar and B. Jothilakshmi: Contra semi generalized star b- continuous functions in
topological spaces 57
Vakeel. A. Khan and Mohd Shafiq: OnsomegeneralisedI-convergentsequencespaces
of interval numbers 66
M. Aliabadi and B. Hassanzadeh: Cartan involution and geometry of semi-simple Lie
groups 78
A. Vadivel and B. Vijayalakshmi: On fuzzy maximal regular semi-open sets and maps
in fuzzy topological spaces 92
D. Sobana, V. Chandrasekar and A. Vadivel: On intuitionistic fuzzy e-compactness
107
Muge Togan, Aysun Yurttas and Ismail Naci Cangul: Zagreb and multiplicative
Zagreb indices of r-subdivision graphs of double graphs 115
S. M. Patil and S. M. Khairnar: Third Hankel determinant for certain subclass of
p-valent functions associated with generalized Ruscheweyh derivative operator 120
iii
H. Liu: A survey on Smarandache notions in number theory I: Smarandache function 132
H. Liu: A survey on Smarandache notions in number theory II: pseudo-Smarandache
function 145
iv
Scientia Magna
Vol. 12 (2017), No. 1, 1-6
On the mean value of exponential divisor
function
Jing Huang1 and Ping Song2
1School of Mathematics and Statistics, Shandong Normal University
Shandong Jinan, China
E-mail: [email protected]
2School of Mathematics and Statistics, Shandong Normal University
Shandong Jinan, China
E-mail: [email protected]
(cid:81)
Abstract Letn>1beaninteger. Theintegerd= s pbi iscalledanexponentialdivisor
(cid:81) i=1 i
of n= s pai, if b |a for every i∈1,2,···,s. Let τ(e)(n) denote the exponential divisor
i=1 i i i (cid:80)
function. In this paper, we study the sum D(1,2,···,2;x) = d(1,2,···,2;n) and get
(cid:124) (cid:123)(cid:122) (cid:125) n≤x (cid:124) (cid:123)(cid:122) (cid:125)
k (cid:80) k
the asymptotic formula for it, where d(1,2,···,2;n)= 1. We get the mean value
(cid:124) (cid:123)(cid:122) (cid:125) n=ab21···b2k
k
for the exponential divisor function, which improves the previous result.
Keywords Dirichlet convolution; asympototic formula; exponential divisor function.
1 Introduction
Many scholars are interested in researching the divisor problem, and they have obtained a
largenumberofgoodresults. However,therearemanyproblemshasn’tbeensolved. Forexam-
ple, F. Smarandache gave some unsolved problems in his book onlyproblems, notsolutions!,
andoneproblemisthat,anumberniscalledsimplenumberiftheproductofitsproperdivisors
is less than or equal to n. Generally speaking, n=p, or n=p2, or n=p3, or pq, where p and
q are distinct primes. The properties of this simple number sequence hasn’t been studied yet.
And other problems are introduced in this book, such as proper divisor products sequence and
the largest exponent (of power p) which divides n, where p≥2 is an integer.
(cid:81)
In this the definition of exponential divisor: suppose n > 1 is an integer, and n = tpai.
(cid:81) i i
If d= tpbi satisfies b |a ,i=1,2,···,t, then d is called an exponential divisor of n, notation
i i i i
d| n. By convention 1| 1.
e e
J. Wu [4] improved the above result got the following result:
(cid:88)
τ(e)(n)=A(x)+Bx12 +O(x29 logx),
n≤x
where (cid:195) (cid:33)
(cid:89) (cid:88)∞ d(a)−d(a−1)
A= 1+ ,
pa
p a=2
2 JingHuangandPingSong No. 1
(cid:195) (cid:33)
(cid:89) (cid:88)∞ d(a)−d(a−1)−d(a−2)+d(a−3)
B = 1+ .
pa2
p a=5
M. V. Subbarao [2] also proved for some positive integer r,
(cid:88)
(τ(e)(n))r ∼A x,
r
n≤x
where (cid:195) (cid:33)
(cid:89) (cid:88)∞ (d(a))r−(d(a−1))r
A = 1+ .
r pa
p a=2
L. Toth [3] proved
(cid:88)
(τ(e)(n))r =Ar(x)+x12P2r−2(logx)+O(xur+ε),
n≤x
where P (t) is a polynomial of degree 2r−2 in t, u = 2r+1−1.
2r−2 r 2r+1+1
Similarly to the generalization of d (n) from d(n), we define the function τ(e)(n):
k k
(cid:89)
τ(e)(n)= d (a ),k ≥2,
k k i
pai(cid:107)n
i
Obviously when k = 2, that is τ(e)(n). τ(e)(n) is obviously a multiplicative function. In this
3
paper we investigate the case k =3, i.e. the properties of the function τ(e)(n).
3
In this paper, we will study the asymptotic formula for the mean value of the r-th power
of the function τ(e)(n), where r >1 is an integer.
3
Theorem 1.1. For every integer r >1, then we have
(cid:88)
(τ3(e)(n))r =Arx+x12R3r−2(logx)+O(xbr+ε),
n≤x
for every ε>0, where b := 1 ,α is as defined in Lemma 2.2, the O−term is related to
r 3−α3r−1 k
r, R (x) is a polynomial of degree 3r−2 and
3r−2
(cid:195) (cid:33)
(cid:89) (cid:88)∞ (d (a))r−(d (a−1))r
A := 1+ 3 3 ,
r pa
p a=2
(cid:88)
where d (n)= 1.
3
n=m1m2m3
2 Some lemmas
In this section, we give some lemmas which will be used in the proof of our theorem.
Lemma 2.2 and Lemma 2.3. can be found in [1] and [5].
Lemma 2.1. For r >1, then we have
(cid:88)∞ (τ(e)(n))r
3 =ζ(s)ζ3r−1(2s)V(s),
ns
n=1
(cid:88)∞ v(n)
where the infinite series V(s):= is absolutely convergent for (cid:60)s> 1.
ns 4
n=1
Vol. 12 Onthemeanvalueofexponentialdivisorfunction 3
Proof. By Euler’s product formula, we can get
(cid:195) (cid:33)
(cid:88)∞ (τ(e)(n))r (cid:89) (τ(e)(p))r (τ(e)(p2))r (τ(e)(p3))r τ(e)(p4))r
3 = 1+ 3 + 3 + 3 + 3 +···
ns ps p2s p3s p4s
n=1 p
(cid:181) (cid:182)
(cid:89) dr(1) dr(2) dr(3) dr(4) dr(5)
= 1+ 3 + 3 + 3 + 3 + 3 +···
ps p2s p3s p4s p5s
p
(cid:181) (cid:182)
(cid:89) 1 3r 3r 6r 3r (2.1)
= 1+ + + + + +···
ps p2s p3s p4s p5s
p
(cid:181) (cid:182)
3r−1 3r
=ζ(s) 1+ + +···
p2s p4s
=ζ(s)ζ3r−1(2s)V(s),
(cid:88)∞ v(n)
where the infinite series V(s):= is absolutely convergent for (cid:60)s> 1.
ns 4
n=1
Lemma 2.2. Suppose k ≥2 is an integer. Then
(cid:88) k(cid:88)−1
Dk(x)= dk(n)=x cj(logx)j +O(xαk+ε),
n≤x j=0
where c is a calculable constant, ε is a sufficiently small positive constant, α is the infimum
j k
of numbers α , such that
k
(cid:88)
∆k(x)= dk(n)−xPk−1(logx)(cid:191)xαk+ε, (2.2)
n≤x
and
131 43
α ≤ , α ≤ ,
2 416 3 94
3k−4
α ≤ , 4≤k ≤8,
k 4k
35 41 7
α ≤ , α ≤ , α ≤ ,
9 54 10 61 11 10
k−2
α ≤ , 12≤k ≤25,
k k+2
k−1
α ≤ , 26≤k ≤50,
k k+4
31k−98
α ≤ , 51≤k ≤57,
k 32k
7k−34
α ≤ , k ≥58.
k 7k
Lemma 2.3. Suppose f(m),g(n) are arithmetical functions such that
(cid:88) (cid:88)J (cid:88)
f(m)= xαjP (logx)+O(xα), |g(n)|=O(xβ),
j
m≤x j=1 n≤x
4 JingHuangandPingSong No. 1
(cid:80)
where α ≥α ≥···≥α >α>β >0,P (t) is a polynomial in t, if h(n)= f(m)g(d),
1 2 J j n=md
then
(cid:88) (cid:88)J
h(n)= xαjQ (logx)+O(xα),
j
n≤x j=1
where Q (t) {j =1,···,J} is a polynomial in t.
j
3 The mean value of d(1,2,· · ·,2;n)
Theorem 3.1. Suppose k ≥2 is an integer, then
(cid:88)
D(1,2(cid:124),·(cid:123)·(cid:122)·,2(cid:125);x)= d(1,(cid:124)2,·(cid:123)·(cid:122)·,2(cid:125);n)=ζk(2)x+x12Qk−1(logx)+O(x3−1αk+ε).
n≤x
k k
(cid:80)
Proof. Recall that d(1,2,···,2;n)= 1, by hyperbolic summation formula, we have
(cid:124) (cid:123)(cid:122) (cid:125) n=ab2···b2
1 k
k
(cid:88) (cid:88)
D(1,2,···,2;x)= d(1,2,···,2;n)= d (m)
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) k
n≤x m2l≤x
k k
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
= d (m) 1+ d (m)− d (m) 1 (3.1)
k k k
m≤y m2l≤x l≤zm2l≤x m≤y l≤z
:=S +S −S ,
1 2 3
where y,z are parameters that will be determined later, and satisfy that y2z = x,1 ≤ y ≤ x.
Now, we deal with S ,S and S , separately
1 2 3
(cid:88) (cid:88) (cid:88) x
S = d (m) 1= d (m)[ ]
1 k k m2
m≤y m2l≤x m≤y
 
(cid:88) d (m) (cid:88)
=x km2 +O dk(m) (3.2)
m≤y m≤y
(cid:88) d (m)
=ζk(2)x−x k +O(y1+ε).
m2
m>y
Using Lemma 2.2 and partial summation formula, we have
 
(cid:90)
(cid:88) d (m) ∞ 1 (cid:88)
k = d d (m)
m2 t2 k
m>y y+ m≤t
 
(cid:90)
∞ 1 k(cid:88)=1
= t2dt cj(logt)j +O(tαk+ε)
y+ j=0
(cid:90)
k(cid:88)=1 ∞ 1
= cj t2d(logt)j +O(y−2+αk+ε)
j=0 y+
k(cid:88)=1
= c y−1[(logy)j +2j(logy)j−1+2j(j−1)(logy)j−2+···+2j(j−1)···1]
j
j=0
+O(y−2+αk+ε).