Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON SÁNDOR TYPE FUNCTIONS
N. ANITHA
Department of Studies in Mathematics
University of Mysore
Manasagangotri
Mysore 570006, India.
volume 6, issue 4, article 127,
2005.
Received 14 June, 2005;
accepted 28 July, 2005.
Communicated by: J. Sándor
EMail: ani_lohith@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
183-05
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Abstract
In this paper we introduce the functions G and G∗ similar to Sándor’s functions
which are defined by,
G(x) = min{m ∈ N : x ≤ em }, x ∈ [1, ∞),
G∗ (x) = max{m ∈ N : em ≤ x}, x ∈ [e, ∞).
A Note on Sándor Type
Functions
We study some interesting properties of G and G∗ . The main purpose of this
paper is to show that
x
π(x) ∼
G∗ (x)
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where π(x) is the number of primes less than or equal to x.
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2000 Mathematics Subject Classification: 40A05, 33E99.
Key words: Asymptotic formula , Infinite Series.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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J. Ineq. Pure and Appl. Math. 6(4) Art. 127, 2005
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1.
Introduction
In his paper [1], J. Sándor discussed many interesting properties of the functions
S and S∗ defined by,
S(x) = min{m ∈ N : x ≤ m!},
x ∈ (1, ∞),
S∗ (x) = max {m ∈ N: m! ≤ x},
x ∈ [1, ∞).
and
A Note on Sándor Type
Functions
He also proved the following theorems:
Theorem 1.1.
log x
S∗ (x) ∼
log log x
Theorem 1.2. The series
∞
X
n=1
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(x → ∞).
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1
n[S∗ (n)]α
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is convergent for α > 1 and divergent for α ≤ 1.
Now we will define functions G(x) and G∗ (x) and discuss their properties.
The functions are defined as follows:
G(x) = min{m ∈ N : x ≤ em },
x ∈ [1, ∞),
G∗ (x) = max{m ∈ N : em ≤ x},
x ∈ [e, ∞).
Clearly,
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J. Ineq. Pure and Appl. Math. 6(4) Art. 127, 2005
G(x) = m + 1,
if x ∈ [em , em+1 ) for
m ≥ 0.
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Similarly,
G∗ (x) = m,
if
x ∈ [em , em+1 ) for
m ≥ 1.
It is immediate that
(
G(x) =
G∗ (x) + 1, if
x ∈ [ek , ek+1 )
G∗ (x),
x = ek+1
if
(k ≥ 1)
(k ≥ 1).
A Note on Sándor Type
Functions
Therefore,
G∗ (x) + 1 ≥ G(x) ≥ G∗ (x).
It can be easily verified that the function G∗ (x) satisfies the following properties:
1. G∗ (x) is surjective and an increasing function.
2. G∗ (x) is continuous for all x ∈ (e, ∞) \ A, where A = ek , k ≥ 1 and
since limx%ek G∗ (x) = k, limx&ek G∗ (x) = k − 1 for k ≥ 1, G∗ (x) is
continuous from the right at x = ek (k ≥ 1), but it is not continuous from
the left.
3. G∗ (x) is differentiable on [e, ∞) \ A, and since
G∗ (x) − G∗ (ek )
lim
= 0,
x − (ek )
x&ek
it has a right derivative at ek .
N. Anitha
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4. G∗ (x) is Reimann integtrable over [a, b] ⊂ R for all a ≤ b.
Also
Z
el
G∗ (x)dx = (e − 1)
ek
l−k
X
(ek + m − 1)(k + m − 1).
m=1
A Note on Sándor Type
Functions
N. Anitha
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2.
Main Result
The main purpose of this paper is to prove the following theorem:
Theorem 2.1.
π(x) ∼
x
.
G∗ (x)
Proof. To prove our theorem first we will prove that
G∗ (x) ∼ log x.
(2.1)
A Note on Sándor Type
Functions
By Stiriling’s formula [2] we have
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n! ∼ ce−n nn+1/2
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i.e.,
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cnn+1/2
e ∼
n!
n
Thus,
n
log e ∼ log
cnn+1/2
n!
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and hence,
n∼n+
1
log n + log c − log n!.
2
Also we have,
log(n!) ∼ n log n ⇒ n ∼ log n
(cf. [1], Lemma 2 ).
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J. Ineq. Pure and Appl. Math. 6(4) Art. 127, 2005
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If x ≥ e then x ∈ [en , en+1 ) for some n ≥ 1.
Since G∗ (x) = n if x ∈ [en , en+1 ), n ≥ 1, we have
n
G∗ (x)
n
≤
≤ .
n+1
log x
n
As
lim
n→∞
n
= 1,
n+1
we have
G∗ (x) ∼ log x.
A Note on Sándor Type
Functions
N. Anitha
From the prime number theorem it follows that
π(x) ∼
x
.
G∗ (x)
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3.
Remark
The following table compares the values of π(x) and
x
:
G∗ (x)
x
π(x)
x
G∗ (x)
10
100
1000
10000
100000
1000000
10000000
5
26
169
1230
9593
78499
664580
3.3333
20.00000
142.857143
1000
8333.3333
71428.571429
588235.294118
Now we prove the following theorem which is similar to Theorem 1.2.
Theorem 3.1. The series
∞
X
n=1
1
n[G∗ (n)]α
is convergent for α > 1 and divergent for α ≤ 1.
A Note on Sándor Type
Functions
N. Anitha
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Proof. By (2.1) we have
A log n ≤ G∗ (n) ≤ B log n
where (A, B ≥ 0) for n ≥ 1.
Therefore it is sufficient to study the convergence of the series
∞
X
n=1
1
.
n(log n)α
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To study the convergence of the above series we use the following result:
If φ(x) is positive for all positive ‘x’ and if
lim φ(x) = 0
x→∞
then the two infinite series
∞
X
φ(n) and
n=1
∞
X
an φ(an )
n=1
A Note on Sándor Type
Functions
behave alike for any positive integer ‘a’.
Therefore the two series
∞
X
n=1
1
n(log n)α
and
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∞
X
n=1
an
(an )[log (an )]α
behave alike.
However, the second series converges for α > 1 and diverges for α ≤ 1.
Hence the theorem is proved.
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References
[1] J. SÁNDOR, On an additive analogue of the function S, Notes Numb. Th.
Discr. Math., 7(2) (2001), 91–95.
[2] W. RUDIN, Principles of Mathematical Analysis, Third ed., Mc Graw-Hill
Co., Japan, 1976.
A Note on Sándor Type
Functions
N. Anitha
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