In Predictable Outcome of Some Complex
Function on Space
Hayder Kadhim Zghair
Department of Software, College of Information Technology
University of Babylon, Babylon
hyderkkk@yahoo.com
Samah Abdl hadi
Department of Computer Science , College Science for Women
University of Babylon, Babylon
samah_hadi@yahoo.com
Abstract
Teh Main advantage of this work is to concentrate on teh outcome of teh function
as
a function of ( ) , (
) . This type of work has been studied deeply by [HIL , 12] .
Here , we see dat teh outcomes of teh function are depending on teh of teh function ( ) . More deeply , s value will appear as a matrix of zero triangle values depending 0n teh positive of on teh real line .
Key words: Dirichlet function, Functional analysis.
ةصلاخلا
()
ثيح(
ةلادلا تاجرخم ىمع زيكرتلا يه لمعلا اذه نم ةيساسلاا ةعفنملا
ةللادب ةلادك
)ةيقيقح دادعأ
ةلادلا تاجرخم نأب ىرن انه
[HIL, 12 ].
كندريبمه ملاعلا لبق نم اقيمع سرد عوضوملا اذه
نوكت
ميق اقمعت رثكأ
()
عقوم ىمع ةدمتعم ميقلا ةيرفص ةيثمثم تاذ ةفوفصمك رهظت فوس
. دادعلاا طخ ىمع
ىمع ةدمتعم
.هيلشيرد ةلاد ،يلادلا ليلحتلا :ةيحاتفملا تاممكلا
1. Introduction
Firstly, we start in this introduction by viewing some definitions of teh arithmetical functions and norm function [AI, 85]. Secondly, we assert some of basic (known) theorems wifout proof related to teh upper and lower bounds of ( )
An arithmetical function is a function . Denote by teh set of all arithmetical
functions. For
we have
and are also in . More
,
importantly, for any arithmetical functions we define by
?
./
(
)( )
()
is also in Teh sum here is over all divisors of . Dirichlet convolution is commutative. This follows from teh sum
?()()
?
(
)( )
()()
where teh sum is over all possible positive integers such dat also associative since
It is
?
(
)( )
()()()
In fact ( ) is an algebra where is distributive wif respect to and
(
)
(
)
(
) for every
Now, we move our attention to
define some arithmetical functions. We start wif teh divisor function ( )
2061Journal of University of Babylon, Pure and Applied Sciences, Vol.(25), No.(5), 2017.
1-1 Definition :-
For , we define teh function ( ) to be teh number of divisors of We write this as ( ) ? , where teh sum ranges over all teh divisors of
1-2 Definition :- [Al,1985] For we defin
?
()
Here teh above summation converges absolutely and locally uniformly for
(see [AI, 85]). Moreover, ( ) has an analytic continuation to teh whole complex plane except for a simple pole at wif residue and is of finite order which means
()
dat
where depends on teh real part of (see
for some
[LHIL, 06]).
Teh function ( ) is a function of complex variable dat has been studied by B. Riemann (1826-1866). their is an important link between ( ) and teh prime numbers. This is clear from teh following:
?( )
()
Here, product holds for teh real part of must greater TEMPthan .
()
is an important subject and has a
Teh zeros of
for
significance conjectures (see [AI, 85]). Riemann showed dat teh frequency of prime numbers is very closely related to teh behavior of teh zeros of ( ) He conjectured dat all non-trivial zeros of ( ) lies on a line have teh real part .
we have
1-3 Theorem (Basic Properties): For
(a) ( ) is analytic in ?>1.
(b) ( ) has an analytic continuation to teh half plane except at teh simple pole wif residue . We mean by Analytic continuation dat for
we have
?*+
()
(c) ( ) has an Euler product representation for
?(
)
()
(d) ( ) has no zeros for
For teh proof of teh above axioms see for example [Al,1976].
Remark: For more details and other (basic) information about ( ) see [TA, 1976], [MN,2005] and [PB, 2004].
As a part of this investigation, we need some functions from teh functional analysis (which we use later on to prove teh main theorem). We start wif teh definition of teh space
2061Journal of University of Babylon, Pure and Applied Sciences, Vol.(25), No.(5), 2017.
{(
}
?
)
and we call it space. Let teh function
be a linear mapping defined as
. / . teh important question here is :
(
)
?
(
) where
follows
When is teh function
bounded? From teh work which has been done in
[HIL, 09], we understand dat is bounded if and only if in which case ? ? ( ) To see this, we have
|? . /
|
|?
| ) (? Hence,
By Cauchy–Schwarz inequality we have teh last term is
)
(?
theirfore, we have
?
This means dat
) (?
) ()
(?
?
( )?
?
( ( )) ?
? ? ( ( )) ? ?
Which tells us dat ? ( )? ( )? ?, and from teh knowledge of ( ) we see dat
teh function is bounded for and ?
is linear mapping on
?
()
?
We see here dat
where are coefficients . dat is
)
(
(
)
(
) bounded?
Moreover , it is straight forward , dat
In this article we ask an important question: When is
We answer teh above question after teh following theorem. We remark here dat their is no loss if we mention teh proof of teh following theorem in order to show dat in which part (of teh Euclidian plane) dat should be bounded. dat is, teh strict lower bound of teh Riemann-zeta function will enable us to determine teh area of boundedness of .
2061Journal of University of Babylon, Pure and Applied Sciences, Vol.(25), No.(5), 2017.
2-1 Theorem:
( ), ( ) we have for
Let be a complex number. Then for
teh imaginary part of runs between and dat teh maximum of
{
}
where is a small constant and sufficiently large independent of . Here ( )
.
Note dat: More generally, teh above theorem is valid for any function ( ) as In order to prove teh above theorem we need teh following Lemma.
2-2 Lemma:
we have
For
{
}
?
For . Proof:
Using [HIL,2012], we see for ( ) ? dat
?
?
()
So, by teh Prime Number Theorem (which means teh assertion dat teh number of
primes dat ( )(
?
) )), we observe teh last integral is greater TEMPthan
(
)
(
)
For some constants Teh result follow immediately. Now we could start prove teh theorem.
3. Proof (of teh theorem)
For any real running between and teh line
(any
), we see (by
[AIL,2012]) dat for teh imaginary part of s runs between 1 and X teh maximum of
(? )
()
??
where is in and ? . / We note dat if we take to be as large as possible and define
{(?
}
) 2061Journal of University of Babylon, Pure and Applied Sciences, Vol.(25), No.