The Behaviors of some Counting Functions of ‎g-primes and g-integers as x goes to Infinity‎

 في هذا البحث نركز على تصرفات الدوال الحسابية الموسعة للأعداد الاولية (x) وللأعداد الصحيحة   (x) وكذلك الرابط بينهما عندما x      . هنا دالة ريمان زيتا    (s) ( =  , (s)  > 1 ), تلعب دورا مهما كرابط بين (x) و   (x) . هذا العمل سيتم من خلال سلوك طريقة العالم بلنزاريو ]بلنزاريو،  1998 [   ليست بالتفاصيل والتي عممت من خلال المعموري ] المعموري ، 2013 [ . بالنهاية سوف نرسم مخططا يحدد العلاقة بين  و  ( حيث  و  هما القوى للحدود الخطأ H1(x)  و H2(x) من (x) و   (x) على التوالي . الغرض من هذا البحث هو تحليل تصرفات (x) و   (x) عندما x      . ملاحظة : من المهم والنافع الاشارة بان جهدنا في هذا البحث ليست تغيير بعض قيم الدوال التي استخدمت في طريقة بلنزاريو حيث ان تغيير اي قيمة مهما كانت صغيرة لإحدى دوال طريقة بلنزاريو ربما تقودنا الى خسارة هدف الموضوع بأكمله . ولهذا نبين ايضا قابلية التغيير المسموح بها في قيم بعض الدوال . كذلك سوف نختم البحث بفتح باب لعمل مستقبلي  In this article  we  focus on the behaviors of  the generalised counting  function of primes (x)  and  the counting  function of integers   (x) as well as  the link between them as  x      . Here the Riemann zeta function  (s) ( =  , (s)  > 1 )  play an  important  role  as  a link between   (x)  and  (x)  .  This  work  will  go  through  the  method  ( not  in  details )  adapted  by Balanzario  [Balanzario , 1998]   and  later  generalised  by  AL- Maamori [AL- Maamori , 2013 ] . Finally we shall draw a diagram in order to determine the relation between   and    , (where  and   are the power of the error terms H1(x) , H2(x) of (x) and (x) respectively) . The aim of this work is to analysis  the behaviour of (x)  and   (x) as  x    .   Note that : ʺ  It’s a beneficial to point out that our effort in this paper is not to exchange the values of some functions of  Balanzarioʹs  method . Since , changing any small value of one of the functions of  Balanzarioʹs method may be leads to loss the aim of the work  ʺ  . Therefore , in this article we show  the ability of  changing  the values of  some functions and in which places in the proof we should sort out

Using.Liouville’s function for Creating a weird numbers from Reals

During 1937 Beurling Showed that any positive infinitely increasing real sequence such that the its first element precisely greater than one, called a Beurling’s primes. Furthermore, the series of Beurling integers (or generalized integers) can be constructed using the fundamental theorem of arithmetic. During the seventieth of the last century, Diamond showed that majority of the arithmetical functions were generalized to deal with the generalization of the primes and integers. This work aims to create some weird numbers from a large enough reals So, the reader has to be familiar with Mobius inversion formula of the Pci function. The challenging of this work is the dealing with an algorithm for generating a weird numbers (or maybe a primitive weird numbers) from a large enough real numbers x. The idea of this work can be used for an application of modeling, data simulation and security subjects

Some Properties of Mobius Function Graph M(−1) in Graphs

In this work, a new graph is called the Mobius Function Graph is introduced. Three ways of determining the prime- counting function by using this graph are presented. Also, some properties of this function are proved. Moreover, the domination, independence, chromatic, and clique number of this graph are determined. Finally, the relationship between the domination number and the independence number is discussed