EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6387 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Effect of Variation in the Order of Zeta Function in Strip (0, 1) on the Upper Bound of Np(x) Zhwan Muhammed Amen1,∗, Faez Al-Maamori2, Mudhafar Fattah Hama1 1 Department of Mathematics, College of Science, University of Sulaimani, Sulaimaniyah 46001, Iraq 2 Department of Information Networks, College of Information Technology, University of Babylon, Babil, Iraq Abstract. During the third decade of the last century, Arne Beurling introduced the generalise primes as any increasing positive real sequence starting with a real number greater than 1 called ”Beurling primes”. Where the fundamental theorem of arithmetics gives Beurling integers. This work study Beurling’s prime systems and concentrates on the upper bound of Beurling zeta function in the region (0, 1). This reflects of course on the size of the error term of Beurling counting function of integers Np(x). 2020 Mathematics Subject Classifications: 11N80, 11M32 Key Words and Phrases: Beurling primes, Beurling integers, Beurling zeta function 1. Introduction The theory of numbers is one of the important branch in mathematics that deals with properties of counting number involving Riemann zeta function. Analytic number theory is that branch of number theory which deals with problems of integers in analytic way and some times to find approximate solutions of number theoritical functions where exact solutions are out of reach. Analytic number theory has a well known results on prime number called Prime Number Theorem which states that the number of primes less than x is about x log x . Since prime number theorem was proved in 1896, independently by Hadamard and de la Vallee Poussin [1] , Mathematitian have wondered which condition on the primes were really necessary to this kind of theorems. During the 1930’s Arne Beurling defined the idea of generalised prime numbers or (Beurling primes): any real sequence P = {p1, p2, p3, ......} satisfying 1 < p1 ≤ p2 ≤ p3 ≤ ., ., ., ., .,≤ pn ≤ ....... and pn −→ ∞ as n −→ ∞. So P called the generalised primes and also he ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6387 Email addresses: zhwan.amen@univsul.edu.iq (Z. M. Amen), faez@itnet.uobabylon@edu.iq (F. A. Al-Maamori), mudhafar.hama@univsul.edu.iq (M. F. Hama) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 2 of 15 formed the generalised integers N (Beurling integers) which is the product of the form N = ∏k i=1 p ai i , where k ∈ N and ai ∈ N ∪ {0}. Therefore, the generalised integer or Beurling integers can be formed by the sense of the Fundamental Theorem of Arithmetic. In this sense, Beurling generalises the notion of prime numbers and natural numbers. From the definition realising that the generalised prime need not be atcual prime , nor even integers. Beurling also defined πp(x) to be the counting function of generalised prime and Np(x) to be the counting function of generalised integers. Beurling also interested to find condition on N which let a prime number theorem holds. In 1937, Beurling proved [2] that if Np(x) = ax+O( x (log x)γ ) for some a ≥ 0 and γ greater than 3/2, then πp(x) ∼ x log x , this is called Beurling Prime Number Theorem. Lator on Diamond [3] modified the definition of counting function and zeta function using Beurling’s definitions. Since last century till now many authors have been dealing with Beurling (generalised) prime system such as Bteman and Diamond[4] and so many pa- pers of Diamond[3, 5–7] , Maliavin[8], Nyman[9], Hall[10], Kahane[11], Lagarias[12] and Zhang[13]. The major reason for that is related to the difficulities of prove or disprove of Riemann Hypothesis as a special case of Beurling generalised prime. This article introduces some concepts of generalised prime counting function πp(x) and concentrates on the behavior of Beurling zeta function ζp(s) in the strip (0, 1) and its effection on the error term of generalised integer counting function Np(x). 2. Preliminaries This section gives some basic concepts and properties that are needed for the aim of this paper. First of all the Chebyshev function which is equivalent to prime counting function [1] is let P be the set of actual prime, The Chebyshev counting function for any positive real x is defined to be ψ(x) = ∑( pk≤x ) log p. where k ∈ N and p ∈ P Prime counting function [1, 14] is a number prime less than or equal to x. That is π(x) = ∑ (p≤x) 1 is a counting function of primes, for a large value x, and also Counting function of integers [1] is N (x) = ∑ (n≤x) 1 for a large value of x, n ∈ N. Riemann zeta function has an important role in analytic number theory and distribu- tion of prime which defined by Riemann [1] as ζ(s) = ∑∞ n=1 1 ns for s ∈ C and Re(s) ≥ 1. Beurling, as we mentioned before, generalises the notion of prime number and the natural number. Beurling defined The generalised prime counting function [15] as πp(x) =∑ p≤x,p∈P 1 and generalised integer counting function Np(x) = ∑ n≤x,n∈N 1. Beurling also generalised the zeta function [15] as ζp(s) = ∑ n∈Np n−s when Re(s) > 1, s ∈ C which is Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 3 of 15 called Beurling zeta function and also has several definitions related to its relation with the counting function of primes and integers: (i) ζp(s) = ∫∞ 1 x−sdN (x), (ii) −ζp(s) ζp(s) = ∫∞ 1 x−sdψ(x), (iii) ζp(s) = exp ∫∞ 1 x−sdπp(x) The above definitions shows the link between ζp(s), πp(x) or ψp(x), and Np(x) which indicated that if we know the behavior of each one, say ψp(x), then we could know the behavior of ζp(s) and Np(x). The following basic definitions are introduced since this work is focusing on upper bound of generalised counting function Np(x). Definition 1. Big-O-Notation[1]. Let g(x) ≥ 0 for all x ≥ a. We write f(x) = O(g(x)) or f(x) � g(x) to mean that the quotient ∣∣∣f(x)g(x) ∣∣∣ is bounded for x ≥ a; that is, there exists a constant M > 0 such that |f(x)| ≤ Mg(x) for all x ≥ a. An equation of the form f(x) = O(g(x)) + h(x) means that f(x)− h(x) = O(g(x)). Definition 2. Asymptotic Notation [1]Let g(x) > 0 for all x ≥ a. If lim x−→∞ f(x) g(x) = 1 we say that f(x) is asymptotic to g(x) as (x→ ∞), and we write f(x) ∼ g(x) as (x→ ∞) . Definition 3. Big-Omega Notation [1] Let g(x) ≥ 0 for all x ≥ a. We write f(x) = Ω(g(x)) or f(x) � g(x) to mean that the quotient ∣∣∣f(x)g(x) ∣∣∣ is bounded for x ≥ a; that is, there exists a constant M > 0 such that |f(x)| ≥Mg(x) for all x ≥ a. Lemma 1. [16] Suppose ψp(x) = x + O(xα) for some α ∈ [0, 1), then ζp(s) has analytic continuation to the half-plane Hα = {s ∈ C : Re(s) > α} except for a simple pole at s = 1 and ζp(s) 6= 0 in this region. 3. Some Known Results Most of authers on this subject had been working on connecting the asymptotic behav- ior of the generalised prime systems and generalised integer counting function involving Beurling zeta function ζp(s) . We mean by the word {system} as follows: As Beurling definition satisfies for any real sequence the condition of Beurlings, this means that there are infinitly many real sequence of Beurling primes which indicate that there are infinitely many explanations of the counting functions and Beurling zeta function related to each other. So, if we assume that B is a set of all arithmetical functions. Let S0 = {h ∈ B : h(1) = 0} and S1 = {g ∈ B : g(1) = 1}, then the order pair (h, g) is called an outer generalised prime system. Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 4 of 15 Inspite of the fact that Beurling answered so many questions about prime number theorem and generalised prime and integer counting functions such as Beurling prime number theorem, there are many more questions regarding these functions. The generalised Riemann zeta fuction also has an important role in the connection of asymptotic behavior of πp(x) and Np(x) since in showing the link between these results the author might observe the Beurlinng zeta function ζp(s) is involved which means that The three functions are related to each others in the sense that an assumption made on πp(x) which is then showing to the property of ζp(s) (this is related to the size of zeta function along the vertical line) and this is then shown to imply a property of Np(x) and vice versa. Moving our attention to list some previous work which are relevant of this work. (i) In 1977, Diamond [7] showed the converse of Beurling’s prime number theorem as he stated: suppose that ∫∞ 2 t−2|Πp(t)− t log t |dt <∞, then there exist a positive constant c such that NP (x) ∼ cx as x→ ∞ (ii) In 1983, Landau [17] proved that if Np(x) = ax+O(xθ), (θ < 1) (1) then πp(x) = li(x) +O(xe−k √ log x) for some k > 0, where li(x) = ∫ x 2 dt log t (iii) In 2006, Diamond, Montogomery and Vorhauer [18] showed that Landau’s result is best possiblre. That is they proved that here is a discrete generalised prime system for which equation (1) holds but πp(x) = li(x) + Ω(xe−q √ logx) for some q > 0 (iv) In 1969, Malliavin [8] showed that for α ∈ (0, 1) and a, c > 0 Np(x) = ax+O(xe−c(log x)β ) implies Πp(x) = li(x) +O(xe−k(log x)α) for some k > 0 where β = 10α (v) In 1970, Diamond [3] showed the converse of Malliavin’s result, as he proved that if Πp(x) = li(x) +O(xe−c(logx)α) holds for α ∈ (0, 1) and some c > 0, then Np(x) = ψx+O(xe−b(log x log log x)β ) for some b > 0 where β = α 1+α . Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 5 of 15 (vi) In 1998, Balanzario [19] showed by example that there exists a continuous generalised prime system for which Πp(x) = li(x) +O(xe−(log x)α) and Np(x) = ρx+Ω± (xe−c(logx)β ) holds for some positive constants ρ and c with α = β = 1 2 . (vii) In 2006, Hilberdink[16] extended Diamond’s result in 6 to the case of α = 1 as follows: suppose ψp(x) = x + O(xα) for some α ∈ (0, 1) then there exists positive constants ρ and c such that Np(x) = ρx+O(xe−c √ log x log log x). (viii) In 2014, Al-Maamori [15] showed by example that there exists a continuous gener- alised prime system for which Πp(x) = li(x) +O(xe−(log x)α) and Np(x) = ρx+Ω± (xe−c(log x)β ) holds for some positive constants ρ and c with α = β. (ix) In 2015, Al-Maamori and Hilberdink [20] showed that Theorem 1. Suppose that for some α ∈ (0, 1) , ζp(s) has an analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β. Further assume that for some c < 1 ζp(σ + it) = O(tc), for some σ ≥ 1− 1 f(log t) where f is positive, strictly increasing continuous function, tending to infinity. Then for γ = 1− c, Np(x) = βx+O(xe− γ 2 h−1(γ−1 log x)) where h(u) = uf(u). The next section focus on the theorem (1) above. In particular, the following work con- centrating on the effect of the constant c in the order of Beurling zeta function on the upper bound of Np(x). Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 6 of 15 4. The Effect of the Constant c on the Upper Bound of Generalised Integr Counting Function Np(x) This work is studying the behavior of Np(x) when Beurling zeta function ζp(s) has known asymptotic behavior. There is no loss of generality if we rewrite the statement of the theorem(1) above in another style: (i) Suppose for some α ∈ (0, 1). (ii) ζp(s) has an analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β. (iii) For some c < 1, we have ζp(σ + it) = O(tc), for some σ ≥ 1− 1 f(log t) with f is positive, strictly increasing continuous function, tending to infinity. (iv) Then for γ = 1− c, Np(x) = βx+O(xe− γ 2 h−1(γ−1 log x)) where h(u) = uf(u). Our aim here is to show the effect of the constant c appearing in the order of Beurling zeta function on the error term of Np(x). For instance assume that theorem(1) exist for c = 1 2 . Its worthwhile to mention that we adapted the same strategy of the proof mentioned in [20, theorem 2.1,p 387], for the purpose of this work. It is more clear to rewrite the theorem (1) with c = 1 2 and one can see how the error term give different result. Theorem 2. Suppose that for some α ∈ (0, 1) , ζp(s) has an analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β. Further assume that for some c < 1 ζp(σ + it) = O(t 1 2 ), for some σ ≥ 1− 1 f(log t) where f is positive, strictly increasing continuous function, tending to infinity. Then for γ = 1 2 , Np(x) = βx+O ( x exp ( −1 4 h−1(2 log x) )) where h(u) = uf(u). Proof. By given assume that the upper bound of ζp(s) = O(t 1 2 ), and to find approxi- mate formula for Zp(x). We know by Perron’s formula [21], Np(y) = 1 2πi ∫ b+i∞ b−i∞ ζp(s)y s s ds, where b > 1 Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 7 of 15 Zp(x) = ∫ x 0 Np(y)dy = ∫ x 0 ( 1 2πi ∫ b+i∞ b−i∞ ζp(s)y s s ds ) dy = 1 2πi ∫ b+i∞ b−i∞ ζp(s)x s+1 s(s+ 1) ds, b > 1 pushing the contour to the left of the line Re(s) = b past the simple pole at s = 1, we get for any T > 0 Zp(x) = β 2 x2 + 1 2πi ∫ λT ζp(s) xs+1 s(s+ 1) ds+ 1 2πi ∫ b+iT 1− 1 f(logt) +iT ζp(s) xs+1 s(s+ 1) ds + 1 2πi ∫ 1− 1 f(logt) −iT b−iT ζp(s) xs+1 s(s+ 1) ds+ 1 2πi ∫ b+i∞ b+iT ζp(s) xs+1 s(s+ 1) ds (2) Figure 1: Contour λT Here λT is the contour s = 1− 1 f(log t) + it for a < |t| ≤ T and s = 1− 1 f(log a) + it for |t| ≤ a. The constant a is chosen such that a > e and 1− 1 f(log a) > α. The integration of the third term of the equation (2) on [1 − 1 f(log t) + iT, b + iT ] is equal to O ( xb+1 T 3/2 log x ) −→ 0 as T −→ ∞. Similarily the fourth term of the a equation (2) is also 0 when T −→ ∞. So, equation (2) becomes Zp(x) = β 2 x2 + 1 2πi ∫ λT ζp(s) xs+1 s(s+ 1) ds where λT is the contour s = 1 − 1 f(log t) + it for |t| > a > e and s = 1 − 1 f(log a) + it for |t| < a. Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 8 of 15 Therefore,∣∣∣Zp(x)− β 2x 2 ∣∣∣ = ∣∣∣∣ 1 2πi ∫ λT ζp(s) xs+1 s(s+1)ds ∣∣∣∣ = O (∫∞ a ∣∣∣ζp(1− 1 f(log t) −it) ∣∣∣ t2 .x 2− 1 f(log t)dt ) +O ( x 2− 1 f(log a) ) = O ( x2 ∫∞ log a exp [ log ζp(1− 1 f(log t) − it)− log t2 + log x − 1 f(log t)dt ]) +O ( x 2− 1 f(log a) ) = O ( x2 ∫∞ log a exp [ log T 1/2 − 2 log T − 1 f(log t) . log x ] dt ) +O ( x 2− 1 f(log a) ) . By using u = log t, to get: = O ( x2 ∫ ∞ log a exp [ − ( 1 2 u+ log x f(u) )] du ) +O ( x 2− 1 f(log a) ) By using some manipulations to the above integral to get:∫ ∞ loga exp [ − ( 1 2 u+ logx f(u) )] du = (∫ A loga + ∫ ∞ A ) exp [ − ( 1 2 u+ logx f(u) )] du for some A > log a. Where the first integral over (log a,A) is∫ A log a exp [ − ( 1 2 + log x f(u) ) du ] ≤ e − log x f(A) ∫ A log a e− 1 2 udu = O ( e − log x f(A) ) Whilest the second integeral over the interval (A,∞) is,∫ ∞ A exp [ − ( 1 2 u+ logx f(u) ) du ] ≤ ∫ ∞ A e− 1 2 udu = O ( e− 1 2 A ) Using the optimality for the two parts above to get: O ( e − log x f(A) ) = O ( e− 1 2 A ) − log x f(A) = −1 2 A which tells as that h(A) = Af(A) = 2 log x which means, A = h−1(2 log x). Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 9 of 15 Hence, ∣∣∣∣Zp(x)− β 2 x2 ∣∣∣∣ = O ( x2exp [ −1 2 h−1(2 log x) ]) (3) we know from given that the function Np is increasing function, so for every 0 < y < x, we have ∫ x 0 Np(u)du− ∫ x−y 0 Np(u)du = ∫ x x−y Np(u)du ≤ yNp(x) on the other hand,∫ x+y 0 Np(u)du− ∫ x 0 Np(u)du = ∫ x+y x Np(u)du ≥ yNp(x). Therefore, Zp(x)−Zp(x− y) y ≤ Np(x) ≤ Zp(x+ y)−Zp(x) y Using equation (3) the left hand side of the above inequality is = 1 y ( β 2 (x2 − (x− y)2) ) +O ( x2 exp [ −1 2 h−1(2 log(x− y)) ]) . That is the left hand side is, = 1 y β 2 ( x2 − (x2 − 2xy + y2) ) +O ( x2exp [ −1 2 h−1(2 log(x− y)) ]) (4) = 1 y (βxy − βy2 2 ) +O ( x2 exp [ −1 2 h−1(2 log(x− y)) ]) . Similarly, the right hand side is 1 y (βxy + βy2 2 ) +O ( x2 exp [ −1 2 h−1(2 log(x)) ]) Now for some ε > 0 and d > 0 we have, h(x)− h(x− d) = xf(x)− (x− d)f(x− d) (by given) = x(f(x)− f(x− d)) + df(x− d) ≥ ε > 0 This means that h(x)− ε ≥ h(x− d) , therefore with y = o(x) (since 0 < y < x) h−1(2 log(x− y)) ≥ h−1(2 log(x− ε)) ≥ h−1(2 log(x− d)) So replacing x− y by x of equation (4), we have for some m > 0 1 y ( βxy − βy2 2 +M ( x2 exp [ −1 2 h−1(2 log x) ])) (5) Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 10 of 15 Assuming B = −1 2h −1(2 log x), so formula (5) appears as: 1 y ( βxy − βy2 2 +M ( x2e B 2 )) Take y = xe B 2 then we have , 1 y ( βxy − βy2 2 +My2 ) = βx− βy 2 +My = βx+ y ( m− y 2 ) = βx+O(y) Hence, Np(x) = βx+O(y) where y = x exp ( −1 4h −1(2 log x) ) Np(x) = βx+O ( x exp ( −1 4 h−1(2 log x) )) . The above work shows that ”how the size of error term of Np(x) affected by changing the size of error term of ζp(s)”. Therefore, by using the same strategies by repeating the above proof, observing the following table: Table 1: Size of error term of Np(x) affected by c between (0, 1) ζp(s) Np(x) O(t0.1) βx+O ( x exp ( − 9 20h −1(109 log x) )) O(t0.5) βx+O ( x exp ( −1 4h −1(2 log x) )) O(t0.9) βx+O ( x exp ( − 1 20h −1(10logx) )) From the above table, one can see that the error terms of Np(x) is always negative when 0 < c < 1 and get smaller as c get closer to 0 and it gets bigger as c closer to 1. Now the interseting point is that what will happen to the size of error term of Np(x) when c > 1. First, we will find the error term of Np(x) when c = 3 2 . The proof has the same step untill we get to the following step: ∣∣∣∣Zp(x)− β 2 x2 ∣∣∣∣ = O ( x2 ∫ ∞ log a exp [ − ( (1− c)u+ log x f(u) )] du ) +O ( x 2− 1 f(log a) ) since c = 3 2 , then 1− c = −1 2 , so the above equation becomes ∣∣∣∣Zp(x)− β 2 x2 ∣∣∣∣ = O ( x2 ∫ ∞ log a exp [ − ( −1 2 u+ log x f(u) )] du ) +O ( x 2− 1 f(log a) ) Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 11 of 15 After doing some manipulations one could see that the first integral over (log a,A) is :∫ A log a exp [ − ( −1 2 u+ log x f(u) )] du = O ( e − log x f(A) ) . ∫ ∞ A exp [ − ( −1 2 u+ log x f(u) )] du = O ( e 1 2 A ) Similarly choosing A opitamally such that O − terms will be: O ( e − log x f(A) ) = O ( e 1 2 A ) A = h−1(−2 log x) Hence ∫ ∞ log a exp [ − ( −1 2 u+ log x f(u) )] du = O ( exp ( 1 2 h−1(−2 log x) )) and Np(x) = βx+O ( x exp ( 1 4 h−1(−2 log x) )) From the above result, one can see that the error term of Np(x) where 1 < c < 2 is positive and big. Its remain here to mention what is the effection of c = 1 and c = 2 on the error term of Np(x) by the following lemmas: Lemma 2. Suppose ζp(s) has an analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β, and ζp(σ+ it) = O(tc) where c = 1 and σ ≥ 1− 1 f(log t) , then for γ = 1− c , Np(x) diverges. Proof. Since c = 1, then γ = 0. after the same calculation as c = 1/2, we get the following equation ∣∣∣∣Zp(x)− β 2 x2 ∣∣∣∣ = O ( x2 ∫ ∞ log a exp(− log x f(u) )du ) as f is increasing and goes to infinity, we get O ( x2 ∫ ∞ log a 1du ) −→ ∞ Hence Np(x) diverges. Lemma 3. Suppose ζp(s) has an analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β, and ζp(σ+ it) = O(tc) where c = 2 and σ ≥ 1− 1 f(log t) , then Np(x) diverges. Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 12 of 15 Proof. Since the third term of equation (2) is O ( xb+1 log x ) which is constant and∣∣∣∣Zp(x)− β 2 x2 ∣∣∣∣ = O ( x2 ∫ ∞ log a x − 1 f(u) exp(u)du ) as f is increasing and goes to infinity, we get O ( x2 ∫ ∞ log a exp(u)du ) −→ ∞ Hence Np(x) diverges. Remark 1. For c > 2, Np(x) goes to ∞ since the third term of equation (2) goes to ∞ which leads to Np(x) to be ∞. It’s worthwhile to mention again that the effection of c on the error term of Np(x) is positive and big when 1 < c < 2, but by adding the condition for f to be even in theorem (1) the error term will decrease as shown in the following lemma Lemma 4. Suppose for some α ∈ (0, 1), ζp(s) has analytic continuation to the half plane Hα except for a simple pole at s = 1 with residue β. Furthermore assume that for 1 < c < 2, ζp(s) = O(tc) for σ ≥ 1− 1 f(log t) , where f is positive, strictly increasing continuous, even function and tends to infinity then Np(x) = βx+O ( x exp ( + γ 2 h−1(k log x) )) , where h(u) = uf(u), γ = 1− c < 0 and k = −γ−1. Proof. By theorem (1) Np(x) = βx+O ( x exp ( −γ 2h −1(γ−1 log x) )) for 0 < c < 1 If 1 < c < 2, then γ = −(1− c) > 0 and γ−1 = 1 1−c < 0 Let −k = γ−1 where k > 0 and h−1 is odd (since f is even). Hence Np(x) = βx+O ( x exp ( −γ 2 h−1(−k log x) )) = βx+O ( x exp (γ 2 h−1(k log x) )) Where γ = 1− c < 0 for 1 < c < 2 The following is the counter example to apply theorem(2). Example 1. Let f(x) = xn log x . Then h(x) = xn+1 log x , h−1(x) ∼ n+1 √ xn log x n+1 , and h−1(log x) ∼ n+1 √ (log x)n log log x n+1 Now if c = 1/2, we have ζp(σ + it) = O(t1/2) for σ ≥ 1− log log t n log t and Np(x) = ρx+O ( x exp ( −1 4 n+1 √ (2 log x)n 2 log log x n+ 1 )) . Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 13 of 15 5. The Connection Between the Constant c and the Real Part σ of Beurling Zeta Function ζp(s) This part concentrated on figuring out the connection between constant c in the order of Beurling zeta function ζp(s) and σ(the real part of Beurling zeta function ζp(s)). In other meaning the reader can see earlier thatt there is a connection between zeta function and beurling integer counting function, so we are interested to figure out this type of connection. fore more details the reader could see ([20], pp.392). Our aim is to apply Theorem (2) and have to show for which region of σ, ζ0(σ + it) = O(t1/2). So in order for |ζ0(σ + it)| � tc to hold for c = 1 2 , we have the following: |ζ0(σ + it)| � exp ( 1 + t100(1−σ) 3 2 (log t) 2 3 ) So, we get exp [( 1 + t100(1−σ) 3 2 ) (log t) 2 3 ] ≤ t 1 2( 1 + t100(1−σ) 3 2 ) (log t) 2 3 ≤ log t 1 2( 1 + e100(1−σ) 3 2 log t ) ≤ 1 2 (log t) 1 3 e100(1−σ) 3 2 log t ≤ 1 + e100(1−σ) 3 2 log t ≤ 1 2 (log t) 1 3 so e100(1−σ) 3 2 log t ≤ 1 2 (log t) 1 3 log ( e100(1−σ) 3 2 log t ) ≤ log ( 1 2 (log t) 1 3 ) 100(1− σ) 3 2 log t ≤ log( 1 2 ) + log ( (log t) 1 3 ) (1− σ) 3 2 ≤ log 1 2 + 1 3 log log t 100 log t log ( e100(1−σ) 3 2 log t ) ≤ log ( 1 2 (log t) 1 3 ) 100(1− σ) 3 2 log t ≤ log( 1 2 ) + log ( (log t) 1 3 ) (1− σ) 3 2 ≤ log 1 2 + 1 3 log log t 100 log t Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 14 of 15 (1− σ) ≤ ( log 1 2 + 1 3 log log t 100 log t ) 2 3 −σ ≤ −1 + ( log 1 2 + 1 3 log log t 100 log t ) 2 3 σ ≥ 1− ( log 1 2 + 1 3 log log t 100 log t ) 2 3 Hence for the above σ, we have ζ0(σ + it) = O(t 1 2 ). One can show the connection between σ and 0 < c < 2 by the following table: Table 2: The Effect of c on σ c ζ0(σ + it) σ 1 10 O(t0.1) ≥ 1− ( log 1 10 + 1 3 log log t 100 log t ) 2 3 1 2 O(t0.5) ≥ 1− ( log 1 2 + 1 3 log log t 100 log t ) 2 3 9 10 O(t0.9) ≥ 1− ( log 9 10 + 1 3 log log t 100 log t ) 2 3 3 2 O(t 3 2 ) ≥ 1− ( log 3 2 + 1 3 log log t 100 log t ) 2 3 From the above table one can see that when c get closer to 1,the region of σ get smaller and also when 1 < c < 2, the region of σ get smaller and smaller. 6. Conclusion In conclusion, the purpose of this work was to concentrate on the impact of the order of Beurling zeta function ζp(s) on the size of error term of Beurling integer counting function Np(x). In particular, we discovered by changing the interval of constant c in Theorem (1), different results has been achieved. First, when constant 0 < c < 1 the error term of Np(x) has different explanation. Second if constant 1 < c < 2, then the error term of Np(x) is big as it shown in Section (4). But then by adding some condition to Theorem (1), the error term of Np(x) became smaller as it shown in lemma (4).Further more, when constant c = 1 and c ≥ 2, Np(x) diverges. In Addition, this work also focused on the connection between constant c in the order of ζp(s) and the real part σ of ζp(s). Z. M. Amen, F. A. Al-Maamori, M. F. Hama / Eur. J. Pure Appl. Math, 18 (4) (2025), 6387 15 of 15 References [1] T. M. Apostol. Introduction to Analytic Number Theory. Springer, 1976. [2] A. Beurling. Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. Acta Mathematica, 68:255–291, 1937. [3] H. G. Diamond. Asymptotic distribution of beurling’s generalised integers. Illinois Journal of Mathematics, 14:12–28, 1970. [4] P. T. Bateman and H. G. Diamond. Asymptotic distribution of beurling’s gener- alised prime numbers. In Studies in Number Theory, pages 152–212. Mathematical Association of America, 1969. [5] H. Diamond. The prime number theorem for beurling’s generalised numbers. Journal of Number Theory, 1:200–207, 1969. [6] H. G. Diamond. A set of generalised numbers showing beurling’s theorem to be sharp. Illinois Journal of Mathematics, 14:29–34, 1970. [7] H. Diamond. When do beurling’s generalised integers have density? Journal für die reine und angewandte Mathematik, 295:22–39, 1977. [8] P. Malliavin. Sur le reste de la loi asymptotique de répartition des nombres premiers généralisés de beurling. Acta Mathematica, 106:281–298, 1961. [9] B. Nyman. A general prime number theorem. Acta Mathematica, 81:299–307, 1949. [10] R. S. Hall. Theorems about Beurling generalised prime and associated zeta function. PhD thesis, University of Illinois, USA, 1967. [11] J. P. Kahane. Sur les nombres premiers généralisés de beurling. Journal de Théorie des Nombres de Bordeaux, 9:251–266, 1997. [12] J. C. Lagarias. Beurling generalised integers with the delone property. Forum Math- ematicum, 11:295–312, 1999. [13] W. Zhang. Beurling primes with rh, beurling primes with large oscillation. Mathe- matische Annalen, 337:671–704, 2007. [14] C. Cesarano, W. Ramirez, and S. Diaz. New results for degenerated generalised apostol-bernoulli, apostol-euler and apostol-genocchi polynomials. WSEAS Transac- tions on Mathematics, 21:604–608, 2022. [15] F. Al-Maamori. Examples of beurling prime systems. Mathematica Slovaca, 67:321–344, 2014. [16] T. Hilberdink and L. Lapidus. Beurling zeta functions, generalised primes, and fractal membranes. Acta Applicandae Mathematicae, 96:21–48, 2006. [17] A. Landau. Neuer beweis des primzahlsatzes und beweis des primidealsatzes. Math- ematische Annalen, 56:645–670, 1903. [18] H. Montgomery, H. Diamond, and U. Vorhauer. Beurling primes with large oscillation. Mathematische Annalen, 334:1–36, 2006. [19] E. P. Balanzario. An example in beurling’s theory of primes. Acta Mathematica, 87:121–139, 1998. [20] F. Al-Maamori and T. Hilberdink. An example in beurling’s theory of generalised primes. Acta Arithmetica, 168:383–395, 2015. [21] E. C. Titchmarsh. The Theory of Functions. Wiley, New York, 2 edition, 1985.