A new generalized prime random approximation procedure and some of its applications
aa r X i v : . [ m a t h . N T ] F e b A NEW GENERALIZED PRIME RANDOM APPROXIMATIONPROCEDURE AND SOME OF ITS APPLICATIONS
FREDERIK BROUCKE AND JASSON VINDAS
Abstract.
We present a new random approximation method that yields the existence ofa discrete Beurling prime system P = { p , p , . . . } which is very close in a certain precisesense to a given non-decreasing, right-continuous, nonnegative, and unbounded function F .This discretization procedure improves an earlier discrete random approximation methoddue to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1–36], andrefined by W.-B. Zhang [Math. Ann. 337 (2007), 671–704].We obtain several applications. Our new method is applied to a question posed by M.Balazard concerning Dirichlet series with a unique zero in their half plane of convergence,to construct examples of very well-behaved generalized number systems that solve a recentopen question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05(2020), 1005–1011], and to improve the main result from [Adv. Math. 370 (2020), Article107240], where a Beurling prime system with regular primes but extremely irregular integerswas constructed. Introduction
In their seminal work [5], Diamond, Montgomery, and Vorhauer established the optimalityof Landau’s abstract prime number theorem [10], partly solving so a long-standing conjec-ture of Bateman and Diamond [1, Conjecture 13B, p. 199]. One of the cornerstones in theirarguments is a probabilistic construction, which they developed in order to produce discreteapproximations to ‘continuous prime distribution functions’ by random generalized primes.Refinements were obtained by Zhang in [15] (cf. [6]). Their discrete random approximationresult, from now on referred to as the DMVZ-method, may be summarized as follows. Theorem 1.1 (Diamond, Montgomery, Vorhauer [5], Zhang [15]) . Let f be a non-negative L loc -function supported on [1 , ∞ ) satisfying (1.1) f ( u ) ≪ u and Z ∞ f ( u ) d u = ∞ . Mathematics Subject Classification.
Key words and phrases.
Discrete random approximation; Diamond-Montgomery-Vorhauer-Zhang proba-bilistic method; Dirichlet series with unique zero in half plane of convergence; Well-behaved Beurling primesand integers; Beurling integers with large oscillation; Riemann hypothesis for Beurling numbers.F. Broucke was supported by the Ghent University BOF-grant 01J04017.J. Vindas was partly supported by Ghent University through the BOF-grant 01J04017 and by the ResearchFoundation–Flanders through the FWO-grant 1510119N. Landau’s original statement is the well-known Prime Ideal Theorem, but his reasoning essentially leadsto the first ever known abstract PNT [1, 9].
Then there exists an unbounded sequence of real numbers < p < p < · · · < p j < . . . suchthat for any t and any x ≥ (cid:12)(cid:12)(cid:12)(cid:12) X p j ≤ x p − i tj − Z x u − i t f ( u ) d u (cid:12)(cid:12)(cid:12)(cid:12) ≪ √ x + s x log( | t | + 1)log( x + 1) . The sequence arising from the DMVZ-method might be regarded as a Beurling primesystem. Indeed, following Beurling [2] (cf. [1, 6]), a set of generalized prime numbers is simplyan unbounded non-decreasing sequence of real numbers P = { p j } ∞ j =1 subject to the onlyrequirement p >
1. We denote as π P ( x ) the function that counts the number of generalizedprimes not exceeding a given number x . The function f can then be interpreted as a template‘prime density measure’ d F ( u ) = f ( u ) d u , whose continuous ‘prime distribution function’ F ( x ) = R x f ( u ) d u is unbounded and satisfies the Chebyshev upper bound ≪ x/ log x . Theimportance of the bound (1.2) lies in the fact that it is often strong enough for transferringmany properties from exp (cid:0)R ∞ x − s d F ( x ) (cid:1) into desired analytic properties of the Beurlingzeta function associated to P , that is, ζ P ( s ) = ∞ Y j =1 (1 − p − sj ) − . We refer to the monograph [6] and the recent article [4] for relevant applications of theDMVZ-method.The main goal of this paper is to establish a direct improvement to the DMVZ-method byobtaining a significantly stronger bound for the difference π P − F than the one delivered byTheorem 1.1. In fact, setting t = 0 in (1.2) yields π P ( x ) − F ( x ) ≪ √ x . We will show that it ispossible to select the sequence P in such way that the much better bound π P ( x ) − F ( x ) ≪ F that are not necessarily absolutelycontinuous with respect to the Lebesgue measure. Theorem 1.2.
Let F be a non-decreasing right-continuous function tending to ∞ , with F (1) = 0 and satisfying the Chebyshev upper bound F ( x ) ≪ x/ log x . Then there exists aset of generalized primes P = { p j } ∞ j =0 such that (cid:12)(cid:12) π P ( x ) − F ( x ) (cid:12)(cid:12) ≤ and such that for any t and any x ≥ (cid:12)(cid:12)(cid:12)(cid:12) X p j ≤ x p − i tj − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ √ x + s x log( | t | + 1)log( x + 1) . If in addition F is continuous, the sequence P can be chosen to be (strictly) increasingand such that (cid:12)(cid:12) π P ( x ) − F ( x ) (cid:12)(cid:12) ≤ . The proof of Theorem 1.2 will be given in Section 2. The essential difference between theDMVZ probabilistic scheme and our proof is that we make a completely different choice ofhow the generalized prime random variables are distributed in order to generate the discreterandom approximations, allowing for a more accurate control on the size of the difference π P ( x ) − F ( x ).The rest of the article is devoted to illustrating the usefulness of Theorem 1.2 throughthree applications. In all these applications, the stronger bound π P ( x ) − F ( x ) ≪ NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 3 of π P ( x ) − F ( x ) ≪ √ x plays a crucial role. Our first application concerns a question posedby M. Balazard (we consider a strengthened version of [12, Open Problem 24]): Question 1.3.
Does there exist a Dirichlet series P ∞ n =1 a n n − s which has exactly one zero inits half plane of convergence?This question is motivated by the fact that if the Riemann hypothesis is true, the Dirichletseries(1.4) ∞ X n =1 µ ( n ) n s = 1 ζ ( s ) , where µ is the (classical) M¨obius function, provides an example of such a Dirichlet series, sinceit would have a unique zero, namely at s = 1, in its half plane of convergence { s : Re > / } .The idea is of course to find an unconditional example. We are not able to answer Question1.3 here for Dirichlet series as in its statement, but, armed with Theorem 1.2, we will provethat Balazard’s question can be affirmatively answered for general Dirichlet series.
Proposition 1.4.
There are an unbounded sequence n < n ≤ n ≤ · · · ≤ n k ≤ . . . and a general Dirichlet series of the form D ( s ) = ∞ X k =0 a k n sk , with a k ∈ {− , , } , such that D ( s ) has abscissa of convergence σ c = 1 / and has a unique zero on { s : Re > / } ,which is located at s = 1 . Our example of a general Dirichlet series satisfying the requirements of Proposition 1.4arises from a Beurling prime system that we shall construct in Section 3. This exampleis actually the Beurling analog of the Dirichlet series (1.4). It turns out that the sameconstructed generalized primes yield a second application, as this generalized number systemalso provides a positive answer to a recent open problem raised by Hilberdink and Neamah(cf. [14, Section 4. Open Problem (1)]) on the existence of well-behaved Beurling numbersystems of a certain best possible type; see Section 3 for details.As a third application, we conclude this article with an improvement to a result recentlyestablished by the authors together with G. Debruyne [4, Theorem 1.1]. In that paper,the authors showed the existence of a prime system with regular primes (i.e., satisfying theRH in the form of the PNT with remainder O ( √ x )) but with integers displaying extremeoscillation. Using Theorem 1.2, we will be able to improve in Section 4 the regularity ofthe primes from an O ( √ x ) error term to one of order O (1), while still keeping the sameirregularity of the integers. 2. The main result
This section is devoted to a proof of Theorem 1.2. Like in the DMVZ-method, our startingpoint is a probabilistic inequality for sums of random variables essentially due to Kolmogorov(see e.g. [11, Chapter V]), which we shall employ to bound the probability of certain events.The following inequality is a slight variant of [5, Lemma 8], and the proof given there canreadily be adapted to yield the ensuing form of the lemma.
F. BROUCKE AND J. VINDAS
Lemma 2.1.
For ≤ j ≤ J , let X j be independant random variables with E ( X j ) = 0 , (cid:12)(cid:12) X j (cid:12)(cid:12) ≤ , and Var( X j ) = σ j < ∞ . Let S = P Jj =1 X j , and σ = P Jj =1 σ j = Var( S ) . Then P ( S ≥ v ) ≤ exp (cid:18) − v σ (cid:19) if v ≤ u σ ;exp (cid:18) − u v (cid:19) if v > u σ . Here u is the positive solution of e u = 1 + u + u , u ≈ . .Proof of Theorem 1.2. Write d F = d F c + d F d , where d F c is a continuous measure, and d F d is purely discrete: d F d = ∞ X n =1 α n δ y n , y n > , α n > , where δ y denotes the Dirac measure concentrated at y and the sequence { y n } ∞ n =1 consistsof distinct points. We will discretize both measures separately . Let us start with thecontinuous part.Set q = 1, q j = min { x : F c ( x ) = j } , for j < j max , where j max = ∞ if F c ( ∞ ) = ∞ ,and j max = ⌊ F c ( ∞ ) ⌋ + 1 if F c ( ∞ ) < ∞ . Let { P j } ≤ j 1. Let C be a constant such that F c ( x ) ≤ C x log( x + 1) . Let J < j max and suppose that q J / log( q J + 1) ≥ log( | t | + 1). Set x = q J and let D =max {√ C, /u } , where u is the number appearing in Lemma 2.1. Applying that lemmato the random variables X j,t − E ( X j,t ), with v = D ( √ x + p x log( | t | + 1) / log( x + 1)), weget P J X j =1 cos( t log P j ) − Z x cos( t log u ) d F c ( u ) ≥ D (cid:18) √ x + s x log( | t | + 1)log( x + 1) (cid:19)! ≤ max exp (cid:18) − D σ (cid:18) x + x log( | t | + 1)log( x + 1) (cid:19)(cid:19) ; exp (cid:18) − u D (cid:18) √ x + s x log( | t | + 1)log( x + 1) (cid:19)(cid:19) . Here σ = J X j =1 Var( X j,t ) ≤ J = F c ( x ) ≤ C x log( x + 1) and r x log( x + 1) ≥ p log( | t | + 1) . d F d is already a purely discrete measure, but does not necessarily arise as the prime counting measureof a discrete Beurling prime system, since { y n } ∞ n =1 may have accumulation points, and since, even if thissequence happens to be discrete, we do not assume that the α n are integers. NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 5 Hence the above probability is bounded by ( x + 1) − ( | t | + 1) − . Applying the same argumentto the random variables − X j,t , ± Y j,t = ± sin( t log P j ), we get the same bounds for thecorresponding probabilities.Let S ( x, t ) = X P j ≤ x P − i tj , S c ( x, t ) = Z x u − i t d F c ( u ) . Then for x = q J with x/ log( x + 1) ≥ log( | t | + 1) P (cid:12)(cid:12) S ( x, t ) − S c ( x, t ) (cid:12)(cid:12) ≥ √ D (cid:18) √ x + s x log( | t | + 1)log( x + 1) (cid:19)! ≤ x + 1) ( | t | + 1) . Let j t = min { j < j max : q j / log( q j + 1) ≥ log( | t | + 1) } , where we set j t = ∞ when the set isempty (which may happen if j max < ∞ and t is sufficiently large). Let A k,j denote the event (cid:12)(cid:12) S ( q j , k ) − S c ( q j , k ) (cid:12)(cid:12) ≥ √ D ( √ x + p x log( | t | + 1) / log( x + 1)). Since ∞ X k =1 X j k ≤ j Suppose now that k ≥ ≤ j < j max , x ∈ ( q j − , q j ]. Then S ( x, k ) = S ( q j − , k ) + O (1) = S c ( q j − , k ) + O (cid:18) √ q j − + s q j − log( k + 1)log( q j − + 1) (cid:19) + O (1)= S c ( x, k ) + O (cid:18) √ x + s x log( k + 1)log( x + 1) (cid:19) . If j max < ∞ and x > q j max , then S ( x, k ) = S ( q j max , k ) = S c ( q j max , k ) + O (cid:18) √ q j max + s q j max log( k + 1)log( q j max + 1) (cid:19) = S c ( x, k ) + O (cid:18) √ x + s x log( k + 1)log( x + 1) (cid:19) . If t ∈ [ k, k + 1] for some k ≥ 0, then by integration by parts, S ( x, t ) = Z x + u − i( t − k ) d S ( u, k ) = S ( x, k ) x − i( t − k ) + i( t − k ) Z x S ( u, k ) u − i( t − k ) − d u = S c ( x, k ) x − i( t − k ) + i( t − k ) Z x S c ( u, k ) u − i( t − k ) − d u + O (cid:18) √ x + s x log( t + 1)log( x + 1) (cid:19) = S c ( x, t ) + O (cid:18) √ x + s x log( t + 1)log( x + 1) (cid:19) . Finally for negative t we obtain the same bounds by taking the complex conjugate.In order to discretize d F d , we can apply the same idea, but with a slight modification,since it may not be possible to partition [1 , ∞ ) into disjoint intervals each having total mass1. We proceed as follows. Set q = 1, q j = min { x : F d ( x ) ≥ j } , for 1 ≤ j < j max , whereagain j max = ∞ if F d ( ∞ ) = ∞ and j max = ⌊ F d ( ∞ ) ⌋ + 1 if F d ( ∞ ) < ∞ . Note that it mayoccur that q j = q j +1 = . . . = q j + k for some k ≥ 1. Set γ = 0 and if γ j − is defined with j < j max , define numbers β j , γ j + k ∈ [0 , 1] as β j = 1 − γ j − − X q j − NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 7 j max be such that q J < q J +1 or J = j max − 1. Suppose also that q J / log( q J + 1) ≥ log( | t | + 1),and set x = q J . We apply Lemma 2.1 to the random variables X j,t − E ( X j,t ); however, inthis case J X j =1 E ( X j,t ) = X y n ≤ x α n cos( t log y n ) − γ J cos( t log q J ) . Nevertheless, we can absorb the last term in the error term by multiplying it by 2. ApplyingLemma 2.1 with v = D ′ ( √ x + p x log( | t | + 1) / log( x + 1)), where D ′ = max {√ C ′ , /u } ,with C ′ a constant such that F d ( x ) ≤ C ′ x/ log( x + 1), we obtain P J X j =1 X j,t − X y n ≤ x α n cos( t log y n ) ≥ D ′ (cid:18) √ x + s x log( | t | + 1)log( x + 1) (cid:19)! ≤ x + 1) ( | t | + 1) . The proof can now be completed, mutatis mutandis, as in the continuous case. (cid:3) We now show that under the assumption that F is absolutely continuous on any finiteinterval, we can ensure that the approximating discrete primes are supported on strictlyincreasing sequences which tend to ∞ sufficiently slowly, while still having the bound π P ( x ) − F ( x ) ≪ π P ( x ) − F ( x ) ≪ √ x delivered by the DMVZ-method. Thefollowing corollary is a direct improvement to [15, Lemma 4]. Corollary 2.2. Suppose f is a non-negative L loc -function supported on [1 , ∞ ) and satisfyingthe conditions (1.1) . Let < v < . . . < v k < v k +1 < . . . , v k → ∞ , be a sequence such that v k +1 − v k ≪ log v k and such that for any t ≥ X v k ≥ h ( t ) ( v k − v k − ) v k log v k ≪ log( t + 1) t , where h ( t ) = log( t + 1) log log( t + e) . Then there exists a generalized prime system P = { p j } ∞ j =1 supported on the sequence { v k } ∞ k =1 such that for any x ≥ and any t (2.2) (cid:12)(cid:12)(cid:12) π P ( x ) − F ( x ) (cid:12)(cid:12)(cid:12) ≪ and (cid:12)(cid:12)(cid:12)(cid:12) X p j ≤ x p − i tj − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ √ x + s x log( | t | + 1)log( x + 1) . Some examples of admissible sequences are v k = (log( k + k )) a (log log( k + k )) b with0 < a < b ∈ R and v k = log( k + k )(log log( k + k )) b with b ≤ Proof. Write d F ( u ) = f ( u ) d u . The idea of the proof is to construct an ‘intermediate’measure d G which is close to d F and supported on the sequence { v k } ∞ k =1 . The primes p j will then be obtained discretizing d G by using Theorem 1.2.We set v = 1 and define the measure d G asd G = ∞ X k =1 α k δ v k , where α k = Z v k v k − d F. Strictly speaking, { p j } ∞ j =1 needs not be a subsequence of { v k } ∞ k =1 , since some primes p j may be repeated. F. BROUCKE AND J. VINDAS By the first requirement on the sequence { v k } ∞ k =1 and the bound d F ( u ) ≪ d u/ log u , we have G ( x ) − F ( x ) ≪ 1. Let now t be arbitrary, and let x be such that x/ log( x + 1) < log( | t | + 1).Then trivially (cid:12)(cid:12)(cid:12)(cid:12) X v k ≤ x α k v − i tk − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ F ( x ) ≪ x log( x + 1) < s x log( | t | + 1)log( x + 1) . If on the other hand x/ log( x + 1) ≥ log( | t | + 1), we proceed as follows: (cid:12)(cid:12)(cid:12)(cid:12) X v k ≤ x α k v − i tk − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ Z v L d F ( u ) + K X k = L +1 Z v k v k − (cid:12)(cid:12)(cid:12) v − i tk − u − i t (cid:12)(cid:12)(cid:12) d F ( u ) . Here K is such that v K ≤ x < v K +1 , and L is the largest integer ≤ K such that v L < h ( | t | ) =log( | t | + 1) log log( | t | + e). Bounding (cid:12)(cid:12) v − i tk − u − i t (cid:12)(cid:12) by | t | ( v k − v k − ) /v k and using the boundd F ( u ) ≪ d u/ log u , we get (cid:12)(cid:12)(cid:12)(cid:12) X v k ≤ x α k v − i tk − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ v L log( v L + 1) + | t | X v k ≥ h ( | t | ) ( v k − v k − ) v k log v k ≪ log( | t | + 1) ≤ s x log( | t | + 1)log( x + 1) , where we used the second property of the sequence { v k } ∞ k =1 and log( | t | + 1) ≤ x/ log( x + 1).Applying Theorem 1.2 to G yields a sequence { p j } ∞ j =1 of primes satisfying (2.2) (by comparingwith d G via the triangle inequality). By construction of the discrete random variables in theproof of Theorem 1.2, the primes p j are contained in the support of d G , that is the sequence { v k } ∞ k =1 . (cid:3) Remark 2.3. It is possible to generalize Theorem 1.2 to functions F with different growth.Indeed, suppose that F ( x ) ≪ A ( x ), where A is non-decreasing, has tempered growth, namely, A ( x ) ≪ x n for some n , and satisfies(2.3) Z x p A ( u ) u d u ≪ p A ( x )(which implies log x ≪ A ( x )). Then the conclusion of theorem holds if we replace the bound(1.3) by (cid:12)(cid:12)(cid:12)(cid:12) X p j ≤ x p − i tj − Z x u − i t d F ( u ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ p A ( x ) (cid:0)p log( x + 1) + p log( | t | + 1) (cid:1) . We remark that (2.3) is satisfied whenever A is of positive increase (see [3, Theorem 2.6.1(b)and Definition of PI in p. 71]).3. Balazard’s question for general Dirichlet series and the existence ofwell-behaved Beurling numbers of type [0 , / , / P = { p j } ∞ j =0 be a Beurling generalized prime system. The associated (multi)set ofBeurling generalized integers N is the semi-group generated by 1 and the numbers p j , which NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 9 we arrange in a non-decreasing fashion (taking multiplicities into account): 1 = n < n ≤ n ≤ . . . ≤ n k ≤ . . . . Besides π P and ζ P , we can associate to the number system familiarnumber theoretic functions [6]. The counting function of the generalized integers is denotedas N P ( x ) = P n k ≤ x . As in classical number theory, one defines the Riemann prime countingfunction as Π P ( x ) = X p νj ≤ x ν = ∞ X ν =1 π P ( x /ν ) ν . The functions N P and Π P are then linked via the zeta function identity(3.1) ζ P ( s ) = Z ∞ − x − s d N P ( x ) = exp (cid:18)Z ∞ − x − s dΠ P ( x ) (cid:19) . The M¨obius function of the generalized number system is determined by its sum function M ( x ) = P n k ≤ x µ ( n k ), where d M is defined as the (multiplicative) convolution inverse ofd N ; equivalently,(3.2) ∞ X k =1 µ ( n k ) n sk = 1 ζ P ( s ) . Let us assume that ζ P has abscissa of convergence 1. Following Hilberdink and Neamah(cf. [14]), we define the three numbers α, β, γ as the unique exponents (necessarily elementsof [0 , x ) = Li( x ) + O ( x α + ε ) ,N ( x ) = ρx + O ( x β + ε ) M ( x ) = O ( x γ + ε ) , hold for some ρ > ε > 0, but no ε < 0. Here we choose to normalize thelogarithmic integral as Li( x ) := Z x − u − log u d u. We then call such a Beurling generalized number system an [ α, β, γ ]-system. The mainresult of [14] (see also [7]) tells us that that Θ = max { α, β, γ } is at least 1/2 and that atleast two of these numbers must be equal to Θ. Hilberdink and Lapidus [8] call a Beurlingnumber system well-behaved if Θ < , / , / / , , / / , / , / , , / / , / , , β, β ] system with β < 1? The following theorem answers thisquestion positively; we actually establish the existence of [0 , / , / We count the primes using Riemann’s counting function Π instead of Chebyshev’s ψ . An error term forΠ can be transported to one for ψ at just the cost of an additional log-factor. For this result it is imperative to consider discrete number systems, since it is obviously false for con-tinuous ones: consider for example Π = Li, for which [6] N ( x ) = x and M ( x ) = 1 − log x , for an easycounterexample. To ensure this it suffices to know that just two of numbers are < 1, as we can deduce from [8, Theorem2.3] and (the proof of) [14, Theorem 2.1]. Theorem 3.1. There is a discrete Beurling generalized prime system P such that (3.4) Π P ( x ) = Li( x ) + O (log log x ) , (3.5) N P ( x ) = x + O (cid:0) x / exp( c (log x ) / ) (cid:1) , M P ( x ) = O (cid:0) x / exp( c (log x ) / ) (cid:1) , for some c > , and (3.6) N P ( x ) = x + Ω ε (cid:0) x / − ε (cid:1) , M P ( x ) = Ω ε (cid:0) x / − ε (cid:1) , for any ε > . It follows at once that (3.2) for the Beurling number system from Theorem 3.1 furnishesan example of a general Dirichlet series having abscissa of convergence σ c = 1 / s = 1, which proves Proposition 1.4. Proof. We apply Theorem 1.2 to F ( x ) = li( x ), where li is such that Li( x ) = P ν ≥ li( x /ν ) /ν .A small computation shows thatli( x ) = ∞ X ν =1 µ ( ν ) ν Li( x /ν ) = ∞ X n =1 (log x ) n n ! nζ ( n + 1) . Here ζ and µ ( ν ) are the classical Riemann-zeta and M¨obius functions, respectively. TheChebyshev bound holds sinceli( x ) ≤ ∞ X n =1 (log x ) n n ! n = Li( x ) ≤ x log x , if x ≫ . We thus find generalized primes P : 1 < p < p < . . . with π P ( x ) = P p j ≤ x x ) + O (1)and satisfying (1.3). To easy the notation, we drop the subscript P from all countingfunctions associated to this generalized prime system, but we make an exception with ζ P ( s )for which the subscript is kept in order to distinguish it from the Riemann zeta function ζ ( s ). The Riemann prime counting function Π of P satisfiesΠ( x ) = j log x log p k X ν =1 ν π ( x /ν ) = j log x log p k X ν =1 ν (cid:0) li( x /ν ) + O (1) (cid:1) = ∞ X ν =1 ∞ X n =1 (log x ) n n ! nζ ( n + 1) 1 ν n +1 − X ν> j log x log p k ∞ X n =1 (log x ) n n ! nζ ( n + 1) 1 ν n +1 + O (log log x )= Li( x ) + O (log log x ) . Also Li( x ) = li( x ) + O (cid:18) √ x log log x log x (cid:19) , so Π( x ) = π ( x ) + O (cid:18) √ x log log x log x (cid:19) . The bound Π( x ) − Li( x ) ≪ log log x implies that Z ( s ) := log ζ P ( s ) − log( s/ ( s − σ > 0. By changing a finite number of primes, wemay assume that Z (1) = 0, so that the corresponding integers have density 1. Using thebound (1.3) we can deduce good bounds for Z in the half plane σ > / 2, which allows oneto deduce the asymptotic relations (3.5) via Perron inversion. The proof is essentially thesame as that of Zang’s theorem [15, Theorem 1], but we will repeat it for convenience of thereader. NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 11 We have that Z ( s ) = Z ∞ x − s d( π ( x ) − li( x )) + Z ∞ x − s d(Π( x ) − π ( x )) − Z ∞ x − s d(Li( x ) − li( x )) . The last two integrals have analytic continuation to σ > / O (( σ − / − ) for σ > / 2. The first integral has analytic continuation to σ > / Z ∞ x − σ d( S ( x, t ) − S ( x, t )) = σ Z ∞ x − σ − ( S ( x, t ) − S ( x, t )) d x ≪ Z ∞ x − σ − / (cid:18) s log( | t | + 1)log x (cid:19) d x ≪ σ − / s log( | t | + 1) σ − / , where have used the same notation as in the proof of Theorem 1.2 for the exponential sumsand integrals. Hence for σ > / C > (cid:12)(cid:12) Z ( s ) (cid:12)(cid:12) ≤ C (cid:18) σ − / s log( | t | + 1) σ − / (cid:19) . Let now x be large but fixed. We want to derive an estimate for N ( x ) by Perron inversion.Actually we will apply the Perron formula to N ( x ) := R x N ( u ) d u , because then the Perronintegral is absolutely convergent. Indeed, we have for any κ > N ( x ) = 12 π i Z κ +i ∞ κ − i ∞ x s +1 ζ P ( s ) s ( s + 1) d s = 12 π i Z κ +i ∞ κ − i ∞ x s +1 e Z ( s ) ( s − s + 1) d s. One then uses the fact that N is non-decreasing, so that N ( x ) − N ( x − ≤ N ( x ) ≤ N ( x + 1) − N ( x ) . Set σ x = 1 / x ) − / . Then uniformly for σ ≥ σ x , (cid:12)(cid:12) Z ( s ) (cid:12)(cid:12) ≤ C (cid:0) (log x ) / + (log x ) / p log( | t | + 1) (cid:1) . We shift the contour to the line σ = σ x . By the residue theorem (recall that Z (1) = 0): N ( x ) ≤ ( x + 1) − x π i Z σ x +i ∞ σ x − i ∞ (( x + 1) s +1 − x s +1 )e Z ( s ) ( s − s + 1) d s = x + 12 + 12 π i Z σ x +i ∞ σ x − i ∞ (( x + 1) s +1 − x s +1 )e Z ( s ) ( s − s + 1) d s. We split the range of integration into two pieces: | t | ≤ x and | t | > x . In the first piece webound ( x + 1) s +1 − x s +1 by | s + 1 | x σ x , whereas in the second one by x σ x +1 . This gives N ( x ) ≤ x + 12 + O (cid:26) x / exp (cid:0) (log x ) / (cid:1) Z x exp (cid:0) C (log x ) / (cid:1) t + 1 d t + x / exp (cid:0) (log x ) / (cid:1) Z ∞ x exp (cid:0) C (log x ) / p log( t + 1) (cid:1) d tt (cid:27) The first integral is bounded by exp (cid:0) C (log x ) / (cid:1) log x and the second one by the term x − exp (cid:0) C (log x ) / (cid:1) . A similar reasoning applies for a lower bound for N , and one seesthat the asymptotic relation in (3.5) for N holds with any c > C + 1. To obtain the asymptotic behavior of M , we apply the same reasoning to N ( x ) + M ( x ),which is also non-decreasing, and which has Mellin transform ζ P ( s ) + 1 ζ P ( s ) = ss − Z ( s ) + s − s e − Z ( s ) , to show that N ( x ) + M ( x ) = x + O (cid:0) x / exp( c (log x ) / (cid:1) . The bound for M in (3.5) thenfollows by combining this asymptotic estimate with that we have already obtained for N .Finally, the oscillation estimates (3.6) follow at once from (3.5) and the result of Hilberdinkand Neamah from [14] quoted above. (cid:3) Remark 3.2. We stress that the strong bound Π P ( x ) − Li( x ) ≪ log log x is crucial in theabove arguments to generate the oscillation estimates (3.6). In particular, if only the weakerbound Π P ( x ) − Li( x ) ≪ √ x had been known (like in Zhang’s generalized number systemfrom [15, Theorem 1], whose construction is based upon application of the DMVZ-method),the Hilberdink and Neamah theorem could not have been used to exclude the possibilitythat the abscissa of convergence σ c of P ∞ k =1 µ ( n k ) n − sk satisfies σ c < / /ζ P ( s ) hasadditional zeros s = σ + i t with σ c < σ ≤ / Remark 3.3. Let P be a generalized prime number system like in Theorem 3.1. Anotherexample of a general Dirichlet series with abscissa of convergence 1 / σ > / L P ( x ) = P n k ≤ x λ ( n k ), can bedefined via the identity ∞ X k =0 λ ( n k ) n sk = ζ P (2 s ) ζ P ( s ) , so that its Dirichlet series has a zero at s = 1. Clearly, we have L P ( x ) = X n k ≤ x M P ( x/n k ) ≪ x / exp( c (log x ) / ) X n k ≤√ x n k ≪ x / exp( c (log x ) / ) log x. Furthermore, the estimate (3.4) and (the proof of) [13, Proposition 19] imply L P ( x ) = Ω( √ x ) , which completes the proof of our claim. Remark 3.4. The bound Π P ( x ) − Li( x ) ≪ log log x implies that ζ P has meromorphiccontinuation to σ > 0, and that it has one simple pole at s = 1 and no other zeros there.The equality β = γ = 1 / ζ P and 1 /ζ P must have infinite order in thestrip 0 < σ < / 2. (However, using convexity arguments one might show that ζ P and 1 /ζ P are of polynomial growth in the region σ > / − / log( | t | + 2).)4. A Beurling number system with highly regular primes but integers withlarge oscillation In [4], the authors showed the existence of a Beurling prime system P for which π P ( x ) = Li( x ) + O ( √ x ) and N P ( x ) = ρx + Ω ± (cid:0) x exp( − c p log x log log x ) (cid:1) for any c > √ 2, where ρ > N P . This was done by firstconsidering a continuous number system (Π c , N c ), for which Π c and N c have the desired NEW GENERALIZED PRIME RANDOM APPROXIMATION PROCEDURE 13 asymptotic behavior, and then discretizing this continuous system with the aid of a variantof the DMVZ-method, supplemented with a specific technique to control the argument ofthe zeta function. The continuous prime system is given byΠ c ( x ) = Li( x ) + ∞ X k =0 R k ( x ) , R k ( x ) = ( sin( τ k log x ) for τ δ k k < x ≤ τ ν k k , . Here { τ k } ∞ k =0 is a rapidly increasing sequence, δ k = (log log τ k + a k ) / log τ k , and { a k } ∞ k =0 and { ν k } ∞ k =0 ⊂ (2 , 3) are bounded sequences chosen such that Π c is (absolutely) continuous. Fora detailed analysis of this continuous example, we refer to [4], where additional technical as-sumptions are imposed on the sequences { τ k } ∞ k =0 , { a k } ∞ k =0 , and { ν k } ∞ k =0 in order to achieve theneeded extremal behavior of its zeta function ζ c ( s ) = R ∞ − x − s d N c ( x ) = exp (cid:0)R ∞ − x − s dΠ c ( x ) (cid:1) .We now show, Theorem 4.1. There exist discrete Beurling prime systems P and P which satisfy π P ( x ) = Li( x ) + O (1) and Π P ( x ) = Li( x ) + O (log log x ) , and for any c > √ , and j = 1 , , (4.1) N P j ( x ) = ρ j x + Ω ± (cid:0) x exp( − c p log x log log x ) (cid:1) , where ρ j > is the asymptotic density of the generalized integer counting function N P j . Remark 4.2. In view of its closeness to the ‘most natural’ continuous number systemΠ ( x ) = Li( x ) and N ( x ) = x , the primes of the system P might be considered to be moreregular than those of P . Note that the zeta function associated to P has no zeros in theright half plane { s : Re s > } , as is also the case for the number system from Theorem 3.1. Proof. The prime systems P and P will be obtained by applying (a variant of) Theorem1.2 to F = Π c and F = π c , respectively, where π c is such that Π c ( x ) = P ∞ n =1 π c ( x /n ) /n . Wehave the formula π c ( x ) = li( x ) + ∞ X k =0 ∞ X n =1 r k,n ( x ) , where (cf. Section 3)li( x ) = ∞ X n =1 (log x ) n n ! nζ ( n + 1) and r k,n ( x ) = µ ( n ) n R k ( x /n ) = ( µ ( n ) n sin (cid:0) ( τ k /n ) log x (cid:1) for ( τ δ k k ) n < x ≤ ( τ ν k k ) n , . Let us first verify that π c satisfies the requirements of Theorem 1.2. We only have to showthat it is non-decreasing, the rest is clear. Set I k,n = h ( τ δ k k ) n , ( τ ν k k ) n i , and let x ≥ n ≥ k such that x ∈ I k,n ;b) if n ≤ n and k , k are such that x ∈ I k ,n ∩ I k ,n , then k ≥ k . An integer distribution function N is uniquely determined by Π. Explicitly one has d N = exp ∗ (dΠ),where the exponential is with respect to the multiplicative convolution of measures (see e.g. [6, Chapter 3]). The first observation is a consequence of I k, ∩ I k +1 , = ∅ , which holds provided that thesequence { τ k } ∞ k =0 grows sufficiently rapidly. The second property follows from the fact thatif k < k , then τ ν k n k < τ (1+ δ k ) n k ≤ τ (1+ δ k ) n k . Now π c is absolutely continuous, so it willfollow that it is non-decreasing if we show that π ′ c is non-negative. We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∞ X k =0 ∞ X n =1 r k,n ( x ) (cid:19) ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ x X k,nx ∈ I k,n τ k n . Let m be the smallest integer ≥ x ∈ I k,m for some k , and let K be such that x ∈ I K,m . By a) and b), the above quantity is bounded by1 x ∞ X n = m τ K n ≤ τ K mx ≤ m τ − δ K K = 2 m e a K log τ K , where the last inequality follows from the fact that x ∈ I K,m . Also,li ′ ( x ) ≥ ζ (2) 1 − x − log x ≥ ζ (2) log x ≥ mζ (2) ν K log τ K , if x ∈ I K,m . Hence π ′ c ( x ) ≥ a k ≥ log(12 ζ (2)) say (recall that ν k ≤ F = Π c and F = π c , we obtain two prime systems P and P with π P ( x ) = Π c ( x ) + O (1) = Li( x ) + O (1) and π P ( x ) = π c ( x ) + O (1), so thatas in the proof of Theorem 3.1, Π P ( x ) = Π c ( x ) + O (log log x ) = Li( x ) + O (log log x ). Todeduce the oscillation estimate (4.1) from the behavior of the zeta functions ζ P and ζ P , twoadditional properties alongside (1.3) were required, namely that S ( x, t ) − S c ( x, t ) ≪ √ x (log τ k ) / , uniformly for | t − τ k | ≤ exp (cid:18) d (cid:18) log τ k log log τ k (cid:19) / (cid:19) , for a certain specified constant d > x and k large enough, and that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) arg ζ P (1 + i τ k l ) ζ c (1 + i τ k l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < π , on an (infinite) subsequence { τ k l } of { τ k } . Both of these additional properties can be obtained as in [4]. For the first one, one modifiesthe proof of Theorem 1.2 and also considers the events B k,j , corresponding to the violationof the above bound for x = q j and t close to τ k . Using the rapid growth of the sequence { τ k } ∞ k =0 , one then shows that the sum of the probabilities of the events B k,j is also finite.For the second property, one adds a finite number of well chosen primes around 80 /π , whichshift the phase of the zeta function at the points 1 + i τ k l to the desired range. We choose toomit further details and instead refer to [4, Section 6] for an account on both methods. (cid:3) References [1] P. T. Bateman, H. G. Diamond, Asymptotic distribution of Beurlings generalized prime numbers , in: Studies in number theory , W. J. LeVeque (ed.), pp. 152–210. Math. Assoc. Amer., Prentice-Hall, En-glewood Cliffs, N.J., 1969.[2] A. 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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent Univer-sity, Krijgslaan 281, 9000 Gent, Belgium Email address : [email protected] Email address ::