The Abstract Prime Number Theorem And A Criterion For The Generalized Riemann Hypothesis

The prime number theorem is one of the most important theorems in analytic number theory which states that the numberÏ€(x) of primes up to x is approximately equal to x/log x when x→+∞. i.e.More precisely we havewhere c₀ is an absolute positive constant.In 1970s, John Knopfmacher[1]-[8] developed the abstract analytic number theory, and established the so-called abstract prime number theorem.To state Knopfmacher's results, we introduce some basic notions and concepts.Let G be a commutative semigroup with an identity element 1. We use P to denote the subset of G consisting of all free generators, namely the prime elements in G. We also assume that there is a norm |·| defined on G. We call (G, |·|) an arithmeticsemigroup if the following conditions are satisfied:(i) The unique factorization principle. Every element a≠1 in G has a unique factorization of the formwhere p_i are distinct elements of P andα_i are positive integers. The uniqueness is understood to be only up to the order of the factors indicated. (iii) |ab|=|a||b| for all a, b∈G,(iv) the total number N_G(x) of elements a∈G of norm |a|≤x is finite, for each real x＞0.In order to study the distribution of prime elements of semigroup G, Knopfmacher introduced the following axiom.Axiom A. There exist positive constants A, 8, andηwith 0≤η＜δ, such thatLet G denote an arithmetical semigroup satisfying Axiom A. LetÏ€_G(x) be the total number of elements p∈P of norm |p|≤x, for each real x＞0, i.eThen the so-called abstract prime number theorem states thatKnopfmacher [9] first proved this theorem by Ikehara's Tauberian Theorem. It can be compared with (0.1). Subsequently Wegmann [10] established a slightly stronger version of abstract prime number theorem, which states that for everyα＞0,where li(x) is the logarithmic integral (?) withHis proof depends on subtle elementary methods (see for example Segal [11]).The aim of this dissertation is to establish a stronger version of the abstract prime number theorem. We first define the abstract zeta functionand give a zero-free region for (g{z) in Chapter 1. Theorem 1.1. There exists an absolute constant c₁＞0 such that (ζ_G(z)≠0 in the domainwhere |t|≥2.Theorem 1.1 plays an important role in the distribution of prime elements of semigroup G. Then we are able to use Theorem 1.1 to establish an asymptotic formula in Chapter 2.Theorem 2.1. For some positive constant c₂ and x≥2, we havewhereΨ(x) is defined as in Chapter 2.As a corollary of Theorem 2.1, we have the following result.Theorem 2.2. As x→∞, we havewhere c₂ is a constant in Theorem 2.1.We know that the prime number theorem in arithmetic progressions is also one of the most important theorems in analytic number theory which states that the numberÏ€(x;k,l) of primes up to x in the progression H_k,l={l,k+l,2k+l,…,1≤l＜k,(k,l)=1} iswhere E₁=1 if there exists a real characterχ₁ modula k such that L(z,χ₁) has a real zeroβ₁＞1-c₄/log k; and E₁ = 0 if otherwise.In 1954, W. Forman and H. N. Shapiro[13] proved an abstract prime number theoremconcerning formation satisfying Axiom A^* (see below for definition). Furthermore, the distribution of prime elements in arithmetic progressions was generalized to that in equivalence classes for formation. To state their result, we first introduce some notions.Let X be a finite abelian group formed with identity-preserving homomorphismsχ:G→C^×. Usuallyχis called a character. Define an equivalence relation -_χon G byLetΓ_x (or simplyΓ) be the set of all different equivalence classes under this relation. The pair (G,Γ) is called an arithmetical formation, and X is the character group of the given formation (G,Γ).Axiom A^*. There exist positive constants A,δ, andηwith 0≤η＜δ, such thatwhere h = cardΓ, andwhere H[x] ={a∈H:|a|≤x}.It follows immediately from this axiom that for fixed x,i.e, G itself satisfies Axiom A. Hence the theory of formation satisfying Axiom A^* is a generalization of the theory of arithmatical semigroups satisfying Axiom A.We assume that (G,Γ) is an arithmetical formation satisfying Axiom A^*.For x＞0, letThe so called abstract prime number theorem for formation states thatIn this dissertation an abstract prime number theorem with sharp error-term for formation is established. Next we denote the abstract L-function forχover G byFor a given formation, the abstract L-function plays a role similar to that of the zeta function of a given arithmetical semigroup. The zero-free region for the abstract L-functionL_G(z,χ) can be used to study the distribution of prime elements of the class group for formation. Hence we establish a zero-free region for L_G(z,χ) in Chapter 3. Theorem 3.1. Let z =σ+it. Ifχis a complex character, then L_G(z,χ)≠0 in the regionIfχis a real character, then L_G(z,χ)≠0 in the regionwhere c₅ is some absolute positive constant.By Theorem 3.1 we can obtain the following two theorems in Chapter 4.Theorem 4.1. Let x≥2. Then for some positive constant c₆, we havewhere E₁=1 if there exists a real characterχ₁ such that L_G(z,χ₁) has a real zeroβ₁>1-c₇/log , and E₁ = 0 if otherwise.Ψ_H(x) is defined as in Chapter 4.Theorem 4.2. We havewhere c₆ is a constant as in Theorem 4.1.If H=H_k,l, thenδ=1 and h=φ(k). We can easily see that Theorem 4.2 is the prime number theorem in arithmetic progressions.In addition, we study the automorphic L-functions attached to the classical auto-morphicforms on GL(2), i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis(GRH) for the above L-functions.According to the well-known Nyman-Beurling criterion, the Riemann Hypothesis is equivalent to the possibility of approximating the characteristic functionχ₁(x) of the interval (0,1] in mean square norm by linear combinations of the fractional parts {1/ax} for the real a greater than 1. In 2003, Baez-Duarte established a strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis. He proved that the above statement remains true if the dilations are restricted to those where the a is positiveinteger and gave a constructive sequence (?) of such approximations, whereμ denotes the Mobius function. Applying Baez-Duarte's idea, we shall generalize Nyman-Beurlingcriterion and give a criterion for the Generalized Riemann Hypothesis(GRH) for the automorphic L-functions attached to the holomorphic cusp form.We state our main result as follows.Theorem 5.1. The Generalized Riemann Hypothesis for the above L-functions is equivalent to the statement that (?).