aa r X i v : . [ m a t h . N T ] J a n AN M -FUNCTION ASSOCIATED WITH GOLDBACH’SPROBLEM KOHJI MATSUMOTO
Abstract.
We prove the existence of the M -function, by whichwe can state the limit theorem for the value-distribution of themain term in the asymptotic formula for the summatory functionof the Goldbach generating function. The Goldbach generating function
One of the most famous unsolved problems in number theory is Gold-bach’s conjecture, which asserts that all even integer ≥ r ( n ) = X l + m = n Λ( l )Λ( m ) , where Λ( · ) denotes the von Mangoldt function. This may be regardedas the Goldbach generating function. In fact, Goldbach’s conjecturewould imply r ( n ) > n ≥
6. Hardy and Littlewood [15]conjectured that r ( n ) ∼ nS ( n ) for even n as n → ∞ , where S ( n ) = Y p | n (cid:18) p − (cid:19) Y p ∤ n (cid:18) − p − (cid:19) ( p denotes the primes). In view of this conjecture, it is interesting toevaluate the sum A ( x ) = X n ≤ x ( r ( n ) − nS ( n )) ( x > . It is known that the estimate A ( x ) = O ( x / ε ) (where, and in whatfollows, ε is an arbitrarily small positive number) is equivalent to theRiemann hypothesis (RH) for the Riemann zeta-function ζ ( s ) (seeGranville [14], Bhowmik and Ruzsa [4], Bhowmik et al. [3]). Mathematics Subject Classification.
Primary 11M41, Secondary 11P32,11M26, 11M99.
Key words and phrases.
Goldbach’s problem, M -function, Riemann zeta-function. The unconditional estimate A ( x ) = O ( x (log x ) − A ) ( A >
0) wasclassically known. In 1991, Fujii published a series of papers [11] [12][13], in which he refined this classical estimate under the RH. Fujii firstproved A ( x ) = O ( x / ) in [11], and then in [12], he gave the followingasymptotic formula A ( x ) = − x / · ℜ Ψ( x ) + R ( x ) , (1.1)where R ( x ) is the error term, andΨ( x ) = X γ> x iγ (1 / iγ )(3 / iγ ) = ∞ X m =1 x iγ m (1 / iγ m )(3 / iγ m ) , (1.2)with γ running over all imaginary parts of non-trivial zeros of ζ ( s )which are positive. We number those imaginary parts as 0 < γ <γ < · · · < γ m < · · · .Concerning the error term R ( x ), Fujii [12] showed the estimate R ( x ) = O ( x / (log x ) / ). Egami and the author [10] raised the conjecture R ( x ) = O ( x ε ) , R ( x ) = Ω( x ) . This conjecture was settled by Bhowmik and Schlage-Puchta [5] in theform R ( x ) = O ( x (log x ) ) , R ( x ) = Ω( x log log x ) . The best upper-bound estimate at present is O ( x (log x ) ). As for themore detailed history, see [2].Properties of the main term on the right-hand side of (1.1) was firstconsidered by Fujii [13]. Let f ( α ) = Ψ( e α ) ( α ∈ R ) . (1.3)In [13], Fujii studied the value-distribution of f ( α ), and proved the fol-lowing limit theorem. Assume that γ ’s are linearly independent over Q (which we call the LIC). Then Fujii stated the existence of the “densityfunction” F ( x ) ( z = x + iy ∈ C ) for whichlim X →∞ X µ { ≤ α ≤ X | f ( α ) ∈ R } = Z Z R F ( x + iy ) dxdy (1.4)holds for any rectangle R in C , where µ {·} means the one-dimensionalLebesgue measure. This is an analogue of the following result of Bohrand Jessen [8] [9] for the value-distribution of ζ ( s ). Let σ > /
2. Bohrand Jessen proved the existence of a continuous function F σ ( z ) for N M -FUNCTION ASSOCIATED WITH GOLDBACH’S PROBLEM 3 whichlim T →∞ T µ {− T ≤ t ≤ T | log ζ ( σ + it ) ∈ R } = Z Z R F σ ( x + iy ) dxdy (1.5)holds for any rectangle R .Fujii gave a sketch of the proof, which is along the same line as in [8].In particular, Fujii indicated explicitly how to construct F ( x + iy ),following the method of Bohr and Jessen [7].2. The theory of M -functions and the statement of themain result The result (1.5) of Bohr and Jessen has been generalized to a widerclass of zeta-functions. The existence of the limit on the left-hand sideof (1.5) is now generalized to a fairly general class (see [25]).It is more difficult to prove the integral expression like the right-handside of (1.5). The case of Dirichlet L -functions L ( s, χ ) is essentially thesame as in the case of ζ ( s ) (see Joyner [22]). The case of Dedekind zeta-functions of algebraic number fields was studied by the author [26] [27][28]. The case of automorphic L -functions attached to SL(2 , Z ) or itscongruence subgroups was established recently in [30] [31].All of those generalizations consider the situation when t = ℑ s varies(like the left-hand side of (1.5)). When we treat more general L -functions, various other aspects can be considered. In 2008, Ihara [16]studied the χ -aspect for L -functions defined on number fields or func-tion fields. His study was then further refined in a series of papers ofIhara and the author [17] [18] [19] [20]. Let us quote a result provedin [18]. Theorem 2.1.
Let s = σ + it ∈ C with σ > / . There exists anexplicitly constructable density function M σ ( w ) , continuous and non-negative, for which Avg χ Φ(log L ( s, χ )) = Z C M σ ( w )Φ( w ) | dw | (2.1) holds, where Avg χ stands for some average with respect to characters, | dw | = dudv/ (2 π ) ( for w = u + iv ) , and Φ is the test function which iseither (i) some continuous function, or (ii) the characteristic functionof a compact subset of C or its complement. The density function M σ is called an M -function. Here we do not givethe details how to define Avg χ , but in [18], two types of averages wereconsidered. One of them is a certain average with respect to Dirichlet KOHJI MATSUMOTO characters, and the other is essentially the same as the average in t -aspect like (1.5). In this sense, F σ in (1.5) may be regarded as anexample of M -functions.Since then, various analogues of Theorem 2.1 were discovered byMourtada and Murty [35], Akbary and Hamieh [1], Lebacque and Zykin[24], Matsumoto and Umegaki [29], Mine [32] [33] [34], and so on.The aim of the present article is to show the following “limit theo-rem”, which is a generalization of Fujii’s (1.4) in the framework of thetheory of M -functions. Theorem 2.2.
We assume the LIC. There exists an explicitly con-structable density function ( M -function ) M : C → R ≥ , for which lim X →∞ X Z X Φ( f ( α )) dα = Z C M ( w )Φ( w ) | dw | (2.2) holds for any test function Φ : C → C which is continuous, or whichis the characteristic function of either a compact subset of C or thecomplement of such a subset. The function M ( w ) is continuous, tendsto when | w | → ∞ , M ( w ) = M ( w ) , and Z C M ( w ) | dw | = 1 . (2.3) Remark . Choosing Φ = R , we recover Fujii’s result (1.4).The above theorem is an analogue of the absolutely convergent casein the theory of M -functions (that is, an analogue of [18, Theorem 4.2]).In this sense, our theorem is a rather simple example of M -functions. Inparticular, complicated mean-value arguments (such as [18, Sections 5–8]) are not necessary. Still, however, our theorem gives a new evidenceof the ubiquity of M -functions.3. The finite truncation
The rest of the present paper is devoted to the proof of Theorem 2.2.We first define the finite truncation of f ( α ). Let b m = (1 / iγ m )(3 / iγ m ), c m = 1 / | b m | , and β m = arg b m . Then f ( α ) = ∞ X m =1 e iαγ m b m = ∞ X m =1 c m e i ( αγ m − β m ) . (3.1)It is to be noted that c m = 1 q + γ m q + γ m ∼ γ m ∼ (cid:18) log m πm (cid:19) (3.2) N M -FUNCTION ASSOCIATED WITH GOLDBACH’S PROBLEM 5 as m → ∞ , hence the above series expression of f ( α ) is absolutelyconvergent.We first consider the finite truncation f N ( α ) = N X m =1 c m e i ( αγ m − β m ) . (3.3)Let T be the unit circle on C , and T N = Q m ≤ N T . Define S N ( t N ) = X m ≤ N c m t m , (3.4)where t N = ( t , . . . , t N ) ∈ T N . Then obviously f N ( α ) = S N ( e i ( αγ − β ) , . . . , e i ( αγ N − β N ) ) . (3.5)The idea of attaching the mapping S N : T N → C to f N goes back tothe work of Bohr [6]. We denote by d ∗ t N the normalized Haar measureon T N , that is the product measure of d ∗ t = (2 π ) − dθ for t = e iθ ∈ T .The following is an analogue of [16, Theorem 1]. Proposition 3.1.
We may construct a function M N : C → R ≥ , forwhich Z C M N ( w )Φ( w ) | dw | = Z T N Φ( S N ( t N )) d ∗ t N (3.6) holds for any continuous function Φ on C . In particular, choosing Φ ≡ we obtain Z C M N ( w ) | dw | = 1 . (3.7) Also for N ≥ the function M N ( w ) is compactly supported, non-negative and M N ( w ) = M N ( w ) .Proof. First consider the case N = 1. Let s n ( t n ) = c n t n . For w = re iθ ∈ C ( r = | w | , θ = arg w ), define m n ( w ) = 1 r δ ( r − c n ) , (3.8) KOHJI MATSUMOTO where δ ( · ) stands for the usual Dirac delta distribution. We have Z C m n ( w )Φ( w ) | dw | (3.9) = Z π Z ∞ m n ( re iθ )Φ( re iθ ) 12 π rdrdθ = 12 π Z π dθ Z ∞ δ ( r − c n )Φ( re iθ ) dr = 12 π Z π Φ( c n e iθ ) dθ = Z T Φ( s n ( t n )) d ∗ t n . In particular, putting n = 1 in (3.9), we find Z C m ( w )Φ( w ) | dw | = Z T Φ( s ( t )) d ∗ t , (3.10)which implies that the case N = 1 of Proposition 3.1 is valid with M = m .Now we prove the general case by induction on N . Define M N ( w ) = Z C M N − ( w ′ ) m N ( w − w ′ ) | dw ′ | (3.11)for N ≥
2. This is compactly supported, and Z C M N ( w )Φ( w ) | dw | = Z C Z C M N − ( w ′ ) m N ( w − w ′ ) | dw ′ | Φ( w ) | dw | = Z C M N − ( w ′ ) | dw ′ | Z C m N ( w − w ′ )Φ( w ) | dw | . The exchange of the integrations is verified because M N is compactlysupported. Putting w ′′ = w − w ′ we see that the inner integral is= Z C m N ( w ′′ )Φ w ′ ( w ′′ ) | dw ′′ | (where Φ w ′ ( w ′′ ) = Φ( w ′′ + w ′ )) , which is, by (3.9), = Z T Φ w ′ ( s N ( t N )) d ∗ t N . N M -FUNCTION ASSOCIATED WITH GOLDBACH’S PROBLEM 7 Therefore Z C M N ( w )Φ( w ) | dw | = Z C M N − ( w ′ ) | dw ′ | Z T Φ w ′ ( s N ( t N )) d ∗ t N = Z T d ∗ t N Z C M N − ( w ′ )Φ w ′ ( s N ( t N )) | dw ′ | = Z T d ∗ t N Z C M N − ( w ′ )Φ s N ( w ′ ) | dw ′ | , where Φ s N ( w ′ ) = Φ( s N ( t N ) + w ′ ) = Φ w ′ ( s N ( t N )). Using the inductionassumption we see that the right-hand side is= Z T d ∗ t N Z T N − Φ s N ( S N − ( t N − )) d ∗ t N − = Z T N Φ s N ( S N − ( t N − )) d ∗ t N . SinceΦ s N ( S N − ( t N − )) = Φ( S N − ( t N − ) + s N ( t N )) = Φ( S N ( t N )) , we obtain the assertion of the proposition. (cid:3) The following two propositions are analogues of [18, Remark 3.2 andRemark 3.3]. For any A ⊂ C , by A we denote the characteristicfunction of A . By Supp( φ ) we mean the support of a function φ . Proposition 3.2.
The formula (3.6) is valid when
Φ = A , where A is either a compact subset of C or the complement of such a subset.Proof. It is enough to prove the case when A is compact. Let φ , φ becontinuous non-negative functions, defined on C , compactly supported,satisfying 0 ≤ φ ≤ A ≤ φ ≤ φ − φ )) < ε (where“Vol” denotes the volume measured by d ∗ t N ). Then Z C M N ( w )( A − φ )( w ) | dw | < C N ε, Z C M N ( w )( φ − A )( w ) | dw | < C N ε, where C N = sup { M N ( w ) } . Therefore, using Proposition 3.1 we have Z C M N ( w ) A ( w ) | dw | − C N ε ≤ Z C M N ( w ) φ ( w ) | dw | = Z T N φ ( S N ( t N )) d ∗ t N ≤ Z T N A ( S N ( t N )) d ∗ t N ≤ Z T N φ ( S N ( t N )) d ∗ t N = Z C M N ( w ) φ ( w ) | dw |≤ Z C M N ( w ) A ( w ) | dw | + C N ε, from which the desired assertion follows. (cid:3) KOHJI MATSUMOTO
In the proof of Proposition 3.1 we have shown that M N is compactlysupported. Now we show more explicitly what is the support. Proposition 3.3.
The support of M N is the image of the mapping S N .Proof. Let A be a compact subset of C . We can use (3.6) with Φ = A because of Proposition 3.2. Then Z A M N ( w ) | dw | = Z T N A ( S N ( t N )) d ∗ t N = Vol( S − N ( A )) , (3.12)which implies the proposition. (cid:3) The finite-truncation version of the theorem
The aim of this section is to prove
Proposition 4.1.
Under the assumption of the LIC, we have lim X →∞ X Z X Φ( f N ( α )) dα = Z T N Φ( S N ( t N )) d ∗ t N (4.1) for any continuous function Φ on C . Then, combining this with Proposition 3.1, we havelim X →∞ X Z X Φ( f N ( α )) dα = Z C M N ( w )Φ( w ) | dw | (4.2)for any continuous Φ, which is the “finite-truncation” analogue of ourmain theorem.In view of (3.5), in order to prove Proposition 4.1, it is enough toprove the following Proposition 4.2.
Under the assumption of the LIC, we have lim X →∞ X Z X Ψ( e i ( αγ − β ) , . . . , e i ( αγ N − β N ) ) dα = Z T N Ψ( t N ) d ∗ t N (4.3) holds for any continuous Ψ : T N → C . This is an analogue of [16, Lemma 4.3.1].
Proof.
Write t N = ( e iθ , . . . , e iθ N ). Then the right-hand side of Propo-sition 4.2 is= 1(2 π ) N Z π · · · Z π Ψ( e iθ , . . . , e iθ N ) dθ · · · dθ N . To show that this is equal to the left-hand side, by Weyl’s criterion(see [23, Chapter 1, Theorem 9.9]), it is enough to show the equalitywhen Ψ = t n · · · t n N N for any ( n , . . . , n N ) ∈ Z N \ { (0 , . . . , } . But in N M -FUNCTION ASSOCIATED WITH GOLDBACH’S PROBLEM 9 this case, since Ψ( e iθ , . . . , e iθ N ) = e i ( n θ + ··· + n N θ N ) , the right-hand sideis clearly equal to 0. The left-hand side is= lim X →∞ X Z X e in ( αγ − β )+ ··· + in N ( αγ N − β N ) dα = lim X →∞ X e − i ( n β + ··· + n N β N ) Z X e iα ( n γ + ··· + n N γ N ) dα. Since we assume the LIC, n γ + · · · + n N γ N = 0 because ( n , . . . , n N ) =(0 , . . . , X →∞ X e − i ( n β + ··· + n N β N ) · e iX ( n β + ··· + n N β N ) − i ( n β + · · · + n N β N )which is also equal to 0. The proposition is proved. (cid:3) The existence of the M -function In this section we prove the existence of the limit function M ( w ) = lim N →∞ M N ( w ) . (5.1)For this purpose we consider the Fourier transform. We follow theargument on pp.644-647 in [18], which is based on the ideas of Ihara [16]and of the author [27].Let ψ z ( w ) = exp( i ℜ ( zw )), and define the Fourier transform of m n as e m n ( z ) = Z C m n ( w ) ψ z ( w ) | dw | . (5.2)Applying (3.9) with Φ = ψ z , we see that the right-hand side of theabove is = Z T ψ z ( s n ( t n )) d ∗ t n = 12 π Z π ψ z ( c n e iθ n ) dθ n = 12 π Z π exp( i ℜ ( z · c n e iθ n )) dθ n . Writing z · c n e iθ n = c n | z | e i ( θ n − τ ) ( τ = arg z ), we have ℜ ( z · c n e iθ n ) = c n | z | cos( θ n − τ ) = c n | z | (cos θ n cos τ + sin θ n sin τ )(5.3)and so e m n ( z ) = 12 π Z π exp( ic n | z | (cos θ n cos τ + sin θ n sin τ ) dθ n . (5.4)Now quote: Lemma 5.1. (Jessen and Wintner [21, Theorem 12])
Let C be a closedconvex curve in C parametrized by x ( θ ) = ( ξ ( θ ) , ξ ( θ )) , z = | z | e iτ ∈ C ,and let g τ ( θ ) = ξ ( θ ) cos τ + ξ ( θ ) sin τ . Assume that ξ , ξ ∈ C and g ′′ τ ( θ ) has (for each fixed τ ) exactly two zeros on C . Then Z C exp( i | z | g τ ( θ )) dθ = O ( | z | − / ) , (5.5) where the implied constant depends on C . In the present case ξ ( θ ) = c n cos θ , x ( θ ) = c n sin θ , and C is thecircle of radius c n . Since g ′′ τ ( θ ) = − c n (cos θ cos τ + sin θ sin τ ) = − c n cos( θ − τ ) , the assumption of the lemma is clearly satisfied, and hence by thelemma we have e m n ( z ) = O n ( | z | − / ) . (5.6)Now define f M N ( z ) = Y n ≤ N e m n ( z ) . (5.7)Then from (5.6) and the obvious inequality | e m n ( z ) | ≤ f M N ( z ) = O N ( | z | − N/ )(5.8)and | f M N ( z ) | ≤ . (5.9)From these inequalities we obtain (i) and (ii) of the following Proposition 5.2.
Let N ≥ . (i) f M N ∈ L t for any t ∈ [1 , + ∞ ] , (ii) | f M N ( z ) | ≤ | f M N ( z ) | for all N ≥ N , (iii) f M N ( z ) converges to a certain function f M ( z ) uniformly in anycompact subset when N → ∞ .Proof of (iii) . It is clear from (5.3) that12 π Z π ℜ ( z · c n e iθ n ) dθ n = 0 . (5.10)Therefore we can write e m n ( z ) − π Z π (exp( i ℜ ( z · c n e iθ n )) − − ℜ ( z · c n e iθ n )) dθ n . (5.11) N M -FUNCTION ASSOCIATED WITH GOLDBACH’S PROBLEM 11 Since | e ix − − ix | ≪ x for any real x (by the Taylor expansion forsmall | x | , and by the fact | e ix | = 1 for large | x | ), we obtain | e m n ( z ) − | ≪ Z π |ℜ ( z · c n e iθ n ) | dθ n ≪ | z | c n . (5.12)Let N < N ′ . Then | f M N ′ ( z ) − f M N ( z ) | ≤ N ′ − N X j =1 | f M N + j ( z ) − f M N + j − ( z ) | = N ′ − N X j − | f M N + j − ( z ) | · | e m N + j ( z ) − |≪ | z | N ′ − N X j =1 c N + j by (5.9) and (5.12). Because of (3.2) we see that the series on the right-hand side converges as N, N ′ → ∞ . Therefore by Cauchy’s criterionwe obtain the assertion (iii). (cid:3) Now we prove the following result, which is an analogue of [18, Propo-sition 3.4].
Proposition 5.3. f M N ( z ) converges to f M ( z ) uniformly in C when N → ∞ . The limit function f M ( z ) is continuous and belongs to L t ( for any t ∈ [1 , ∞ ]) , and the above convergence is also L t -convergence.Proof. Let 0 < ε <
1. By Proposition 5.2 (i) we can find R = R ( N ) > Z | z |≥ R | f M N ( z ) | t | dz | < ε (5.13)for any 1 ≤ t < ∞ and (noting (5.8))sup | z |≥ R | f M N ( z ) | < ε. (5.14)(Here R is independent of t , because by (5.9) the inequality (5.13) for t = 1 implies (5.13) for other finite values of t .) Because of Proposition5.2 (ii), the above inequalities are valid also for f M N ( z ) for all N ≥ N .Taking N → ∞ in the above inequalities, we find that f M ∈ L t (1 ≤ t ≤ ∞ ). Let N ′ > N . Then | f M N ′ ( z ) − f M N ( z ) | t = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y N When N → ∞ , M N ( w ) converges to M ( w ) uni-formly in w ∈ C . The limit function M ( w ) is continuous, non-negative,tends to when | w | → ∞ , M ( w ) = M ( w ) , and Z C M ( w ) | dw | = 1 . (5.25) The functions M and f M are Fourier duals of each other. This is an analogue of [18, Proposition 3.5], and the proof is exactlythe same. 6. Completion of the proof Now we finish the proof of our main Theorem 2.2. Among the state-ment of Theorem 2.2, the properties of M ( w ) is already shown in theabove Proposition 5.4. Therefore the only remaining task is to prove(2.2).First consider the case when Φ is continuous. We have already shownthe “finite-truncation” version of (2.2) as (4.2). We will prove that itis possible to take the limit N → ∞ on the both sides of (4.2).From (3.4) we see that the image of the mapping S N is includedin the disc of radius P ∞ m =1 c m for any N . Therefore by Proposition3.3 we find that the support of M N for any N is also included in thesame disc, hence is the support of M . The image of f is clearly alsobounded. Therefore, to prove (2.2), we may assume that Φ is compactlysupported, hence is uniformly continuous.Then, as N → ∞ , Φ( f N ( α )) tends to Φ( f ( α )) uniformly in α . Also, M N ( w )Φ( w ) tends to M ( w )Φ( w ) uniformly in w , because of Proposi-tion 5.4. This yields that, when we take the limit N → ∞ on (4.2), wemay change the integration and this limit. Therefore we obtain (2.2)for continuous Φ.Finally, similarly to the proof of Proposition 3.2, we can deduce theassertion in the case when Φ is a characteristic function of a compactsubset or its complement. This completes the proof of Theorem 2.2. Remark . Consider the Dirichlet seriesΨ( s, x ) = X γ> x iγ (1 / iγ ) s (3 / iγ ) s (6.1) where s ∈ C . Obviously Ψ(1 , x ) = Ψ( x ). Because of (3.2), the se-ries (6.1) is absolutely convergent when ℜ s > / 2. It is easy to seethat we can extend Theorem 2.2 to Ψ( s, x ) in this domain of absoluteconvergence. Remark . A generalization of the theory of the Goldbach generatingfunction to the case with congruence conditions was first considered byR¨uppel [36], and the generalized form of Ψ( x ) in this case (written interms of the zeros of Dirichlet L -functions) was determined by Suzuki[37]. (See also [2] [3] [4].) 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Suzuki, A mean value of the representation function for the sum of twoprimes in arithmetic prgressions, Intern. J. Number Theory (2017), 977–990. K. Matsumoto: Graduate School of Mathematics, Nagoya Univer-sity, Chikusa- ku, Nagoya 464-8602, Japan Email address ::