ADDITIVE DECOMPOSABILITY OF MULTIPLICATIVELY DEFINED SETS Christian Elsholtz. 1. Introduction

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1 Functiones et Aroximatio XXXV 006, 6 77 ADDITIVE DECOMPOSABILITY OF MULTIPLICATIVELY DEFIED SETS Christian Elsholtz To Professor Eduard Wirsing on the occasion of his 75th birthday Abstract: Let T denote the set of integers which are comosed of rime factors from a given set of rimes T only Suose that A+B T, where T and T differ at finitely many elements only Also assume that = τ log x+o We rove that log x, T AB = Olog τ holds In the case τ we give an examle where both A and B are of order of magnitude log 4, which shows that this is close to best ossible Keywords: inverse Goldbach roblem, additive decomositions of sets, sums of two suares Introduction In his two-volume monograh [] on additive number theory, Ostmann, Eduard Wirsing s PhD advisor, studied the structure of sumsets A sumset is defined by A + B = {a + b : a A, b B} Other monograhs on this toic are those by Mann [8], Freĭman [0], athanson [0] and a forthcoming one by Tao and Vu [5] Inverse uestions, eg whether a given set can be additively decomosed, lead to difficult roblems Definition See Ostmann [], vol, Let S and S denote sets of ositive integers We say that S and S are asymtotically eual, if there exists an integer 0 such that S [ 0, ] = S [ 0, ] Definition See [], vol, 5 Let S be a set of ositive integers We say that S is additively irreducible if there are no two sets of ositive integers A, B, with at least two elements each, such that A + B = S 000 Mathematics Subject Classification: Primary P3, 36; Secondary E5 rimitiv in the German original

2 6 Christian Elsholtz Definition 3 See [], vol, 5, and Wirsing [7] Let S be a set of ositive integers We say that S is asymtotically additively irreducible or we say that no asymtotic additive decomosition exists if there are no two sets of ositive integers A, B, with at least two elements each, such that A + B is asymtotically eual to S In his first ublication Wirsing [7] roved that almost all sets of ositive integers are asymtotically additively irreducible For a recise definition of almost all we refer to Wirsing s aer The set of all rimes is additively irreducible The essential reason is that the set of rimes starts with {, 3, } The existence of an additive decomosition imlied that a attern n, n + of two consecutive rimes occurred again, which is of course imossible Ostmann derived a number of generalizations Moreover he stated the following Conjecture 4 Ostmann, [] vol, 3 The set of rimes P is asymtotically additively irreducible This roblem has attracted considerable attention, and several authors have roved that in a conceivable asymtotic decomosition of P both summands A and B would need to be infinite, see Hornfeck [5], Mann [8], Laffer and Mann [6] It has reeatedly been mentioned in roblem collections such as Erdős [6], [7], Oberwolfach 998 [3], and Wolke s survey [3] The roblem itself has resisted a solution so far For artial results in addition to those discussed below see for examle Hofmann and Wolke [4], Puchta [3] or the resent author [3], [4], and [5] In analogy to Goldbach s roblem, it seems that a combination of additive and multilicative roerties leads to very difficult roblems The roblem of the asymtotic additive decomosition of the rimes has also been called the inverse Goldbach roblem, eg in Wirsing s Oberwolfach lecture of 97 [30] and his roblem at the 998 Oberwolfach conference [3] Also, several authors indeendently roved the following Theorem 5 If A + B P, then the following bound on the counting functions holds: AB Here f g is the Vinogradov notation, meaning f = Og When taking extra care of finitely many elements, this imlies: Corollary 6 If an asymtotic additive decomosition of the set of rimes exists, say P = A + B, then the following bound on the counting functions holds: AB Indeed, the first author to rove Theorem 5 was Eduard Wirsing He resented his result at the 97 Oberwolfach meeting [30], but it was never ublished totalrimitiv in the German original

3 Additive decomosability of multilicatively defined sets 63 Indeendently, similar results were roved by Pomerance, Sárközy, and Stewart [], by Hofmann and Wolke [4], and by Bshouty and Bshouty [] The result of Bshouty and Bshouty is stated in a somewhat different way A slightly simlified version is as follows: Theorem 7 Let A = {a,, a n }, B = {b,, b n } and A + B P, then n 4 a n a b n b Therefore, if A, B [, ], then n ote that this theorem only covers the case A = B = n The imlied methods would be weaker when alied to the case of different sizes of A and B As uite a few different ideas go into these roofs of different authors and as Wirsing s original account is unublished we decided to discuss these methods when roving the new results below Theorem 8 Let T denote a set of rimes with log τ log x < C x T Here 0 < τ < denotes a real constant and C denotes a ossitive real constant Let T = {n : n T} Let A + B T, where T [ 0, ] = T [ 0, ], for sufficiently large 0 Then AB τ,c log τ The uer bound is at least sometimes close to best ossible, as discussed below We will give two roofs, one following Wirsing s method to rove Theorem 5 This is based on the analytic version of the large sieve method The second follows the method of Pomerance, Sárközy, and Stewart [], and Hofmann and Wolke [4], which are very similar It is based on the arithmetic version of the large sieve method We then discuss very simle roofs of Theorem 7 The first roof by Bshouty and Bshouty [] makes use of a divisibility roerty of the Vandermonde determinant We then eliminate the Vandermonde argument and relace it by a count of rime factors which resembles a roof of Gallagher s larger sieve, and eventually we show that Gallagher s larger sieve can be directly alied to rove Theorem 7 We then rove the following new bounds in the original roblem of Ostmann Theorem 9 Suose that there is an asymtotic additive decomosition of the set of rimes, A + B = P, then log 3 A log The same bounds hold for B

4 64 Christian Elsholtz This imroves uon the earlier bounds log 5 A log 4 in Elsholtz [4] Let us recall that bounds of this tye imly that there is no ternary asymtotic additive decomosition of the rimes P = A +B+C, where A, B and C have at least two elements each Examles Examle Assuming the rime k -tule conjecture, for any admissible set A = {a,, a k } ie that for no rime the image A mod occuies all residue classes there exists an infinite set B with counting function B such that A + B P Unconditionally, for each finite large c a,,a k log k and each k there exists a set A = {a,, a k } [, ] such that there exists a corresonding set B [, ] with B and A + B P see [] klog k Similarly, let A + B T, where T is as in Theorem 8 Using the method of [] see also [8] one can show that there exists a set B [, ] with B k Moreover, for each finite, one can find sets A and B with A log τk B +o log τ log log This shows that for small fixed k the uer bound AB = O τ,c log τ of Theorem 5 is off by a logarithmic factor only As k increases one would exect much better uer bounds In the roblem A + B P, if A > log r where r one can rove using the methods of [5] that B +or But if A grows as fast as the best uer bound on B that one can currently rove is O only One might exect much stronger uer bounds if A and B are of about the same size If A B and A + B P one might exect that for examle A = O ε for arbitrary ositive ε On the other hand, Theorem 8 is close to best ossible if A and τ, as shown by the following examle Examle Let A = {n : n } and B = {n : n and n mod 4} ote that A and B c log, since by Lemma 3 or Lemma 4 the number of n comosed of rime factors mod 4 only is asymtotically c log o element a i + b j A + B contains any rime factor 3 mod 4 since for such one would have: a i + b j 0 mod imlies that both a i 0 mod and b j 0 mod, which is not the case by construction Hence A + B T, where T = { P : = or mod 4}, and τ =

5 Additive decomosability of multilicatively defined sets 65 In this examle we have AB c log, which is uite close to the general uer bound O log given by Theorem 8 If τ > the same examle shows that the constructive lower bound differs from the uer bound by a logarithmic factor only just allow further rimes in T We do not exect that such examles exist if τ <, and for these cases I exect that if A B, then the size of these sets is considerably smaller than Since often constructions of large sets with certain roerties are more difficult in the case of A = B, let us note that with B and T as above we even have B + B T Moreover, some of the combinatorial constructions only work for finite sets, whereas the sets considered here are infinite Variations of the examle above are: artition P 4,3 = { P : 3 mod 4} = T T Let T i, x = τ i log log x+c i +o, for i =, Here τ +τ = Thus an exlicit examle with τ = τ = 4 is T = P 8,3 = { P : 3 mod 8} and T = P 8,7 = { P : 7 mod 8} Let A be the set of all suares comosed of rimes of P 4, T and let B be the set of suares comosed of rimes of P 4, T Then A c and B c log τ, so log τ that again AB c c And again by construction: for all rimes 3 mod 4: log a i + b j 0 mod For the determination of the counting function we made use of the following lemmas Lemma 3 Wirsing, [8] Let T denote a set of rimes and 0 < τ If x T = τ log log x + C + o, then {n, n T} C T log τ In view of the rime number theorem for arithmetic rogressions, Wirsing s result contains a well known result by Landau as a secial case Lemma 4 Landau [7] Let a,, a r be r distinct reduced residues modulo m Then {n, n a, a,, a r mod m} C m,a,a,,a r log r ϕm

6 66 Christian Elsholtz 3 Proof of Theorem 8 3 A roof using the analytic large sieve, based on Wirsing s aroach We need the following lemma: Lemma 3 Let T denote a set of rimes, let 0 < τ < be a real constant and let C be a ositive real constant If log τ log x < C, x then T n n T for some ositive constant C τ,c µ n C τ,c log τ, In order to rove this we make use of another result of Wirsing [9] which we adat from Schwarz and Silker [4], age 76 Lemma 3 Wirsing Let f be a real non-negative multilicative arithmetical function satisfying f G for all rimes and x f log with some constants G > 0, τ > 0 and k τ log x, f k k < If 0 < τ, then, in addition, the condition f k x = O log x is assumed to hold Then n x k, k x x e γτ fn = + o log x Γτ x + f + f + Here γ is the Euler-Mascheroni constant { if T Proof of Lemma 3 ow with f = and f 0 otherwise, k = 0, if k, all hyotheses of the lemma are satisfied and we are left with analysing

7 Additive decomosability of multilicatively defined sets 67 the roduct: x + f + f + = ex ex x, T x, T log + = exτ log log x + O C τ,c log x τ Here we used that from x, T log τ log x < C it follows by artial summation that x, T = τ log log x + O C For the remainder of this section we adat Wirsing s roof of Theorem 5 to the current roblem Let us state the analytic large sieve ineuality, see euation 0 of Montgomery [9] Lemma 33 Let A [, ], ex = exπix and S A := a A eax Then = r,= S A r + A Let S = P\T and S = {n : n S} ote that by Lemma 3 S τ,c log holds, and that for sufficiently large elements a + b A + B τ and S are corime We shall give an uer bound and a lower bound of r r S A S S B, r,= combining these will rove the theorem Lemma 34 S r,= r r S A S B + A B,

8 68 Christian Elsholtz Proof Using the Cauchy-Schwarz ineuality and the large sieve ineuality above we find: S S S r,= r,= r,= r r S A S B r r S A S B S A + A B r S Lemma 35 See 664 of Hardy and Wright [3] r,= This is a well known Ramanujan sum e r = µ Lemma 36 If a + b T and a + b >, then S r,= Proof Consider the identity: r,= S A r r,= S B r r r S A S B A B C τ,c log τ r S B = a A b B r,= a + br e Since a + b T with a + b > and S are corime by construction, the values a + br are just a rearrangement of the corime residue classes modulo so that r,= a + br e = r,= e r = µ

9 Additive decomosability of multilicatively defined sets 69 Therefore, by Lemma 3, r r S A S S B = A B r,= S µ A B C τ,c log τ Proof of Theorem 8 Let A [, ], A 0 := A [0, ], A := A [, ], and similarly for B Instead of studying the four roducts A i B j, with i, j {0, }, it is sufficient to rove the uer bound A B = O τ,c log τ without loss of generality for A [, ], B [, ] To see this observe that A 0 B 0 = O and that one can assume that A 0 B A B 0 Also note that a + b > > 0 so that one can assume that a + b T Combining the uer and lower bounds in Lemma 34 and 36 gives, with = : A B τ,c log τ Remark Since we do not sum over all but over S only, one might hoe for a stronger version of the large sieve ineuality relacing + by something smaller This would imrove Lemma 34 There is some discussion on this issue on age 564 of [9] 3 The roof using the arithmetic large sieve, following Pomerance, Sárközy and Stewart, and Hofmann and Wolke The roofs according to Pomerance, Sárközy, and Stewart [] and Hofmann and Wolke [4] are very similar We use the arithmetic version of the large sieve in a version due to Montgomery [9]: Lemma 37 Let C denote a set of integers which avoids ω residue classes modulo Here ω : P with 0 ω Then the following uer bound on the counting function holds: C +, where L = µ L Lemma 38 If A, B Z/Z, then A + B min, A + B ω ω Assume that A + B T, where T [ 0, ] = T [ 0, ] Let ν A = A mod and ν B = B mod denote the number of residue classes modulo that do occur in A and B, and let ω A = ν A and ω B = ν B denote the number of residue classes modulo that do not occur in A and B, resectively

10 70 Christian Elsholtz As above, without loss of generality, we consider A [, ] and B [, ] Every a + b A + B is larger than and does not contain any rime factor S = P\T, so that ν A+B holds for these rimes The Cauchy-Davenort ineuality further imlies ν A+B ν A + ν B, so that By definition and therefore ν A + ν B ω A + ν A =, ω B + ν B =, ω A + ω B, for S Lemma 39 Let S [, ], then Proof This follows from ω A ω B ω A ω B ω A ω B ω A ω B ω A ω B ω B ω A = Proof of Theorem 8 Using Montgomery s large sieve, the Cauchy-Schwarz ineuality, Lemma 3, and utting =, it follows that AB + µ ω A ω A µ S 4 µ S τ,c log τ 4 + µ ωa ω B ω A ω B τ,c log τ ω B ω B

11 Additive decomosability of multilicatively defined sets 7 4 The roof of Theorem 7 4 The roof of Bshouty and Bshouty This is a very simle roof Let us first do some rearation a a a n Lemma 4 Let S = Sa,, a n = are integers Then n i= i! divides det S a a a n a n a n an n, where the a i Proof ote that n i= in i = n n n = n i= i! Also recall that det S = i<j n a j a i is the well known Vandermonde determinant; the lemma states a less well known divisibility roerty of it Comare roblem 70 of [9] It can be roved as follows: a a a n a a a n det M = a n an =!!3! n! an n a a a a a a n + a a a a a a n + a n a n a n a n a n a n n + Exanding the roducts and adding suitable linear combinations of the receding columns shows that a a n a a n det M =!!3! n! a n an n Since det M is an integer it follows that det S is divisible by!!3! n! This roves the Lemma ow assume A = {a,, a n } [, ], B = {b,, b n } [, ] and A + B P T := n i= i! divides det Sa,, a n, b,, b n = a j a i b j b i a i + b j i<j n i<j n i n j n

12 7 Christian Elsholtz We will get the reuired relation between n and by finding uer and lower bounds on T Since T consists of rime factors < n only, T also divides S := a j a i b j b i a i + b j, i<j n i<j n a i+b j<n which imlies that T S It is imortant to note that S is considerably smaller n than det Sa,, a n, b,, b n Let us note that all but at most O log n terms of the form a i + b j are rimes > n This can be seen as follows: for fixed a i, all a i + b,, a i + b n are distinct rimes By Chebychev s ineuality, πn n log n so that only O n log n of the above a i + b j can be rimes less n than n Similarly, the size of the index i is bounded by i = O log n, giving n O ossibilities with a i + b j < n that log n Estimating now T from above and below using Stirling s formula shows n e n T S n n cn log n Taking the n -th root, this imlies that n 4 A variation, reminiscent of Gallagher s sieve The roof in this section is insired by Bshouty and Bshouty s roof but may be more natural While studying the roduct a j a i seems natural, studying the determinant of Sa,, a n, b,, b n aears to be a trick Here we study how often the rime factors < n occur in the roduct i<j n a j a i b j b i The roof below is selfcontained and does not make direct use of any sieve method But in a similar sirit it is ossible to rove Gallagher s larger sieve Lemma 4 Comare [5], age 45 If S be an odd rime, such that no a + b is divisible by, then ν A + ν B 4 With ν A + ν B see section 3 above we find that ν A + ν B ν A + ν A, which takes for its minimum at ν A = ± Lemma 43 Let r,, r k be nonnegative integer arameters with k i= r i = C, for a ositive constant C Then fr,, r k = k ri i= is minimised if the ri are as eual as ossible, ie r i C k, C k + Sketch roof We consider the r i to be continuous variables and use Lagrange s multilier method Here the minimum is attained for r i = C k For the discrete

13 Additive decomosability of multilicatively defined sets 73 analogue any deviation from the mean value which is larger than necessary increases the value of f : assume r = C k + + δ, where δ 0 There must be some r = C k δ, where δ > 0 Such a air r, r cannot be art of the minimal solution since it can be reduced to r = C k + δ and r = C k + δ, since r + r r r = δ + δ > 0 Similarly, assuming that a minimal solution would contain r = C k δ, r = C k + δ with δ 0, δ > 0 leads to a contradiction, which roves the Lemma The rime factor occurs in the roduct i<j n a j a i, whenever two a i s are in the same residue class modulo Let w i<j n a j a i, then w A A ν A, as can be seen as follows: ut r i = {a A : a i mod } Only ν A of the r i are ositive Further i=0 r i = n is constant and w ri i=0 By the lemma above, the right hand side takes its minimum if the distribution of A in the ν A residue classes is as eual as ossible, ie: w A ν A A ν A ν A This imlies: n A ν A A + B ν B B i<j n a j a i i<j n A A + B B With A = B = n and Lemma 4 it follows that n n n n Writing n n n = ex n log n + O, and ta- = ex n n king the n -th root it follows that n log b j b i 43 A roof based on Gallagher s larger sieve Here we show that a direct alication of Gallagher s larger sieve also roves Theorem 7 Gallagher s larger sieve is essentially already incororated into the ad hoc argument of the last section For convenience we use a variant of Gallagher s larger sieve, introduced by Croot and the author in [] Lemma 44 Variant of [] Let S [, ] denote a set of rimes or owers of rimes such that A [, ] lies in at most ν A residue classes modulo, for each S Then, S Λ 3 ex A max, ex Here Λ is the von Mangoldt function S Λ ν A

14 74 Christian Elsholtz Alying this to A = B = n with = gives: if is the maximum of any of the uer bounds on A or B, then n and so A B = n = Otherwise, combining Lemma 4 and Lemma 44 with S = P [, ], gives: ex S Λ A B ex S Λ ex S Λ ν A ex S Λ ν B ex ex log log ν A + log ν B ex log 5 Proof of Theorem 9 Assume that there exists an asymtotic additive decomosition of the set of rimes: A + B = P Here we rove sharer sieve bounds than were reviously roved in [4] The main idea of the roof in [4] was that an inverse alication of Gallagher s larger sieve alied to the smaller seuence A say shows that this seuence occuies many residue classes modulo many rimes y This information can successfully be injected into Montgomery s sieve to give an uer bound of B log 4 One of the obstacles to get a still better bound was that the sieve level u to which we used the distribution modulo rimes needed to be the same y = m = 4 for both arts of the sieve Even though we count in finite intervals, the original roblem is of course about infinite sets So, it is ossible to do both sieve arts indeendently It was already shown in [4], that A ε ε Since this holds for all, we can show, u to any level, by an inverse alication of Gallagher s larger sieve that for a ositive roortion of the rimes the number ν A of occuied classes modulo is large Let us artition the set S = [ 0, ] P into two arts: S = { S : ν A 0 log }, and S = { S : ν A > 0 log } By the rime number theorem π = log + log + O log, which imlies for 3 sufficiently large, S log Hence at least one of S log or S log holds We show that the first condition cannot hold, whence the second condition holds We use again variant of Gallagher s larger sieve, see Lemma 44 Let = 4 Assume that S log, which imlies 3 S Λ

15 Additive decomosability of multilicatively defined sets 75 Λ ex S A max, Λ ex S ν A max, ex 0 log S Λ max, =, ex 0 log which contradicts A ε ε For the roblem of infinite sets A, B and therefore A ε ε for all large, this imlies that S holds for all large log Since a + b > is a rime, we know that a + b 0 mod, for rimes This imlies that ωb ν A We ut = and use Montgomery s large sieve in the secial case, where the in the denominator are rimes in S : 3 B + S ω B 0 log log log Since AB log this imlies that A log By symmetry, the same uer and lower bounds hold on A and B 3 It should be noted that the arameter in both arts of the sieve alication is not necessarily the same The information about the distribution of the used classes modulo rimes u to in the first sieve argument ossibly makes use of elements a + b which are outside the interval [, ] that is used for the second sieve alication This trick obviously cannot work for the finite version of the roblem I would like to thank Eduard Wirsing for discussions on the roblem, for giving me access to his unublished manuscrit [30], and for comments on an earlier version of this aer I am also grateful to igel Watt for a discussion on section 3 of this aer, and to Ernie Croot for many discussions on sieve methods References [] D Bshouty and H Bshouty, A note on rime n-tules, Rocky Mountain J Math 7 997, [] E Croot and C Elsholtz, On variants of the larger sieve, Acta Math Hungarica , [3] C Elsholtz, A Remark on Hofmann and Wolke s Additive Decomositions of the Set of Prime, Arch Math 76 00, [4] C Elsholtz, The Inverse Goldbach Problem, Mathematika 48 00, 5-58

16 76 Christian Elsholtz [5] C Elsholtz, Some remarks on the additive structure of the set of rimes, umber theory for the millennium, I Urbana, IL, 000, 49 47, Proceedings of the Millennial Conference on umber Theory Bennett etal, AK Peters, 00 [6] P Erdős, uelues roblèmes de théorie des nombres, Monograhies de L Enseignement Mathématiue, no 6, 8-35, Université Geneva, 963 [7] P Erdős, Problems and results in number theory, umber Theory Day, ew York, 976, Sringer, Lecture otes in Math 66, 43 7 [8] P Erdős, CL Stewart and R Tijdeman, Some Diohantine euations with many solutions, Comositio Math , [9] DK Faddeev and IS Sominskii, Problems in Higher Algebra, WH Freeman and Comany, San Francisco 965 [0] GA Freĭman, Foundations of a structural theory of set addition, translated from the Russian version 966 Translations of Mathematical Monograhs, Vol 37 American Mathematical Society, Providence, 973 [] PX Gallagher, A Larger Sieve, Acta Arith 8 97, 77-8 [] GH Hardy and JK Littlewood, Some Problems of Partitio umerorum, III On The Exression of a umber as a Sum of Primes, Acta Math 44 93, -70 [3] GH Hardy and E M Wright, An Introduction to the Theory of umbers, fifth edition, Clarendon Press, Oxford, 989 [4] A Hofmann and D Wolke, On Additive Decomositions of the Set of Primes, Arch Math , [5] B Hornfeck, Ein Satz über die Primzahlmenge, Math Z , 7 73, see also the Zentralblatt review by Erdős and the correction in Math Z 6, 955, age 50 [6] WB Laffer and HB Mann, Decomosition of sets of grou elements, Pacific J Math [7] E Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner Verlag, Leizig, Berlin 909 [8] HB Mann, Addition theorems: The addition theorems of grou theory and number theory, Wiley Interscience, ew York-London-Sydney, 965 [9] H Montgomery, The Analytic Princile of the Large Sieve, Bull Amer Math Soc , [0] M athanson, Additive umber Theory, Inverse roblems and the geometry of sumsets Graduate Texts in Mathematics, 65 Sringer-Verlag, ew York, 996 [] H-H Ostmann, Additive Zahlentheorie, volumes, Sringer-Verlag, Berlin- -Heidelberg-ew York, 956, rerint 968 [] C Pomerance, CL Stewart and A Sárközy, On Divisors of Sums of Integers, III, Pacific J Math , [3] J-P Puchta, On additive decomositions of the set of rimes Arch Math 78 00, 4 5 [4] W Schwarz and J Silker, Arithmetical functions, Cambridge University Press, 994

17 Additive decomosability of multilicatively defined sets 77 [5] T Tao, V Vu, Additive Combinatorics, Cambridge University Press, in rearation [6] RC Vaughan, Some Alications of Montgomery s Sieve, J umber Theory 5 973, [7] E Wirsing, Ein metrischer Satz über Mengen ganzer Zahlen Arch Math 4 953, [8] E Wirsing, Über die Zahlen, deren Primteiler einer gegebenen Menge angehören Arch Math 7, 63 7, 956 [9] W Wirsing, Das asymtotische Verhalten von Summen über multilikative Funktionen, II, Acta Math Acad Sci Hung 8, [30] E Wirsing, Über additive Zerlegungen der Primzahlmenge, lecture at Oberwolfach, an abstract can be found in Tagungsbericht 8/97 An unublished written version 998 exists with the title On the additive decomosability of the set of rimes, A Memo [3] E Wirsing, Problem at the roblem session of the Oberwolfach conference Elementare und analytische Zahlentheorie 998 Abstract booklet of the Oberwolfach Conference 0/998 [3] D Wolke, Das Goldbach sche Problem, Math Semesterber 4 994, Address: Deartment of Mathematics, Royal Holloway, Egham, Surrey TW0 0EX, UK christianelsholtz@rhulacuk Received: 4 December 005; revised: June 006