THE NUMBER OF GOLDBACH REPRESENTATIONS OF AN INTEGER

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 3, March 01, Pages S ) Article electronicall published on Jul 0, 011 THE NUMBER OF GOLDBACH REPRESENTATIONS OF AN INTEGER ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI Communicated b Matthew A. Papanikolas) Abstract. Let Λ be the von Mangoldt function and Rn) h+kn Λh)Λk) be the counting function for the Goldbach numbers. Let N and assume that the Riemann Hpothesis holds. We prove that N Rn) N N ρ+1 ρρ +1) + ON log3 N), ρ where ρ 1/+iγ runs over the non-trivial eros of the Riemann eta-function ζs). This improves a recent result b Bhowmik and Schlage-Puchta. 1. Introduction Let Λ be the von Mangoldt function and Rn) Λh 1 )Λh ) h 1 +h n be the counting function for the Goldbach numbers. This paper is devoted to studing the behaviour of the average order of magnitude of Rn) forn [1,N], where N is a large integer. We have the following Theorem 1.1. Let N and assume the Riemann Hpothesis RH) holds. Then N Rn) N N ρ+1 ρρ +1) + O N log 3 N ), ρ where ρ 1/ +iγ runs over the non-trivial eros of the Riemann eta function ζs). The first result of this kind was proved in 1991 b Fujii, who subsequentl improved it see [4] [6]) until reaching the error term O N log N) 4/3). Then Granville [8]-[9] gave an alternative proof of the same result and, finall, Bhowmik and Schlage-Puchta [] were able to reach the error term O N log 5 N ).In[]the also proved that the error term is ΩN log log N). Our result improves the upper bound in Bhowmik and Schlage-Puchta [] b a factor of log N. In fact, this seems to be the limit of the method in the current state of the circle-method technolog; see the remark after the proof. Received b the editors November 11, 010 and, in revised form, December 16, Mathematics Subject Classification. Primar 11P3; Secondar 11P55. Ke words and phrases. Goldbach-tpe theorems, Hard-Littlewood method. 795 c 011 American Mathematical Societ Reverts to public domain 8 ears from publication

2 796 ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI If one admits the presence of some suitable weight in our average, a sharper result can be obtained. For example, using the Fejér weight we could work with LN; α) N n N N n )enα) T N; α) instead of T N; α) in 4.). The ke propert is that, for 1/N < α 1/, the function LN; α) decas as α instead of α 1, and so the dissection argument in 4.5) is now more efficient and does not cause an loss of logs. Such a phenomenon is well-known from the literature about the existence of Goldbach numbers in short intervals; see, e.g., Languasco and Perelli [1]. In fact we will obtain Theorem 1.1 as a consequence of a weighted result. Letting ψx) m x Λm), we have Theorem 1.. Let N and assume the Riemann Hpothesis RH) holds. Then [ ] 1.1) max Rn) ψn) n) e n/n N log 3 N. [,N] The ke reason wh we are able to derive Theorem 1.1 from Theorem 1. via partial summation is that the exponential weight in 1.1) just varies in the range [e 1/N,e 1 ], and so it does not change the order of magnitude of the functions involved. We will use the original Hard and Littlewood [10] circle method setting, i.e., the weighted exponential sum 1.) Sα) Λn)e n/n enα), where ex) expπix),sinceitletsusavoidtheuseofgallagher slemmalemma 1 of [7]) and hence, in this conditional case, it gives slightl sharper results; see Lemma.1 below. Such a function was also used b Linnik [13, 14]. The new ingredient in this paper is Lemma.5 below in which we unconditionall detect the existence of the term ρ N ρ+1 /ρρ + 1)) connecting it to a suitable average of the difference ψn) n. In the previousl mentioned papers this is obtained b appling the explicit formula for ψn) twice. The ideas that led to Theorems 1.1 and 1. also work for the sum of k 3 primes, i.e., for the function R k n) Λh 1 ) Λh k ). We can prove the following. h h k n Theorem 1.3. Let k 3 be an integer, N and assume the Riemann Hpothesis RH) holds. Then N R k n) N k k N ρ+k 1 ) k! ρρ +1) ρ + k 1) + O k N k 1 log k N, ρ where ρ 1/+iγ runs over the non-trivial eros of ζs). The proof of Theorem 1.3 is completel similar to the one of Theorems 1.1 and 1.. We just remark that the main differences are in the use of the explicit formula

3 GOLDBACH REPRESENTATIONS OF AN INTEGER 797 for ψ j t) : 1 t n) j Λn), j! n t where j is a non-negative integer, and of the following version of Lemma 1 of [11]: Lemma. Assume the Riemann Hpothesis RH) holds. Let N, 1/N πiα and α [, 1/]. Then Sα) 1 N 1/ 1+N α ) 1/) log N. Another connected problem we can address with this technique is a short-interval version of Theorem 1.1. We can prove the following. Theorem 1.4. Let H N and assume the Riemann Hpothesis RH) holds. Then N+H Rn) HN + H N + H) ρ+1 N ρ+1 + O N log N log H ), ρρ +1) nn ρ where ρ 1/+iγ runs over the non-trivial eros of ζs). Also in this case we do not give a proof of Theorem 1.4. We just remark that the main difference is in the use of the exponential sum N+H nn enα) instead of N enα).. Setting of the circle method For brevit, throughout the paper we write.1) 1 N πiα, where N is a large integer and α [, 1/]. The first lemma is an L -estimate for the difference Sα) 1/. Lemma.1 Languasco and Perelli [1]). Assume RH. Let N be a sufficientl large integer and be as in.1). For0 ξ 1/, we have ξ Sα) 1 dα Nξlog N. ξ This follows immediatel from the proof of Theorem 1 of [1] since the quantit we would like to estimate here is R 1 + R 3 + R 5 there. Lemma.1 is the main reason wh we use Sα) instead of its truncated form Sα) N Λn)enα) as in Bhowmik and Schlage-Puchta []. In fact Lemma.1 lets us avoid the use of Gallagher s Lemma [7], which leads to a loss of a factor log N in the final estimate compare Lemma.1 with Lemma 4 of []). For a similar phenomenon in a slightl different situation, see also Languasco [11]. The next four lemmas do not depend on RH. B the residue theorem one can obtain Lemma. Eq. 9) of [1]). Let N, 1 n N and be as in.1). We have e nα) 1 dα ne n/n + O1) uniforml for ever n N.

4 798 ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI Lemma.3. Let N be a sufficientl large integer and be as in.1). We have Sα) 1 dα N Nlog log N + O N) 1/). 1 Proof. B the Parseval theorem and the Prime Number Theorem we have Sα) dα Λ m)e m/n N log N + ON). 1 m1 Recalling that the equation at the beginning of page 318 of [1] implies 1 dα N π arctanπn), the lemma immediatel follows using the relation a b a + b Rab) and the Cauch-Schwar inequalit. Let.) V α) e m/n emα) m1 m1 e m 1 e 1. Lemma.4. If satisfies.1), thenv α) 1 + O1). Proof. We recall that the function w/e w 1) has a power-series expansion with radius of convergence π see for example Apostol [1], page 64). In particular, uniforml for w 4 < π we have w/e w 1) 1 + O w ). Since satisfies.1) we have 4, and the result follows. Now let.3) T ; α) ) 1 enα) min ;. α Lemma.5. Let N be a large integer, N and be as in.1). We have.4) / T ; α) Sα) 1/) dα e n/n ψn) n)+o N log N) 1/). We remark that Lemma.5 is unconditional, and hence it implies, also using Lemma.6, that the abilit of detecting the term depending on the eros of the Riemann ζ-function in Theorem 1.1 does not depend on RH. Proof. Writing Rα) Sα) 1/, b Lemma.4 we have.5) / T ; α) Rα) dα / / T ; α) Rα)V α)dα ) 1/ + O T ; α) Rα) dα T ; α) Rα)V α)dα + O N log N) 1/),

5 GOLDBACH REPRESENTATIONS OF AN INTEGER 799 since, b the Parseval theorem and Lemma.3, the error term above is / ) 1/ / T ; α) dα Rα) 1/ dα) N log N) 1/. Again b Lemma.4 we have and hence.5) implies / Rα) Sα) 1 Sα) V α)+o1), / T ; α) Rα) dα T ; α) ) Sα) V α) V α)dα ) 1/.6) + O T ; α) V α) dα + O N log N) 1/). The Cauch-Schwar inequalit and the Parseval theorem impl that / / ) 1/ / ) 1/ T ; α) V α) dα T ; α) dα V α) dα.7) B.6)-.7), we have.8) / T ; α) Rα) dα e m/n) 1/ N) 1/. m1 / T ; α) ) Sα) V α) V α)dα + O N log N) 1/). Now, b 1.) and.) we can write Sα) V α) Λm) 1)e m/n emα) m1 so that / T ; α) ) Sα) V α) V α)dα Λm 1 ) 1)e m 1/N.9) m 1 1 Λm 1 ) 1)e m 1/N m 1 1 n 1 e n/n m 1 1 Λm 1 ) 1) / e m /N m 1 m 1 e m /N em 1 + m n)α)dα { 1 if m 1 + m n, 0 otherwise e n/n ψn 1) n 1)), since the condition m 1 + m n implies that both variables are <n.nowψn) ψn 1) + Λn) sothat e n/n ψn 1) n 1)) e n/n ψn) n)+o).

6 800 ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI B.8)-.9) and the previous equation, we have / T ; α) Rα) dα e n/n ψn) n)+o +N log N) 1/) e n/n ψn) n)+o N log N) 1/) since N, and hence.4) is proved. Lemma.6. Let M>1be an integer. We have that n) ψn) M ρ+1 ρρ +1) + OM). ρ Proof. We recall the definition of ψ 0 t) asψt) Λt)/ ift is an integer and as ψt) otherwise. Hence ψn) ψ 0 n)+ 1 Λn) ψ 0 n)+om) b the Prime Number Theorem. Using the fact that ψ 0 n) ψ 0 t) for ever t n, n + 1), we also get n+1 M ψ 0 n) ψ 0 t) dt ψ 0 t) dt + OM). Remarking that we can write.10) ψn) n) M 0 n n M 0 0 t dt + OM), ψ 0 t) t) dt + OM) M ψ 0 t) t) dt + OM), since the integral on 0, ] gives a contribution of O1). For t we will use the explicit formula see eq. 9)-10) of 17 of Davenport [3]).11) ψ 0 t) t t ρ ρ ζ ζ 0) 1 1 log 1t ) + R ψ t, Z), γ Z where.1) R ψ t, Z) t ) t Z log tz)+logt)min 1;. Z t The term ζ ζ 0) 1 log ) 1 1 t gives a contribution of OM) to the integral over [,M] in.10). We now need an L 1 estimate of the error term defined in.1). Let.13) EM,Z) : M R ψ t, Z) dt. The first term in.1) gives a total contribution to EM,Z) whichis.14) M Z log MZ).

7 GOLDBACH REPRESENTATIONS OF AN INTEGER 801 The second term in.1) gives a total contribution to EM,Z) whichis.15) log M [ n+1/ n n log M + log M min 1; n+1/z n n n+1 1/Z n+1/ 3 Z n t ) n+1 dt + min 1; Zt n) n+1/ n+1/ dt + n+1/z ) t dt Zn +1 t) + n +1 Z t dt Zt n) + t ) ] dt Zn +1 t) n+1 n+1 1/Z Z )) log M log M log Z. Z Combining.13)-.15), for Z M log M we have that.16) EM,Z) M. Now inserting.11) and.16) into.10) we obtain.17) ψn) n) M ρ+1 ρρ +1) + OM). γ >Z γ Z The lemma follows from.17) b remarking that M ρ+1 + ρρ +1) M log t t dt M log Z Z Z M log M dt since Z M log M. 3. Proof of Theorem 1.1 We will obtain Theorem 1.1 as a consequence of Theorem 1.. B partial summation we have N [ ] N {[ Rn) ψn) n) e n/n Rn) ψn) n) ]e } n/n 3.1) e N [ ] Rn) ψn) n) 1 N N 0 e n/n { [ ] } Rn) ψn) n) e n/n e /N d + O1). Inserting 1.1) in 3.1) we get N [ ] Rn) ψn) n) N log 3 N, and hence 3.) N Rn) N N n + ψn) n)+o N log 3 N ).

8 80 ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI Theorem 1.1 now follows b inserting Lemma.6 and the identit N n N /+ ON) in 3.). 4. Proof of Theorem 1. Let N. We first recall the definition of the singular series of the Goldbach problem: Sk) 0fork odd and Sk) p> 1 1 ) p 1) p k p> p 1 p for k even. Hence, using the well-known estimate Rn) nsn) n log log n, we remark that 4.1) [ ] Rn) ψn) n) e n/n n log log n log log. Therefore, it is clear that 1.1) holds for ever [,N 1/ ]. Assume now that [N 1/,N] and let α [, 1/]. Writing Rα) Sα) 1/ and recalling.3) we have, sa, 4.) e n/n Rn) 1 1 T ; α) dα + I 1 )+I )+I 3 ). Sα) e nα) dα 1 T ; α) Rα) 1 dα + Sα) T ; α) dα 1 T ; α) Rα) dα Evaluation of I 1 ). B Lemma. we obtain 4.3) I 1 ) T ; α) 1 dα ne n/n + O). 1 e nα) dα ) ne n/n + O1) Estimation of I ). B.4) of Lemma.5 we obtain 4.4) I ) e n/n ψn) n)+o N log N) 1/).

9 GOLDBACH REPRESENTATIONS OF AN INTEGER 803 Estimation of I 3 ). Using.3) and Lemma.1 we have that I 3 ) 1 1 T ; α) Rα) dα Rα) dα + 1 Olog ) N log N + k1 k Rα) k+1 k α 1 dα + 1 Rα) dα Rα) Olog ) N log N + k N k+1 log N k1 4.5) N log N log. End of the proof. Inserting 4.3) and 4.4)-4.5) into 4.) we immediatel have e n/n Rn) ne n/n + e n/n ψn) n)+o N log N log ). Hence [ ] e n/n Rn) ψn) n) N log N log, and the maximum of the right hand side is attained at N. Thus we can write [ ] 4.6) max Rn) ψn) n) e n/n N log 3 N. [N 1/,N] Combining 4.1) and 4.6) we get that Theorem 1. is proved. Remark. Let { 1 fα) f N α) N 1/ log N if α log N) 1, 0 if α > log N) 1. Then f satisfies both Lemma.1 and Lemma.3 in the sense that for all ξ [0, 1/], and However, / 1/ ξ ξ / fα) dα Nξlog N) fα) dα 1 N log N. fα) dα 1 / log N α 4 Nlog N) 1/ dα α α 1 4 Nlog N) log/ log N) Nlog N) 3 for N 1/ and sufficientl large N. This means that the crucial bound for I 3 ) in 4.5) is essentiall optimal in the present state of knowledge and that it cannot be dα

10 804 ALESSANDRO LANGUASCO AND ALESSANDRO ZACCAGNINI improved without deeper information on Sα) 1, such as the stronger analogue of Lemma.1 that follows from a suitable form of Montgomer s Pair-Correlation Conjecture. Acknowledgment The authors would like to thank Alberto Perelli for a discussion. References [1] T. Apostol, Introduction to Analtic Number Theor, Springer-Verlag 1976). MR :789) [] G. Bhowmik, J.-C. Schlage-Puchta, Mean representation number of integers as the sum of primes, Nagoa Math. J., ), MR [3] H. Davenport, Multiplicative Number Theor, Springer-Verlag, 3rd ed. 000). MR f:11001) [4] A. Fujii, An additive problem of prime numbers, ActaArith., ), MR k:11106) [5] A. Fujii, An additive problem of prime numbers. II, Proc. Japan Acad. Ser. A Math. Sci., ), MR b:1113a) [6] A. Fujii, An additive problem of prime numbers. III, Proc. Japan Acad. Ser. A Math. Sci., ), MR b:1113b) [7] P.X.Gallagher,A large sieve densit estimate near σ 1, Invent. Math., ), MR :4775) [8] A. Granville, Refinements of Goldbach s conjecture, and the generalied Riemann hpothesis, Funct. Approx. Comment. Math., ), MR b:11179) [9] A. Granville, Corrigendum to Refinements of Goldbach s conjecture, and the generalied Riemann hpothesis, Funct. Approx. Comment. Math., ), MR c:1115) [10] G. H. Hard, J. E. Littlewood, Some problems of Partitio Numerorum ; III: On the expression of a number as a sum of primes, ActaMath.,44 193), MR [11] A. Languasco, Some refinements of error terms estimates for certain additive problems with primes, J. Number Theor, ), MR h:11130) [1] A. Languasco, A. Perelli, On Linnik s theorem on Goldbach number in short intervals and related problems, Ann. Inst. Fourier, ), MR g:11097) [13] Y. Linnik, A new proof of the Goldbach-Vinogradow theorem, Rec. Math. N.S., ), 3 8 Russian). MR :317c) [14] Y. Linnik, Some conditional theorems concerning the binar Goldbach problem, Iv.Akad. Nauk SSSR Ser. Mat., ), Russian). MR :847b) Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 3511 Padova, Ital address: languasco@math.unipd.it Dipartimento di Matematica, Università di Parma, Parco Area delle Sciene 53/a, Campus Universitario, 4314 Parma, Ital address: alessandro.accagnini@unipr.it