VINOGRADOV S THREE PRIME THEOREM

Transcription

1 VINOGRADOV S THREE PRIME THEOREM NICHOLAS ROUSE Abstract I sketch Vinogradov s 937 proof that every sufficiently large odd integer is the su of three prie nubers The result is dependent on nuerous interediate results, soe of which I prove and others of which have proofs too long to give here The ain technique is decoposition into ajor and inor arcs Contents The von Mangoldt function and counting representations 2 A first bound 2 3 Soe definitions 2 4 Working with ajor arcs 3 5 The inor arcs 6 Vinogradov s Theore 2 Acknowledgents 3 References 3 The von Mangoldt function and counting representations Definition We define the von Mangoldt function as { log p if n = p k for a prie nuber p, Λn = otherwise The otivation for introducing this function is another function Definition 2 rn = k +k 2+k 3=n Λk Λk 2 Λk 3 The function rn is the nuber of ways to express n as the su of three nubers that each are either prie or a power of a prie nuber with weight of logp logp 2 logp 3 attached to that value Clearly, if there are no representations of a given integer in our case, we shall be interested in odd integers as the su of three prie nubers, then rn = However, rn does not guarantee that such a representation exists, only that one of prie powers does However, fro partial suation we can see that the prie powers contribute less than the actual prie nubers, so a bound on rn will be sufficient Date: August 3, 23

2 2 NICHOLAS ROUSE Consider the su 2 A first bound Sα = k N Λne 2πkiα, for soe arbitrary constants n and N We can recover soething resebling rn if we take the cube of the su Sα 3 = Λk Λk 2 Λk 3 e 2πk+k2+k3iα k,k 2,k 3 N This looks siilar to rn but there is the issue that k, k 2, and k 3 are each less than a constant N rather than their su equal to another constant n We will decopose the su so that k, k 2 and k 3 = n is part of the su Sα 3 = n k +k 2+k 3=n k,k 2,k 3 N Λk Λk 2 Λk 3 e2πniα Again, this is not quite rn because if n > N, the inner su will not necessarily count all representations appropriately weighted, of course However, this is not really an issue as we can denote the inner su rn, N and point out that it is identical to rn for n N, so we ll write Sα 3 as Sα 3 = n rn, Ne 2πniα We have a Fourier series, so we can find the coefficients of the Fourier series 2 rn = Sα 3 e 2πniα dα, R/Z where R/Z is the quotient group of the real nubers odulo the integers As a brief reark, R/Z is isoorphic to the unit circle, so the integral can be understood as integrating over the circle Appropriately, the technique of setting up such an integral is called the Hardy-Littlewood circle ethod There is no obvious way to bound this integral, but the approach of Vinogradov is to consider the integral over subintervals of R/Z The subintervals are the ajor arcs denoted M and the inor arcs denoted Intuitively, the ajor arcs are the subintervals near a rational nuber with a sall denoinator Minor arcs are everything else in R/Z We will define ajor and inor arcs ore precisely 3 Soe definitions Definition 3 For constants n and B, let P = log B n and Q = n/ log 2B n For any q P and a such that a q where the greatest coon divisor of a and q henceforth denoted a, q is, we define Ma, q = {α R/Z α a q Q } Moreover, let M be the union of all such Ma, q and be the copleent of M in R/Z Lea 32 is nonepty

3 VINOGRADOV S THREE PRIME THEOREM 3 Proof Any two ajor arcs are disjoint To prove this, we take a q a q have for sufficiently large N a q a q qq P 2 > 2 Q Since the ajor arcs are then not all of R/Z, is nonepty Then we If we can bound 2 when integrating over the ajor and inor arcs individually, we will succeed in bounding the whole integral For the sake of exposition, however, we will first state soe basic definitions fro group theory and analytic nuber theory for objects we will use Definition 33 For integers a and q, the congruence class or residue class of a od q is denoted as ā q and defined as a q = {a + kn k Z} Moreover, the integer a is said to be the representative integer or siply the representative for the residue class Definition 34 The integers odulo q denoted Z/qZ is the set of all congruence classes a od q That is Z/qZ = {a q a Z} Definition 35 A Dirichlet character to odulus q is any function χ : Z C with the following properties: If p and q are not relatively prie, then χp = That is, if p, q, χp = 2 If p, q =, then χp = 3 If q and q 2 are any two positive integers, then χq q 2 = χq χq 2 If q =, the Dirichlet character is called the trivial character and denoted χ A Dirichlet character χ is called priitive odulo q if for every divisor d of q there exists an integer x od d and x, q =, such that χx Moreover, the set of all Dirichlet characters odulo q is denoted Ẑ/qZ 4 Working with ajor arcs We start with an individual Ma, q For any character χ to odulus q, we can consider the Gauss su τχ = χe 2πi/q We can consider another su φq Z/qZ χ Ẑ/qZ χnτχ, where φq is the Euler totient function and is equal to the nuber of positive integers less than or equal to q such that n, q = and χ is siply the coplex conjugate of the Dirichlet character By the first property of definition 35 we have, χnτχ = if n, q φq χ Ẑ/qZ

4 4 NICHOLAS ROUSE However, if n, q =, then we have fro the Gauss su, φq χ Ẑ/qZ χnτχ = φq χ Ẑ/qZ Z/qZ If we take nh od q we then have, χnχe 2πi/q = φq φq φq Then we have, χ Ẑ/qZ Z/qZ = φq h Z/qZ 4 e 2πn/q = φq e 2πinh/q = e 2πn/q χnχe 2πi/q χ Ẑ/qZ Z/qZ χ Ẑ/qZ χh χ Ẑ/qZ χnτχ We return to the function Sα and take α = a/q + β Sα = Λke 2πkia/q+β + Olog 2 N k N k,q= χhe 2πinh/q The error ter accounts for fact that this su restricts Sα to k values such that k, q = At any rate, we can anipulate this su using 4 Sα = φq = φq Λk k N χ Ẑ/qZ χkaτχe 2πikβ + Olog 2 N τχχa χkλke 2πikβ + Olog 2 N χ Ẑ/qZ k N Here we can use suation by parts 42 χkλke 2πikβ = e N 2πiNβ χnλn 2πiβ e 2πiuβ χnλndu k N n N n u We introduce ψx, χ, which we define as ψx, χ = n x χnλn We can rewrite 42 as 43 χkλke 2πikβ = e 2πiNβ ψn, χ 2πiβ N k N We have reached the first very iportant lea e 2πiuβ ψu, χdu

5 VINOGRADOV S THREE PRIME THEOREM 5 Lea 44 Let χ be a nontrivial Dirichlet character to odulus q If q log M x for a positive constant M, then ψx, χ = Oxe CM logx, for a positive constant CM which is a function of solely M This lea is actually equivalent to the prie nuber theore for arithetic progressions This lea is the first but not last result whose proof I ust oit for space considerations We can apply the lea to 43 to obtain a bound for the nontrivial Dirichlet characters χkλke 2πikβ = O + β NNe c logn k N However, our lea does not treat the contribution fro the trivial Dirichlet character We can work with another ψ function this one is known as the suatory von Mangoldt function or the second Chebyshev function ψx = n x Λn Note that χ = Then we have that ψx, χ ψx { if n, q = if n, q n x n,q> Λn = Ologq logx The prie nuber theore gives us an additional bound for ψx, ψx = x + Oxe c logx We can set ψu, χ = u + Ru to obtain 45 Ru = Oue c logu We define another function T β = k N We again use suation by parts to obtain T β = e 2πiNβ 2πiNβ Applying this result to 42, we have e 2πikβ N e 2πiNβ u du χ kλke 2πikβ = T β + e 2πiNβ RN 2πiβ k N We use 45 to get T β + e 2πiNβ RN 2πiβ N N e uβ Rudu e uβ Rudu = T β + O + β NNe c logn

6 6 NICHOLAS ROUSE So it is clear that the su for the trivial Dirichlet character differs fro the su for nontrivial Dirichlet characters given by the lea by only the ter T β We can cobine the contributions fro both the trivial and nontrivial Dirichlet characters to obtain 46 Sα = φq τχ T β + O + β NNe c logn + χ Z/qZ χ χ That is an estiate, but two facts about Gauss sus will iprove it First, we need a definition τχχa O + β NNe c logn Definition 47 The Möbius function µ : N {,, } is defined as if q is not divisible by a perfect square and has an even nuber of prie factors, µq = if q is not divisible by a perfect square and has an odd nuber of prie factors, if q is divisible by a perfect square Lea 48 Let τχ be a Gauss su and µq be the Möbius function, then Proof τχ = q τχ = µq e 2πi/q χ e 2πi/q = = q,q= Setting d = we have q = µd e 2πi d/q = µq d= q/d Lea 49 Let τχ be a Gauss su, then τχ q Proof We will prove this fact for priitive Dirichlet characters and oit the longer proof regarding ipriitve characters Taking χ priitive we have χnτχ = χhe 2πinh/q We can consider the square χn 2 τχ 2 = n Z/qZ So then we have that h Z/qZ h Z/qZ h 2 Z/qZ φq τχ 2 = q h Z/qZ and it follows that τχ q χh χh 2 n Z/qZ χh 2 = qφq, e 2πih h2n/q

7 VINOGRADOV S THREE PRIME THEOREM 7 With these two facts, we can significantly clean up 5 to obtain Sα = µq φq T β + O + β N qne c logn Returning to the definition of ajor arcs, we can set q P and β Q because α is in the ajor arc Ma, q So we have Sα = µq φq T β + ONe c logn If we take the cube, we can return to the Fourier series that we are interested in Sα 3 e 2πiNα dα = µq /Q φq 3 e 2πiaN/q T β 3 e 2πiNβ dβ+on 2 e c logn Ma,q /Q The c erely accounts for the possibility that there is different constant here This integral accounts for a single ajor arc If we want to find the contribution of all the ajor arcs, we will have to su over all of the q /Q M Sα 3 e 2πiNα dα = q P The su µq φq 3 a= a,q= q a= a,q= e 2πiaN/q e 2πiaN/q /Q T β 3 e 2πiNβ dβ+on 2 e c logn is called Raanujan s su and is denoted c q We now ake a detour into soe iportant definitions and results fro nuber theory The penultiate proof ust be oitted for space, but all others will be supplied Definition 4 An arithetic function is a function f : N C Moreover, we define addition for arithetic functions f and g as f + gn = fn + gn Definition 4 An arithetic function f is ultiplicative if fn = ffn whenever, n = Lea 42 If f is a ultiplicative function of n and li fp k =, p k where p k is a prie power so the liit goes through all prie powers, then li fn = n Proof The hypothesis iplies that there are only finitely any prie powers such that fp k We define A = fp k fp k

8 8 NICHOLAS ROUSE Clearly, A Choose ɛ such that < ɛ < A Then there are only finitely any prie powers such that fp k ɛ/a If we consider a fixed p k that divides an integer n, there will only be finitely any integers n such that fp k ɛ/a It follows that if n is sufficiently large, there is at least one prie power p k that divides n and fp k < ɛ/a Take p,, p r+s+t as pairwise distinct pries with the following properties fp ki i for i =,, r 2 ɛ/a fp ki i < for i = r +,, r + s 3 fp ki i < ɛ/a for i = r + s +,, r + s + t 4 t We can write n as n = r i= Then we have r r+s fn = fp ki i i= i=r+ p ki i r+s i=r+ fp ki p ki i r+s+t i=r+s+ r+s+t i i=r+s+ Lea 43 For ɛ > and sufficiently large n, n ɛ < φn < n p ki i fp ki i < Aɛ/A t ɛ Proof Clearly, if n >, then φn < n, and for prie nuber p, have p ɛ φp = p ɛ p p = p p ɛ p p It follows that We point out that n ɛ φn Lea 44 Let p ɛ li p φp = 2 p ɛ p p is ultiplicative and invoke Lea 4 to obtain c q = n ɛ li n φn q a= a,q= e 2πiaN/q c q is ultiplicative in q 2 If, for p prie, p α is the highest power of p that divides n, then φp β if β α, c p β n = p α if β = α +, if β > α + 2, so we

9 VINOGRADOV S THREE PRIME THEOREM 9 Lea 45 Let GN = q= µqc q n φq 3 GN converges absolutely and uniforly in N and has Euler product GN = p 2 + p 3 p N Moreover, there exist positive constants c and c 2 such that for all positive odd N p N c < GN < c 2 Proof Clearly we have c q N < φn For positive ɛ and sufficiently large q, by Lea 42, we have µqc q n φq 3 < φq 2 < q 2 ɛ So GN converges absolutely and uniforly in N Moreover, using Lea 43 with β =, we have { p if p N, c p N = if p N Since the Möbius function, the Euler totient function, and Raanujan s su are all ultiplicative, we can evaluate the Euler product for GN GN = µp j c p j N + φp j 3 = c pn φp 3 p j= p = p 2 + p 3 p N It follows that for all N > there exist c and c 2 such that c < GN < c 2 As an observation, note that GN = when N is even We can use this estiate to bound another part of the su µqc q n φq 3 = O φq 2 = Olog B+ N, q>p q>p where B is a constant that we will choose later We can cobine these two results to bound the su that actually appears in the integral over the ajor arcs µqc q N 46 φq 3 = GN + Olog B+ N q P We have obtained suitable bounds for a ter in the integral over the ajor arcs, but we still have to treat another ter We will do this by splitting the integral into an integral with ore tractable liits of integration and a ter we can bound ore easily /Q /Q T β 3 e 2πiNβ dβ = T β 3 e 2πiNβ dβ + O T β 3 dβ /Q p N /Q

10 NICHOLAS ROUSE We invoke that we defined T β as a geoetric series and can evaluate it as such We define x R/Z as the distance between x and the nearest integer T β = e2πin+b e 2πiβ e 2πiβ = Oin N, β R/Z We can evaluate the rightost ter /Q /Q T β 3 e 2πiNβ dβ = T β 3 e 2πiNβ dβ + OQ 2 We express Q in ters of N as in the definition of ajor arcs T β 3 e 2πiNβ dβ + OQ 2 = T β 3 e 2πiNβ dβ + O N 2 log 4B N The reaining integral counts representations of N as the su of three positive integers So we can have a bound for it T β 3 e 2πiNβ dβ = 2 N N 2 = 2 N 2 + ON Cobining these results we have a coplete bound for the integral /Q 47 tβ 3 e 2πiNβ dβ = N O N 2 log 4NB N /Q Cobining results we obtain a desirable bound for the integral over the ajor arcs 48 Sα 3 e 2πiNα dα = 2 GNN 2 N 2 + O log B N M 5 The inor arcs As the nae perhaps iplies, the inor arcs are less iportant than the ajor ones In fact, Vinogradov s theore relies on the inor arcs contributing sufficiently sall aounts that the ajor arcs doinate The integral we ust consider then is over the inor arcs We can begin with soe straightforward bounds Sα 3 e 2πiNα dα ax Sα Sα 2 dα Of course, the whole interval [, ] will contribute ore on the integral than just the inor arcs, so we can bound the integral over the inor arcs above with that interval and obtain a ore workable integral ax Sα Sα 2 dα ax Sα Sα 2 dα We can treat this integral using the definition of Sα Sα 2 dα = Λk Λk 2 e 2πiαk k2 dα k N k 2 N Taking k = k 2 = k we have a bound 5 Λk Λk 2 e 2πiαk k2 dα = Λk 2 = ON logn k N k N k 2 N

11 VINOGRADOV S THREE PRIME THEOREM We have to consider ax Sα We will start by working with the definition of inor arc Take α and q P α a q > Q = log2b N N We can generalize to a stateent about all eleents of the inor arcs 52 inf q P α a q > log2b N N We are in need of the last lea whose proof is too lengthy to give here Lea 53 For α R, C >, fixed k, and sufficient large N, if then inf dα R/Z k log28c+4 N, d 6 log 8C+4N N N n N µne 2πinα = Olog C+ N The stateent of the lea involves the Möbius function We will have to anipulate it to work with the von Mangoldt function We can work with the definitions of these two functions Λn = n logdµ d d n The Möbius function is not defined on values other than the natural nubers, but we can fill in the function by taking µx = when x N We can rewrite the von Mangoldt function in ters of this new Möbius function Λn = d n logdµ nd We can rewrite Sα Sα = n N Λne 2πinα Note that this is siply the original definition of Sα We replace the von Mangoldt function with the Möbius function Λne 2πinα = n logdµ e 2πinα d n N n N n N d n d n We define = n d and further anipulate the su n logdµ d e 2πinα = logd µe 2πidα d N N d We transfor 52 to soething that resebles the conditions on the lea by considering the distance fro qα to the nearest integer inf q log B N qα R/Z > log4b N N

12 2 NICHOLAS ROUSE We can recover the conditions on the lea with B = 9C + 4 We can bound Sα in the inor arcs Sα < N/d logd d N log B 9 5 N/d N log B 9 6 N d = O N log B 9 6 N d N We can bring this bound with 5 to obtain a bound over the inor arcs 54 Sα 3 e 2πiNα N 2 dα = O log B 9 6 N 6 Vinogradov s Theore We can cobine these results into Vinogradov s theore Theore 6 For A >, rn = 2 GNN 2 + O Proof Fro 2 we have rn = R/Z N 2 log A N Sα 3 e 2πniα dα Fro ajor and inor arc decoposition, we split the integral obtaining, Sα 3 e 2πniα dα = Sα 3 e 2πniα dα + Sα 3 e 2πniα dα R/Z M Fro 44, we have Sα 3 e 2πiNα dα = 2 GNN 2 N 2 + O log B N M Fro 54 we have Sα 3 e 2πiNα dα = O N 2 log B 9 6 N We take B = 9A + 6, and note that the power on the logarith is uch saller in the denoinator of the error ter fro the inor arcs than that fro the ajor arcs, so we have rn = Sα 3 e 2πniα dα + Sα 3 e 2πniα dα = 2 GNN 2 N 2 + O log A N M Corollary 62 Every sufficiently large odd integer is the su of three pries

13 VINOGRADOV S THREE PRIME THEOREM 3 Proof By Lea 44 GN is bounded for all odd N, so for sufficiently large N, N 2 is uch greater than GN, so rn, the nuber of representations of n as the su of three prie powers, is bounded below for sufficiently large N Moreover, we can see by partial suation that the contribution fro sus of proper prie powers is uch less than that of prie nubers Therefore, for sufficiently large values of N, N has a representation as the su of three prie nubers Acknowledgents This paper would not have been possible without the help of y entor Mohaad Rezaei Not only did he ake e aware of the theore this paper covers, but he guided e to the iddle path away fro a paper that would have been too siple by assuing the Generalized Rieann Hypothesis and a paper that would be five ties as long as the present one if I had attepted to prove all of the results stated Moreover, Mohaad was invaluable in answering questions about this paper As far as written atheatical sources go, this paper largely follows the path set by Ian Petrow, the first source in the bibliography Suppleental aterial about nuber theory is ostly fro the next two sources, Nathanson and Apostol The reaining three web sources were useful in understanding the ethods and steps featured in Petrow s paper Of course, it would not do not to thank Peter May for his efforts in running the REU that allowed e to spend y suer doing atheatics References [] Ian N Petrow Vinogradov s Three Pries Theore ipetrow/v3pdf [2] Melvyn B Nathanson Additive Nuber Theory: The Classical Bases Springer 996 [3] To M Apostol Introduction to Analytic Nuber Theory Springer-Verlag 976 [4] Noah Snyder Dirichlet L-functions, Dirichlet Characters and pries in arithetic progressions nsnyder/tutorial/lecture2pdf [5] Andreas Strönergsson Analytic Nuber Theory: Lecture Notes Based on Davenport s Book astrobe/analtalt8/www notespdf [6] Andreas Strönergsson Applications of the Hardy-Littlewood Circle Method baczkowskid/talks/nuber theory/waring3pdf