1 = 2 d x. n x n (mod d) d n

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1 HW2, Problem 3*: Use Dirichlet hyperbola metho to show that τ 2 + = 3 log + O. This ote presets the ifferet ieas suggeste by the stuets Daiel Klocker, Jürge Steiiger, Stefaia Ebli a Valerie Roiter for solvig this problem. PROOF: Let = # { m Z/Z : m mo } eote the umber of solutios moulo of the cogruece mo. The actual applicatio of the Dirichlet hyperbola metho is oly i the first part of the proof, amely i Claim. We have the followig ietity τ 2 + = 2 Proof. We ca rewrite τ2 + as τ 2 + = = O. 2 + = mo followig from Dirichlet hyperbola metho. The factor 2 arises sice every ivisor of 2 + with [ 2 + ] = has a complimetary ivisor q with q = 2 + a q > a vice versa there eists a bijectio betwee the ivisors of 2 + with a the ivisors >. Sice the coitio mo is perioic i with perio the above sum ca be rewritte as 2 + O where eotes the umber of the solutios of 2 + = 0 i Z/Z. Hece τ 2 + = 2 + O. Aalyses of The harest part of the proof is to uersta the fuctio a the sums a. Here is where the ieas split a we will preset three ifferet methos, the tet followig closely the solutios of the stuets.. Iea of Valerie The first metho is presete i most cosice way, however it is the least elemetary of all that will be give here, requirig some kowlege i algebraic umber theory.

2 The fuctio ρ is multiplicative i, while f = τ 2 + is ot. From Fermat s two square theorem, we kow that ρp = 2 if p mo 4 a ρp = 0 if p 3 mo 4. With Hesel s lemma ρ ca be compute at prime powers. The multiplicity of ρ follows from the Chiese Remaier Theorem. We ow wat to estimate the sums a. Usig a result from [] we get P = 6H + O log where P is the umber of solutios of the equatio 2 + b + c = 0 with = b 2 4c < 0 a H is the Hurwitz class umber. The Hurwitz class umber is efie as H = A,B,C reuce B 2 4AC= 2 wa, B, C, where A, B, C is a poistive efiite quaratic form A 2 + By + Cy 2 with the same iscrimiat as the equatio 2 + b + c a wa, B, C is the size of the automorohism group of the form A, B, C. We have that wa, A, A = 6, wa, 0, A = 4 a for other reuce forms wa, B, C = 2. The mai iea of McKee s proof is to fi a eplicit epressio for usig the automorphism group of a quaratic form a the, usig this eplicit epressio, reformulatio as a este sum of Möbius fuctios, which after some chages of ier sums the ca be compute eplicitly. I our case we have the equatio 2 + = 0, = 4 a H 4 = 2. Hece = O log = O. Usig summatio by parts we get = 3 log + O. 2 Combiig everythig we fially get τ 2 + = 2.2 Iea of Stefaia + O = 3 log + O. This metho uses the itriguig relatio of the fuctio to the Gauss circle problem. Defie the fuctios r = # {, y Z 2 :, y = a 2 + y 2 = } coutig the proper represetatios of as a sum of two squares, a P = # {, y Z 2 :, y =, > 0, y 0 a 2 + y 2 = } 2

3 which couts the proper represetatios i the first quarat. Recall that the Gauss circle problem couts the lattice poits i a circle of raius by the asymptotic formula = + O y 2 = As the esity of coprime itegers is 6/ 2, oe epects that the primitive circle problem gives r = 6 + O/2+ε. 3 The mai statemet importig i the cotet of the circle problem is the followig Lemma. Theorem 3.2, [2] We have r = 4ρ for every positive iteger. I particular ρ = P. Proof. Cosier ay solutio of 2 + y 2 =, where > 0. Of the four solutios, y, y,,, y, y, eactly oe of them has both the first a the seco cooriate positive. Thus, let P be efie as above, we will have r = 4P a we will ow prove that ρ = P. Suppose that is a give positive iteger, we shall ehibit oe-to-oe correspoece betwee the represetatios 2 + y 2 = with > 0, y 0,, y = a the solutios s of the cogruece s 2 mo. We will o this i three steps. First we efie a fuctio from the appropriate pairs, y to the appropriate resiue class s mo. Seco, we will show that the fuctio is oe-to-oe. Thir, we prove that the fuctio is oto. To efie the fuctio, suppose that a y are itegers such that 2 + y 2 =, > 0 a y 0, a, y =. The, =, so there eists a uique s mo such that s y mo. More precisely, if is chose so that mo, the s y mo. Sice 2 y 2 m mo, o multiplyig both sies by 2 we euce that s 2 mo. We ow show that our fuctio is oe-to-oe. Suppose that for i =, 2 we have = 2 i + y2 i with i > 0 y i 0, i, y i = a i s i y i mo. We show that if s s 2 mo the = 2 a y = y 2. Suppose s s 2 mo, as y 2 s y y 2 2 y s 2 mo it follows that y 2 2 y mo sice s i, =. But 0 < 2 i so that 0 < i a similary 0 < y i. From this iequalities we eeuce that 0 y 2 a similary 0 2 y. As these two umbers are cogruet moulo a both lie i the iterval [0, we ca coclue that y 2 = 2 y. Thus 2 y. But, y = a 2, y 2 =, so it follows that 2 a 2. As the i are positive we euce that = 2 a hece y = y 2. This completes the proof that the fuctio is oe-to-oe. To complete the proof we just ee to show that our fuctio is oto. That is, for each s such that s mo there is a represetatio 2 + y 2 = with > 0 a y 0,, y = a s y mo. Suppose that such s is give, the there is a iteger c such that 2s 2 4c = 4. Thus g, y = 2 + 2sy + cy 2 is a positive efiite biary quaratic form of iscrimiat 4. I the proof of Theorem 3.20[2] it is show that all these forms are equivalet. Thus there is a matri M Γ that takes the form f, y = 2 + y 2 to the form g. From efiitio 3.7a[2] we see that m 2 + m 2 2 = a m, m 2 = sice etm = m m 22 m 2 m 2 =. From 3.7b we see that s = m m 2 + m 2 m 22. Hece: m 2 = m 2 + m m 2 m 22 = m 2 2m 2 + m m 2 m 22 mo sice m 2 m 2 2 mo = m 2 2m 2 + m 2 + m 2 m 2 sice m m 22 m 2 m 2 = = m 2 3

4 I the case m > 0 a m 2 0 it is suffices to take = m a y = m 2. I case these iequalities o ot hol, the we take the poit, y to be oe of the poit m 2, m, m, m 2, m 2, m. From the cogruece m s m 2 mo, s 2 mo we euce that m 2 s m mo. Thus s y mo i ay of these case. This complete the proof that r = 4ρ. Equatio 2 shows us the asymptotic behaviour of the umber of iteger poits isie a circle of raius. I the followig lemma we will fi the asymptotic behaviour of iteger poits, y isie a circle of raius such that, y =, i.e. we will prove 3. Lemma 2. We have I particular Q := P = = r = 6 + O log. = = Proof. From the efiitio of Q we ca write Q = ρ = O log. u,v u 2 +v 2 u,v= a B := u,v u 2 +v 2 = u,v u 2 +v 2 u,v= = u,v u 2 +v 2 / 2 u,v = = Q 2 where u = u, v = v 4 Applyig a certai versio of the Möbius Iversio Formula to 4 we have Q = = = µb 2 µ 2 + O by 2 µ 2 + O µ = 6 + O + O log sice = 6 + O log. µ log 4

5 We kow that r = 4P = 4ρ, hece = P = ρ = O log = O 5 Corollary. We have ρ = 3 log + O. 2 Proof. First we will prove that = r = 6 log + O. Applyig Abel Trasformatio with h = = r a g = we get r = t r + t 2 r t From Lemma 2 we have = = r = = 6 log + O + = 6 log + O = 6 t + Ot 3 2 log t t It follows immeiately that ρ = 3 log + O. 2 Fially we have the tools to prove our asymptotic formula. By Corollary a 5 oe easily sees that τ 2 + = 3 log + O..3 Iea of Daiel a Jürge The proofs of Daiel a Jürge are essetially the same, with the ifferece that Daiel restrais from the otio for Dirichlet L-series, a rather uses Abel s iequality for a certai estimate. Later this will become clearer. First cosier the fuctio µ, where µ is the Möbius fuctio, i.e. this is the squarecoutig fuctio which is if is squrefree a 0 otherwise. Also cosier the multiplicative fuctio χ : N C efie by if mo 4 χ = if 3 mo 4 0 otherwise. The basic steps i the aalyses of a the sums from Claim ivolvig this fuctio are:. Show ρ is multiplicative a compute ρp α for all primes p. 2. Show ρ = µ χ, where eotes Dirichlet covolutio of arithmetic fuctios. 5

6 3. Estimate µ, µ, χ a χ. 4. Estimate ρ a ρ. First we ote that ρ =, ρ2 = a ρ2 k = 0 for k 2. By the first supplemet for quaratic resiues we kow for primes p: { 2, if p = 4 + ρp = 0, if p = By Hesel s Lemma we get for every solutio of a 2 mo p a solutio of a 2 mo p k for k 2. Therefore we get for prime powers also { ρp k 2, if p = 4 + = 0, if p = Claim 2. Let µ be the Möbius fuctio. The we have = ab= µa χb = µ χ. Proof. By the Chiese Remaier Theorem is multiplicative, but ot strogly multiplicative. Obviously by efiitio, so the Dirichlet series D ρ is absolutely coverget for Res > 2 its absolute covergece abscissa is but we will ot pursue this here. Therefore for Res > 2 we ca write ρ s = p = + ρp p s + ρp2 + ρ2 2 s p>2 = + 2 s p=4+ = + 2 s p=4+ = + 2 s p=4+ = + 2 s = p = p = p=4+ p 2s ρp p 2s +... p s + ρp2 + 2 p s + 2 p 2s k p s 2 k p s + p s p s + p s χpp s + p s χpp s µ s By multiplicatio of Dirichlet series we get = a p χ s µa χ a = µa χb. ab= 6

7 So we see that µ a χ. = ab µa a Claim 3. Let ζs be the Riema zeta fuctio. The we have: µ χb b. Therefore we shoul look closer at the sums = log ζ2 + O Proof. From the lecture 4.. Claim4 we kow that µ = ζ2 + O. Usig Abel trasformatio we get µ = µ + µ t 2 t t = t ζ2 + O + ζ2 + O t t 2 t = ζ2 + O + ζ2 t t + O t t t = log ζ2 + O. We efie the Dirichlet L-series at as So whe we wat to calculate Claim 4. We have L, χ = χ. χ, we ca epress it as L, χ plus a error term. χ = L, χ + O. Proof. Let A := χ. By efiitio of χ it follows that A, so we have A = O for every. We kow that So it remais to show that > So we get < y lim L, χ = χ χ + > χ = O. Usig Abel trasformatio we get χ = Ay y A + = O y < y sice the limit oes t epe o y. χ = > χ y = O At t 2 t, Leibiz showe that so Claim 4 reas as L, χ = = 4, χ = 4 + O. 7

8 Daiel s proof iffers from Jürge s oly at this last claim, more precisely i its proof, which he oes usig Abel s iequality. Note that metioig Dirichlet L-fuctio is ot really ecessary for the proof, but we rather use it for a comfortable otatio. Further we kow that ζ2 = 2 /6. Puttig this together we ca fi ow a epressio for ρ : Claim 5. We have Proof. Usig the claims before we get ρ Claim states that τ2 + = 2 ρ = 3 log + O. 2 = µa χb a b ab = µa χb a b a b a log a = ζ2 + O 4 + O = 3 log + O. 2 + O. Fially we ee to look at. This is ot har, because from Claim 2 a the fact that χ follows = a µa b a χb = a µa O = O. Puttig all together we get the result: 3 τ 2 + = 2 log + O + O 2 Refereces = 3 log + O. [] McKee, James O the average umber of ivisors of quaratic polyomials, Math. Proc. of the Cambrige Philosophical Society 7, p , 995. [2] Nive, Iva ; Zuckerma, Herbert S. ; Motgomery, Hugh L. A itrouctio to the theory of umbers. Fifth eitio. Joh Wiley & Sos, Ic., New York, 99. 8