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Math 259: Introduction to Analytic Number Theory pseudo-syllabus 0. Introduction: What is analytic number theory? 1. Distributionof primes before complex analysis: classical techniques (Euclid, Euler); primes in arithmetic progression via Dirichlet characters and L-series; further elementary but trickier methods (C(cid:20)eby(cid:20)sev; Selberg sieve). 2. Distribution of primes using complex analysis: (cid:16)(s) and L(s;(cid:31)) as functions of a complex variable, and the proofof the PrimeNumberTheorem andits extensionto Dirichlet; blurbfor C(cid:20)ebotarev density; functional equations; the Riemann hypothesis, extensions, generalizations and consequences. 3. Lower bounds on discriminants, conductors, etc. from functional equations; geometric analogs. 4. Analytic estimates on exponential sums (van der Corput etc.); prototypical applications: Weylequidistribution,upperboundsonj(cid:16)(s)jandjL(s;(cid:31))jonverticallines,latticepointsums. 5. Analytic bounds on coe(cid:14)cients of modular forms and functions; applications to counting partitions, representations of integers as sums of squares, etc. Prerequisites WhileMath 259 willproceedat a pace appropriatefora graduate-level course, its prerequisites are perhaps surprisingly few: complex analysis at the level of Math 113, and linear algebra and basic number theory (up to say arithmetic in the (cid:12)eld Z=p and Quadratic Reciprocity). Some considerably deeper results (e.g. estimates on Kloosterman sums) will be cited but may be regarded as black boxes for our purposes. If you know about algebraic number (cid:12)elds or modular forms or curves over (cid:12)nite (cid:12)elds, you'll get more from the course at speci(cid:12)c points, but these points will be in the nature of scenic detours that are not required for the main journey. Texts There is no textbook for the class; lecture notes will be handed out periodically. This class is an introduction to several di(cid:11)erent (cid:13)avors of analytic methods in number theory, and I know of no one work that covers all this material. Supplementary readings such as Serre's A Course in Arithmetic and Titchmarsh's The Theory of the Riemann Zeta-Function will be suggested as we approach their respective territories. O(cid:14)ce Hours 335 Sci Ctr, Thursdays 2:30{4 PM; or e-mail me at elkies@math to ask ques- tions or set up an alternative meeting time. Grading There will be no required homework, though the lecture notes will contain recom- mended exercises. If you are taking Math 256 for a grade (i.e. are not a post-Qual math graduate student exercising your EXC option), tell me so we can work out an evaluation and grading procedure. This will most likely be either an expository (cid:12)nal paper or an in-class presentation on some aspect of analytic number theory related to but just beyond what we cover in class. Which grading method is appropriate will be determined once the class size has stabilized after \Shopping Period". The supplementary references will be a good source for paper or presentation topics. Math 259: Introduction to Analytic Number Theory Introduction: What is analytic number theory? One may reasonably de(cid:12)ne analytic number theory as the branch of mathe- matics that uses analytical techniques to address number-theoreticalproblems. However, this \de(cid:12)nition", while correct, is scarcely more informative than the phrase it purports to de(cid:12)ne. (See [Wilf 1982].) What kind of problems are suited to \analytical techniques"? What kind of mathematical techniques will be used? What style of mathematics is this, and what will its study teach you beyond the statements of theorems and their proofs? The next few sections brie(cid:13)y answer these questions. The problems of analytic number theory. The typical problem of ana- lytic number theory is an enumerative problem involving primes, Diophantine equations, or similar number-theoretic manifestations, usually asking for large values of some parameter. Examples of problems which we'll address in this course are: (cid:15) How many 100-digit primes are there, and how many of theme have the last digit 7? More generally, how do the prime-counting functions (cid:25)(x) and (cid:25)(x;amodq) behave for large x? [For the 100-digit problems we'd 99 100 need x=10 and 10 , q =10, a=7.] (cid:15) Given a prime p > 0, a nonzero cmodp, and integers a1;b1;a2;b2 with ai <bi,howmanypairs(x1;x2)ofintegersaretheresuchthatai <xi <bi (i = 1;2) and x1x2 (cid:17) cmodp? For how small an H can we guarantee that if bi(cid:0)ai >H then there is at least one such pair? (cid:15) Isthereanintegernsuchthatthe(cid:12)rstelevendigitsofn!are31415926535? Are there in(cid:12)nitely many such n? How many such n are there of at most 1000digits? (cid:15) Given integers n;k, how many ways are there to represent n as a sum of k squares? For instance, how many integer solutions has the equation b 2 2 2 2 2 2 2 100 a +b +c +d +e +f +g +h =10 ? As often happens in mathematics, working on such down-to-earth questions quickly leads us to problems and objects that appear at (cid:12)rst glance to belong to completely di(cid:11)erent areasof mathematics: P1 s (cid:15) Analyze the Riemann zeta function (cid:16)(s) := n=11=n and Dirichlet L-functions such as (cid:0)s (cid:0)s (cid:0)s (cid:0)s (cid:0)s (cid:0)s (cid:0)s L(s):=1(cid:0)3 (cid:0)7 +9 +11 (cid:0)13 (cid:0)17 +19 +(cid:0)(cid:0)+(cid:1)(cid:1)(cid:1) as functions of a complex variable s. 1 (cid:15) Provethat the \Kloostermansum" p(cid:0)1 (cid:18) (cid:19) X 2(cid:25)i (cid:0)1 K(p;a;b):= exp (ax+bx ) p x=1 (cid:0)1 p (with x being the inverse of x mod p) has absolute value at most 2 p. (cid:15) Show that if a function f : R!R satis(cid:12)es reasonable smoothness condi- tions then for large N the absolute value of the exponential sum N X exp(if(n)) n=1 (cid:18) grows no faster than N for some (cid:18) <1 (with (cid:18) depending on the condi- tions imposed on f). (cid:15) Investigate the coe(cid:14)cients of modular forms such as Y1 8 8 n 8 2n 8 2 3 4 5 6 (cid:17) (cid:17)2 =q (1(cid:0)q ) (1(cid:0)q ) =q(cid:0)8q +12q +64q (cid:0)210q (cid:0)96q (cid:1)(cid:1)(cid:1): n=1 (Fortunatelyitwillturnoutthattheroutefrom(say)(cid:25)(x)to(cid:16)(s)isnotnearly n n n as long and tortuous as that from x + y = z to deformations of Galois 1 representations::: ) Thetechniquesofanalyticnumbertheory. Ahallmarkofanalyticnumber theory is the treatment of number-theoretical problems (usually enumerative, as noted above) by methods often relegated to the domain of \applied mathe- matics": elementarybut clevermanipulation of sumsand integrals;asymptotic anderroranalysis;Fourierseriesandtransforms;contourintegralsandresidues. Will there is still good new workto be done along these lines, much contempo- rary analytic number theory also uses advanced tools from within and outside numbertheory(e.g.modularformsbeyondtheupperhalf-plane,Laplacianspec- tral theory). Nevertheless, in this introductory course we shall emphasize the classicalmethods characteristicof analytic numbertheory, on the groundsthat theyarerarelytreatedinthisDepartment'scourses,whileourprogramalready o(cid:11)ers ample exposure to the algebraic/geometric tools. As already noted in the pseudo-syllabus, we shall on a few occasions invoke results that depend on deep(non-analytic)techniques,butweshalltreatthemasdeusexmathematica, developing only their analytic applications. The style of analytic number theory. It has often been said that there 2 are two kinds of mathematicians: theory builders and problem solvers. In 1Seeforinstance[Stevens1994]. 2Actually there are three kinds of mathematician: those who can count, and those who cannot. [AttributedtoJohnConway] 2 the mathematics of our century these two styles are epitomized respectively by A.Grothendieck and P.Erdo(cid:127)s. The Harvard math curriculum leans heavily to- wards the systematic, theory-building style; analytic number theory as usually practiced falls in the problem-solving camp. This is probably why, despite its illustrious history (Euclid, Euler, Riemann, :::) and present-day vitality, ana- lyticnumbertheoryhasrarelybeentaughthere|inthepasttwelveyearsthere have been only a handful of undergraduate seminars and research/Colloquium talks, and no Catalog-listed courses at all. Now we shall see that there is more to analytic number theory than a bag of unrelated ad-hoc tricks, but it is true that fans of contravariant functors, ad(cid:18)elic tangent sheaves, and (cid:19)etale cohomol- ogy will not (cid:12)nd them in the present course. Still I believe that even ardent structuralists will bene(cid:12)t from this course. First, speci(cid:12)c results of analytic number theory often enter as necessary ingredients in the only known proofs of important structural results. Consider for example the arithmetic of elliptic curves: the many applications of Dirichlet's theorem on primes in arithmetic progression,anditsgeneralizationtoC(cid:20)ebotarev'sdensitytheorem,3 includethe recentworkofKolyvaginandWiles; in [Serre1981]sievemethodsareelegantly 4 applied to investigate the distribution of traces of an elliptic curve; in [Merel 1996] a result (Lemme 5) on the x1x2 (cid:17)cmodp problem is required to bound the torsion of elliptic curves over number (cid:12)elds. Second, the ideas and tech- niquesapplywidely. Sieveinequalities,forinstance,arealsousedinprobability to analyze nearly independent variables; the \stationary phase" methods that we'll use to estimate the partition function are also used to estimate oscilla- tory integrals in quantum physics, special functions, and elsewhere; even the van der Corput estimates on exponential sums have recently found applica- tion in enumerative combinatorics [CEP 1996]. Third, the habit of looking for asymptotic results and error terms is a healthy complement to the usual quest for exact answers that we can tend to focus on too exclusively. Finally, An ambitioustheory-buildershould regardthe absencethus farofa GrandUni(cid:12)ed Theory of analytic number theory not as an insult but as a challenge. Both machinery- and problem-motivated mathematicians should note that some of the moreexciting recent workin number theorydepends critically oncontribu- tions from both sides of the stylistic fence. This coursewill introduce the main analytic techniques needed to appreciate, and ultimately to extend, this work. References [CEP 1996] Cohn, H., Elkies, N.D., Propp, J.: Local statistics for random 3We shalldescribe C(cid:20)ebotarev's theorembrie(cid:13)yinthe course but not develop itindetail. GivenDirichlet'stheoremandtheasymptoticformulafor(cid:25)(x;amodq),theextraworkneeded to get C(cid:20)ebotarev isnot analytic but algebraic: the development ofalgebraic numbertheory andthearithmeticofcharactersof(cid:12)nitegroups. ThusafulltreatmentofC(cid:20)ebotarevdoesnot alasbelonginthiscourse. 4My thesis work on the case of trace zero (see e.g. [Elkies 1987]) also used Dirichlet's theorem. 3 domino tilings of the Aztec diamond, Duke Math J. 85 #1 (Oct.96), 117{166. [Elkies1987]Elkies,N.D.: Theexistenceofin(cid:12)nitelymanysupersingularprimes for every elliptic curve over Q. Invent. Math. 89 (1987), 561{567. [Merel 1996] Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437{449. [Serre1981]Serre,J.-P.: Quelquesapplicationsduth(cid:19)eor(cid:18)emededensit(cid:19)edeCheb- otarev. IHES Publ. Math. 54 (1981), 123{201. [Stevens 1994] Stevens, G.: Fermat's Last Theorem, PROMYS T-shirt, Boston University 1994. [Wilf 1982] Wilf, H.S.: What is an Answer? Amer. Math. Monthly 89 (1992), 289{292. 4 Math 259: Introduction to Analytic Number Theory 1 Elementary approachesI: Variations on a theme of Euclid The (cid:12)rst asymptotic question to ask about (cid:25)(x) is whether (cid:25)(x)!1 as x!1, that is, whether there are in(cid:12)nitely many primes. That the answer is Yes was (cid:12)rst shownbythe justly famed argumentin Euclid. While often presentedasa proofbycontradiction,theargumentcanreadilyberecastasane(cid:11)ective(albeit Qn rather ine(cid:14)cient) construction: given primes p1;p2;:::;pn, let Pn = k=1pn, de(cid:12)ne Nn = PN +1, and let pn+1 be the smallest factor of Nn. Then pn+1 is a prime no larger than Nn and di(cid:11)erent from p1;:::;pn. Thus fpkgk>1 is an in(cid:12)nite sequence of distinct primes, Q.E.D. Moreover this argument also gives an explicit upper bound on pn, and thus a lower bound on (cid:25)(x). Indeed we may take p1 =2 and observe that Yn Yn pn+1 (cid:20)Nn =1+ pn (cid:20)2 pn: k=1 k=1 2n(cid:0)1 if equality were satis(cid:12)ed at each step we would have pn = 2 . Thus by 2 induction we see that 2n(cid:0)1 pn (cid:20)2 2n(cid:0)1 (and of course the inequality is strict once n>1). Therefore if x(cid:21)2 then 3 pk <x for k=1;2;:::;n and so (cid:25)(x)(cid:21)n, so we conclude (cid:25)(x)>log2log2x: ThePn+1trickcanevenbeusedtoproveotherspecialcasesoftheresultthat (a;q) = 1) (cid:25)(x;amodq)!1 as x!1. Of course the case 1mod2 is trivial given Euclid. For (cid:0)1modq with q = 3;4;6, start with p1 = q(cid:0)1 and de(cid:12)ne Nn =qPn(cid:0)1. Moregenerally,foranyquadraticcharacter(cid:31)therearein(cid:12)nitely many primes p with (cid:31)(p)=(cid:0)1; e.g. given an odd prime q0, there are in(cid:12)nitely many primes p which are quadratic nonresidues of q0. [I'm particularly fond of this argument because I was able to adapt it as the punchline of my doctoral thesis; see [El].] The case of (cid:31)(p)=+1, e.g. the result that (cid:25)(x;1mod4)!1, 4 2 is only a bit trickier. For that case, let p1 =5 and Nn =4Pn +1, and appeal 1Speci(cid:12)cally, [Eu, IX, 20]. For moreon the history ofwork on the distribution of primes uptoabout1900,see[Di,XVIII]. 2Curiously the same bound is obtained from a much later elementary proof: between 1 and 22n there are n pairwisecoprime numbers,the (cid:12)rst n(cid:0)1 Fermatnumbers 22m +1 for 0(cid:20)m<n,[whyaretheypairwisecoprime?],sonecessarilyatleastnprimesaswell. 3Q: What sound does a drowning analytic number theorist make? A: log log log log :::[R.Murty,viaB.Mazur] 4Butenough sothat aproblemfromarecent QualifyingExamforourgraduate students askedtoprovethattherearein(cid:12)nitelymanyprimescongruentto1mod4. 1 2 2 to Fermat's theorem on the prime factors of x +y . Again this even yields an 5 explicit lower bound on (cid:25)(x;1mod4), namely (cid:25)(x;1mod4)>Cloglogx for some positive constant C. [Exercises: Exhibit an explicit value of C. Use cyclotomic polynomials to show more generally that for any q0, prime or not, 6 there exist in(cid:12)nitely many primes congruentto 1 mod q0, and that indeed the number of such primes <x grows at least as fast as some multiple of loglogx. Modify this trick to show that there are in(cid:12)nitely many primes congruent to 4 mod 5, again with a loglog lower bound.] But Euclid'sapproachand its variations,howeverelegant,arenot su(cid:14)cient for our purposes. For one thing, numerical evidence suggests | and we shall soon prove | that log2log2x is a gross underestimate on (cid:25)(x). For another, one 7 cannot prove all cases of (a;q)=1)(cid:25)(x;amodq)!1 using only variations on the Euclid argument. Our next elementary approaches will address at least the (cid:12)rst de(cid:12)ciency. References [Di] Dickson, L.E.: History of the Theory of Numbers, Vol. I: Divisibility and Primality. Washington: Carnegie Inst., 1919. [Eu] Euclid, Elements. [El]Theexistenceofin(cid:12)nitelymanysupersingularprimesforeveryellipticcurve over Q, Invent. Math. 89 (1987), 561{568; See also: Supersingular primes for elliptic curves overreal number (cid:12)elds, Compositio Math. 72 (1989), 165{172. 5Evenadrowninganalyticnumbertheoristknowsthatloglogandlog2log2 areasymptot- icallywithinaconstantfactorofeachother. Whatisthatfactor? 6AresultattributedtoEulerin[Di,XVIII]. 7Thisisnot atheorem,ofcourse;foronethinghowdoesonepreciselyde(cid:12)ne\variationof theEuclidargument"? ButI'llbequiteimpressedifyoucaneven(cid:12)ndaEuclid-styleargument forthein(cid:12)nitudeofprimescongruent to2mod5ormod7. 2 Math 259: Introduction to Analytic Number Theory Elementary approaches II: the Euler product It was Euler who (cid:12)rst went signi(cid:12)cantly beyond Euclid's proof, by recasting anotherhighlightofancientGreeknumbertheory,thistimeuniquefactorization into primes, asagenerating-functionidentity. That is, fromthe fact that every Q positiveintegernmaybewrittenuniquelyas pprimepcp,withcp anonnegative integer that vanishes for all but (cid:12)nitely many p, Euler obtained: 0 1 X1 n(cid:0)s = Y @X1 p(cid:0)cpsA= Y 1 : (E) 1(cid:0)p(cid:0)s n=1 pprime cp=0 pprime The sum on the left-hand side of (E) is now called the zeta function 1 1 X1 (cid:0)s (cid:16)(s)=1+ + +(cid:1)(cid:1)(cid:1)= n ; 2s 3s n=1 the formula(E) iscalledthe Euler product for(cid:16)(s). Sofarwehaveonlyproved (E)asaformalidentity. But,sinceallthetermsandfactorsinthein(cid:12)niteseries andproductsarepositive,(E) actuallyholdsasanidentitybetween convergent P1 (cid:0)s Rse1ries(cid:0)asnd products provided that n=1n converges. By comparison with x dx (i.e. by the \Integral Test" of elementary calculus) we see that the 1 this happens if and only if s>1. Moreover,from Z x+1 (cid:0)s 1 (cid:0) 1(cid:0)s 1(cid:0)s(cid:1) y dy = x (cid:0)(x+1) x s(cid:0)1 we obtain the inequality 1(cid:0)s 1(cid:0)s (cid:0)s x (cid:0)(x+1) (cid:0)s (x+1) < <x ; s(cid:0)1 1 summing this overx=1;2;3;::: we obtain lower and upper bounds on (cid:16)(s): 1 1 <(cid:16)(s)< +1: ((cid:3)) s(cid:0)1 s(cid:0)1 2 In particular (cid:16)(s)!1 as s!1 from above. This yields Euler's proof of the in(cid:12)nitude of primes: if there were only (cid:12)nitely many then the product (E) would remain (cid:12)nite for all s>0. 1In factmoreaccurate estimatesareavailable fromthe\Euler-Maclaurinformula",aswe shallseeinduecourse. 2ActuallyEulersimplysubstituteds=1in(E)toclaimacontradictionwiththehypothesis that thereareonly(cid:12)nitelymanyprimes,but itiseasy enough toconvert that ideaintothis legitimateproof. Ifyouhappentoknowthat(cid:25)istranscendental,oronlythat(cid:25)2isirrational, then from(cid:25)2 =6(cid:16)(2) (another famousEuleridentity) you mayalso obtain the in(cid:12)nitude of primes,though naturallytheresultingboundson(cid:25)(x)aremuch worse. 1 Infactthatproductisactuallyin(cid:12)niteforsaslargeas1,afactthatyieldsmuch tighter estimates on (cid:25)(x) than were available starting from Euclid's proof. For (cid:18) instance wecannothaveconstantsC;(cid:18) with (cid:18) <1 suchthat (cid:25)(x)<Cx for all x,becausethentheEulerproductwouldconvergefors>(cid:18). Togofurtheralong these lines it is convenient to take logarithms in (E), convertingthe product to a sum: X (cid:0)s log(cid:16)(s)= (cid:0)log(1(cid:0)p ); p andtoestimatebothsidesass!1+. By(*)theleft-handsidelog(cid:16)(s)isbetween log1=(s(cid:0)1) and logs=(s(cid:0)1); since 0 < logs < s(cid:0)1, we conclude that (cid:16)(s) is within s(cid:0)1 of log1=(1(cid:0)s). In the right-hand side we approximate each (cid:0)s (cid:0)s 1 (cid:0)2s summand (cid:0)log(1(cid:0)p ) by p ; the error is at most p , which summed P 2 1 (cid:0)2 over all primes is less than 2 pp <(cid:16)(2)=2. The point is not the numerical bound on the errorbut the fact that it remains (cid:12)nitely bounded as s!1+. We thus have: X (cid:0)s 1 p =log +O(1) (1<s<2): (E') s(cid:0)1 p The O(1) here is our (cid:12)rst encounter with the \Big Oh" notation; in general if g is nonnegative then \f = O(g)" is shorthand for \there exists a constant C suchthatjfj(cid:20)Cg foreachallowedevaluationoff;g". ThusO(1) isabounded P (cid:0)s function, so (E') means that there exists a constant C such that pp is 1 within C of log for all s2(1;2). [The upper bound of 2 on s is not crucial, s(cid:0)1 since we're concerned with s near 1, but we must put some upper bound on s for(E')tohold|doyouseewhy?] Anequivalentnotation,moreconvenientin somecircumstances,isf (cid:28)g (org (cid:29)f). Forinstance,alinearmapT between Banachspacesiscontinuousi(cid:11)Tv=O(jvj)i(cid:11)jvj(cid:29)jTvj. EachinstanceofO((cid:1)) or (cid:28) or (cid:29) is presumed to carry its own implicit constant C. If the constant depends on someparameterthat parameterappearsasasubscript; e.g. forany (cid:15) (cid:15) (cid:15) > 0 we have logx = O(cid:15)(x ) (equivalently logx (cid:28)(cid:15) x ) on x 2 [1;1). For basic properties of O((cid:1)) see the Exercises at the end of this section. Now (cid:25)(x) still does not occur explicitly in the sum in (E'). We thus rewrite (cid:0)s R1 (cid:0)1(cid:0)s this sum as follows. Express the summand p as an integral s y dy. p Summing over all p we (cid:12)nd that y occurs in the interval of integration (p;1) i(cid:11) p<y, i.e. with multiplicity (cid:25)(y). Thus the sum in (E') becomes an integral involving (cid:25)(y), and we (cid:12)nd: Z 1 (cid:0)1(cid:0)s 1 s (cid:25)(y)y dy =log +O(1) (1<s<2): (E") 1 s(cid:0)1 Tworemarksareinorderhere. First,thatthetransformationfrom(E')to(E") isanexampleofamethodweshalluseoften,knowneitheraspartialsummation or integration by parts. To explain the latter name, consider that the sum in R1 (cid:0)s (E') may be regarded as the Stieltjes integral y d(cid:25)(y), which integrated 1 2 by parts yields (E"); that is how we shall write this transformationhenceforth. Second, that the integral in (E") is just s times the Laplace transform of the u (cid:0)u function (cid:25)(e ) on u 2 [0;1): via the change of variable y = e that integral R1 u (cid:0)su becomes (cid:25)(e )e du. In general if f(u) is a nonnegative function whose 0 R1 (cid:0)su Laplace transform Lf(s) := 0 f(u)e du converges for s > s0 then the behaviorofLf(s)ass!s0+detectsthebehavioroff(u)asu!1. Inourcase, s0 =1, so we expect that (E") will give us information on the behaviorof (cid:25)(x) for large x. 3 We note now that (E") is consistent with (cid:25)(x)(cid:24)x=logx, that is, that Z 1 (cid:0)s y 1 dy =log +O(1) (1<s<2): 2 logy s(cid:0)1 Let the integral be I(s). Di(cid:11)erentiating under the integral sign we (cid:12)nd that 0 R1 (cid:0)s 1(cid:0)s I (s)=(cid:0) y dy =2 =(1(cid:0)s)=1=(1(cid:0)s)+O(1). Thus for 1<s<2 we 2 have Z 2 Z 2 0 d(cid:27) 1 I(s)=I(2)(cid:0) I ((cid:27))d(cid:27) =+ +O(1)=log +O(1) s s (cid:27)(cid:0)1 s(cid:0)1 as claimed. This does not provethe Prime Number Theorem, but it does show 0 that, for instance, if c < 1 < C then there are arbitrarily large x;x such that 0 0 0 (cid:25)(x)>cx=logx and (cid:25)(x )<Cx=logx. Exercises, mostly on the Big Oh (a.k.a. (cid:28)) notation: 1. If f (cid:28) g and g (cid:28) h then f (cid:28) h. If f1 = O(g1) and f2 = O(g2) then f1f2 = O(g1g2) and f1 +f2 = O(g1 +g2) = O(max(g1;g2)). Given a positive function g, the functions f such that f =O(g) constitute a vector space. Rx Rx 2. If f (cid:28) g on [a;b] then f(y)dy (cid:28) g(y)dy for x 2 [a;b]. (We already a a 0 used this toobtain I(s)=log(1=(s(cid:0)1))+O(1) from I (s)=1=(1(cid:0)s)+O(1).) In general di(cid:11)erentiation does not commute with \(cid:28)" (why?). Nevertheless, 0 P1 (cid:0)s 2 prove that (cid:16) (s)[=(cid:0) n=1n logn] is (cid:0)1=(s(cid:0)1) +O(1) on s2(1;1). 3. So far all the implicit constants in the O((cid:1)) or (cid:28) we have seen are e(cid:11)ective: we didn't bother to specify them, but we could if we really had to. Moreover the transformations in exercises 1,2 preserve e(cid:11)ectivity: if the input constants aree(cid:11)ectivethen soaretheoutput ones. However,itcanhappenthat weknow that f = O(g) without being able to name a constant C such that jfj (cid:20) Cg. Here is a prototypical example: suppose x1;x2;x3;::: is a sequence of positive reals which we suspect are all (cid:20) 1, but all we can show is that if i 6= j then xixj <xi+xj. Prove that xi are bounded, i.e. xi =O(1), but that as long as wedonot(cid:12)ndsomexi greaterthan1,wecannotusethistoexhibitaspeci(cid:12)cC 3Weshiftthelowerlimitofintegrationtoy=2toavoidthespurioussingularityof1=logy aty=1,andsuppressthefactorsbecauseonlythebehaviorass!1mattersandmultiplying bysdoesnota(cid:11)ectittowithinO(1). 3

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