Lectures on analytic number theory

100 Pages·1998·0.892 MB·English

by  Elkies N.D.

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