The Prime Number Theorem as a Capstone in a Complex Analysis Course
TTHE PRIME NUMBER THEOREMAS A CAPSTONE IN A COMPLEX ANALYSIS COURSE
STEPHAN RAMON GARCIAA
BSTRACT . We present a detailed proof of the prime number theorem suitable for a typ-ical undergraduate- or graduate-level complex analysis course. Our presentation is partic-ularly useful for any instructor who seeks to use the prime number theorem for a seriesof capstone lectures, a scaffold for a series of guided exercises, or as a framework for aninquiry-based course. We require almost no knowledge of number theory, for our aim is tomake a complete proof of the prime number theorem widely accessible to complex analysisinstructors. In particular, we highlight the potential pitfalls and subtleties that may catchthe instructor unawares when using more terse sources.
1. I
NTRODUCTION
The prime number theorem is one of the great theorems in mathematics. It unexpectedlyconnects the discrete and the continuous with the elegant statement lim x Ñ8 π p x q x { log x “ , in which π p x q denotes the number of primes at most x . The original proofs, and mostmodern proofs, make extensive use of complex analysis. Our aim here is to present, forthe benefit of complex analysis instructors, a complete proof of the prime number theoremsuitable either as a sequence of capstone lectures at the end of the term, a scaffold for aseries of exercises, or a framework for an entire inquiry-based course. We require almostno knowledge of number theory. In fact, our aim is to make a detailed proof of the primenumber theorem widely accessible to complex analysis instructors of all stripes.Why does the prime number theorem belong in a complex-variables course? At variousstages, the proof utilizes complex power functions, the complex exponential and loga-rithm, power series, Euler’s formula, analytic continuation, the Weierstrass M -test, locallyuniform convergence, zeros and poles, residues, Cauchy’s theorem, Cauchy’s integral for-mula, Morera’s theorem, and much more. Familiarity with limits superior and inferior isneeded toward the end of the proof, and there are plenty of inequalities and infinite series.The prime number theorem is one of the few landmark mathematical results whoseproof is fully accessible at the undergraduate level. Some epochal theorems, like theAtiyah–Singer index theorem, can barely be stated at the undergraduate level. Others, likeFermat’s last theorem, are simply stated, but have proofs well beyond the undergraduatecurriculum. Consequently, the prime number theorem provides a unique opportunity for Mathematics Subject Classification.
Key words and phrases. prime number theorem, complex analysis, complex variables, Riemann zeta function,Euler product formula.SRG supported by NSF Grant DMS-1800123. a r X i v : . [ m a t h . HO ] J u l STEPHAN RAMON GARCIA students to experience a mathematical capstone that draws upon the entirety of a course andwhich culminates in the complete proof of a deep and profound result that informs muchcurrent research. In particular, students gain an understanding of and appreciation for theRiemann Hypothesis, perhaps the most important unsolved problem in mathematics. Onestudent in the author’s recent class proclaimed, “I really enjoyed the prime number theorembeing the capstone of the course. It felt rewarding to have a large proof of an importanttheorem be what we were working up towards as opposed to an exam.” Another added, “Ienjoyed the content very much. . . I was happy I finally got to see a proof of the result.”Treatments of the prime number theorem in complex analysis texts, if they appear atall, are often terse and nontrivial to expand at the level of detail needed for our purposes.For example, the standard complex analysis texts [4, 7, 20, 26, 28, 31–33, 38] do not includeproofs of the prime number theorem, although they distinguish themselves in many otherrespects. A few classic texts [1,6,25,41] cover Dirichlet series or the Riemann zeta functionto a significant extent, although they do not prove the prime number theorem. Bak andNewman [3, Sec. 19.5] does an admirable job, although their presentation is dense (fivepages). Marshall’s new book assigns the proof as a multi-part exercise that occupies half apage [27, p. 191]. Simon’s four-volume treatise on analysis [35] and the Stein–Shakarchianalysis series [37] devote a considerable amount of space to topics in analytic numbertheory and include proofs of the prime number theorem. Lang’s graduate-level complexanalysis text [21] thoroughly treats the prime number theorem, although he punts at acrucial point with an apparent note-to-self “(Put the details as an exercise)”.On the other hand, number theory texts may present interesting digressions or tangen-tial results that are not strictly necessary for the proof of the prime number theorem. Theysometimes suppress or hand wave through the complex analysis details we hope to exem-plify. All of this may make navigating and outlining a streamlined proof difficult for thenonspecialist. We do not give a guided tour of analytic number theory, nor do we dwellon results or notation that are unnecessary for our main goal: to present an efficient proofof the prime number theorem suitable for inclusion in a complex analysis course by aninstructor who is not an expert in number theory. For example, we avoid the introductionof general infinite products and Dirichlet series, Chebyshev’s function ψ and its integratedcousin ψ , the von Mangoldt function, the Gamma function, the Jacobi theta function, Pois-son summation, and other staples of typical proofs. Some fine number theory texts whichcontain complex-analytic proofs of the prime number theorem are [2, 8, 11, 13, 18, 39].No instructor wants to be surprised in the middle of the lecture by a major logical gapin their notes. Neither do they wish to assign problems that they later find are inaccuratelystated or require theorems that should have been covered earlier. We hope that our pre-sentation here will alleviate these difficulties. That is, we expect that a complex analysisinstructor can use as much or as little of our proof as they desire, according to the level ofrigor and detail that they seek. No step is extraneous and every detail is included.The proof we present is based on Zagier’s [42] presentation of Newman’s proof [29](see also Korevaar’s exposition [19]). For our purposes their approach is ideal: it involvesa minimal amount of number theory and a maximal amount of complex analysis. Thenumber-theoretic content of our proof is almost trivial: only the fundamental theorem ofarithmetic and the definition of prime numbers are needed. Although there are elementaryproofs [12, 34], in the sense that no complex analysis is required, these are obviouslyunsuitable for a complex analysis course. HE PRIME NUMBER THEOREM 3
This paper is organized as follows. Each section is brief, providing the instructor withbite-sized pieces that can be tackled in class or in (potentially inquiry-based) assignments.We conclude many sections with related remarks that highlight common conceptual issuesor opportunities for streamlining if other tools, such as Lebesgue integration, are available.Proofs of lemmas and theorems are often broken up into short steps for easier digestion oradaptation as exercises. Section 2 introduces the prime number theorem and asymptoticequivalence p„q . We introduce the Riemann zeta function ζ p s q in Section 3, along with theEuler product formula. In Section 4 we prove the zeta function has a meromorphic contin-uation to Re s ą . We obtain series representations for log ζ p s q and log | ζ p s q| in Section5. These are used in Section 6 to establish the nonvanishing of the zeta function on thevertical line Re s “ . Section 7 introduces Chebyshev’s function ϑ p x q “ ř p ď x log p andestablishes a simple upper bound (needed later in Section 10). In Section 8, we prove thata function related to log ζ p s q extends analytically to an open neighborhood of the closedhalf plane Re s ě . Section 9 provides a brief lemma on the analyticity of Laplace trans-forms. Section 10 is devoted to the proof of Newman’s Tauberian theorem, a true festivalof complex analysis. Section 11 uses Newman’s theorem to establish the convergence ofa certain improper integral, which is shown to imply ϑ p x q „ x in Section 12. We end inSection 13 with the conclusion of the proof of the prime number theorem. Acknowledgments.
We thank Ken Ribet, S. Sundara Narasimhan, and Robert Sachs forhelpful comments. 2. P
RIME N UMBER T HEOREM
Suppose that f p x q and g p x q are real-valued functions that are defined and nonzero forsufficiently large x . We write f p x q „ g p x q if lim x Ñ8 f p x q g p x q “ and we say that f and g are asymptotically equivalent when this occurs. The limit lawsfrom calculus imply that „ is an equivalence relation.Let π p x q denote the number of primes at most x . For example, π p . q “ since , , , ď . . The distribution of the primes appears somewhat erratic on the smallscale. For example, we believe that there are infinitely many twin primes; that is, primeslike and which differ by (this is the famed twin prime conjecture). On the otherhand, there are arbitrarily large gaps between primes: n ! ` , n ! ` , . . . , n ! ` n is a stringof n ´ composite (non-prime) numbers since n ! ` k is divisible by k for k “ , , . . . , n .The following landmark result is one of the crowning achievements of human thought.Although first conjectured by Legendre [22] around 1798 and perhaps a few years earlierby the young Gauss, it was proved independently by Hadamard [16] and de la Vall´eePoussin in 1896 [9] with methods from complex analysis, building upon the seminal 1859paper of Riemann [30] (these historical papers are reprinted in the wonderful volume [5]). Theorem 2.1 (Prime Number Theorem) . π p x q „ Li p x q , in which Li p x q “ ż x dt log t STEPHAN RAMON GARCIA
20 40 60 80 1000510152025 (A) x ď
200 400 600 800 100050100150 (B) x ď , (C) x ď , (D) x ď , F IGURE
1. Graphs of Li p x q versus π p x q on various scales. is the logarithmic integral . The predictions afforded by the prime number theorem are astounding; see Figure 1.Unfortunately, Li p x q cannot be evaluated in closed form. As a consequence, it is con-venient to replace Li p x q with a simpler function that is asymptotically equivalent to it.L’Hˆopital’s rule and the fundamental theorem of calculus imply that lim x Ñ8 Li p x q x { log x L “ lim x Ñ8 x log x ´ x p x qp log x q “ lim x Ñ8 ´ x “ and hence Li p x q „ x log x . However, the logarithmic integral provides a better approximation to π p x q ; see Table 1.We will prove the prime number theorem in the following equivalent form. Theorem 2.2 (Prime Number Theorem) . π p x q „ x log x . Our proof incorporates modern simplifications due to Newman [29] and Zagier [42].However, the proof is still difficult and involves most of the techniques and tools from atypical complex analysis course. There is little number theory in the proof; it is almost allcomplex analysis. Consequently, it is an eminently fitting capstone for a complex analysiscourse. As G.H. Hardy opined in 1921 [24]:
No elementary proof of the prime number theorem is known, and one may askwhether it is reasonable to expect one. Now we know that the theorem is roughlyequivalent to a theorem about an analytic function, the theorem that Riemann’szeta function has no roots on a certain line. A proof of such a theorem, not
HE PRIME NUMBER THEOREM 5 x π p x q Li p x q x { log x T ABLE
1. The logarithmic integral Li p x q is a better approximation to the prime countingfunction π p x q than is x { log x (entries rounded to the nearest integer). fundamentally dependent on the theory of functions [complex analysis], seemsto me extraordinarily unlikely. In 1948 Erd˝os [12] and Selberg [34] independently found proofs of the prime numbertheorem that avoid complex analysis. These “elementary” proofs are more difficult andintricate than the approach presented here; see [10,11,17,23,40] for the details and [15,36]for an account of the murky history of the elementary proof.
Remark 2.3.
A common misconception is that f p x q „ g p x q implies that f p x q ´ g p x q tends to zero, or that it remains small. The functions f p x q “ x ` x and g p x q “ x areasymptotically equivalent, yet their difference is unbounded. Remark 2.4.
The prime number theorem implies that p n „ n log n , in which p n denotesthe n th prime number. Since π p p n q “ n , substitute q “ p n and obtain lim n Ñ8 n log np n “ lim n Ñ8 ˆ π p p n q log p n p n ˙ ˆ log n log p n ˙ “ lim n Ñ8 log n log p n “ lim q Ñ8 log π p q q log q “ lim q Ñ8 log ´ π p q q log qq ¯ ` log q ´ log log q log q “ lim q Ñ8 ˆ log 1log q ` ´ log log q log q ˙ “ . Remark 2.5.
Another simple consequence of the prime number theorem is the density of t p { q : p, q prime u in r , [14].3. T HE R IEMANN ZETA FUNCTION
The
Riemann zeta function is defined by ζ p s q “ ÿ n “ n s , for Re s ą . (3.1)The use of s for a complex variable is standard in analytic number theory, and we largelyadhere to this convention. Suppose that Re s ě σ ą . Since | n s | “ | e s log n | “ e Re p s log n q “ e p log n q Re s “ p e log n q Re s “ n Re s ě n σ STEPHAN RAMON GARCIA it follows that ˇˇˇˇˇ ÿ n “ n s ˇˇˇˇˇ ď ÿ n “ n σ ă 8 . The Weierstrass M -test ensures that (3.1) converges absolutely and uniformly on Re s ě σ .Since σ ą is arbitrary and each summand in (3.1) is analytic on Re s ą , we concludethat (3.1) converges locally uniformly on Re s ą to an analytic function.In what follows, p denotes a prime number and a sum or product indexed by p runs overthe prime numbers. Here is the connection between the zeta function and the primes. Theorem 3.2 (Euler Product Formula) . If Re s ą , then ζ p s q ‰ and ζ p s q “ ź p ˆ ´ p s ˙ ´ . (3.3) The convergence is locally uniform in Re s ą .Proof. Since | p ´ s | “ p ´ Re s ă for Re s ą , the geometric series formula implies ˆ ´ p s ˙ ´ “ ÿ n “ ˆ p s ˙ n “ ÿ n “ p ns , in which the convergence is absolute. Since a finite number of absolutely convergent seriescan be multiplied term-by-term, it follows that ˆ ´ s ˙ ´ ˆ ´ s ˙ ´ “ ˆ ` s ` s ` ¨ ¨ ¨ ˙ ˆ ` s ` s ` ¨ ¨ ¨ ˙ “ ` s ` s ` s ` s ` s ` s ` s ` ¨ ¨ ¨ , in which only natural numbers divisible by the primes or appear. Similarly, ź p ď ˆ ´ p s ˙ ´ “ ˆ ` s ` s ` s ` s ` s ` ¨ ¨ ¨ ˙ ˆ ` s ` s ` ¨ ¨ ¨ ˙ “ ` s ` s ` s ` s ` s ` s ` s ` s ` s ` ¨ ¨ ¨ , in which only natural numbers divisible by the primes , , or appear. Since the primefactors of each n ď N are at most N , and because the tail of a convergent series tends to , it follows that for Re s ě σ ą ˇˇˇˇˇ ζ p s q ´ ź p ď N ˆ ´ p s ˙ ´ ˇˇˇˇˇ ď ÿ n ą N ˇˇˇˇ n s ˇˇˇˇ ď ÿ n “ N n σ Ñ as N Ñ 8 . This establishes (3.3) and proves that the convergence is locally uniform on Re s ą . Since each partial product does not vanish on Re s ą and because the limit ζ p s q is not identically zero, Hurwitz’ theorem ensures that ζ p s q ‰ for Re s ą . (cid:3) Remark 3.4. By ś p p ´ p ´ s q ´ we mean lim N Ñ8 ś p ď N p ´ p ´ s q ´ . This definitionis sufficient for our purposes, but differs from the general definition of infinite products (interms of logarithms) one might see in advanced complex-variables texts. Let Ω Ď C be nonempty, connected, and open, and let f n be a sequence of analytic functions that convergeslocally uniformly on Ω to f (which is necessarily analytic). If each f n is nonvanishing on Ω , then f is eitheridentically zero or nowhere vanishing on Ω . HE PRIME NUMBER THEOREM 7
Remark 3.5.
The convergence of ś p p ´ p ´ s q ´ and the nonvanishing of each factordoes not automatically imply that the infinite product is nonvanishing (this is frequentlyglossed over). Indeed, lim N Ñ8 ś Nn “ “ N “ even though each factor is nonzero.Thus, the appeal to Hurwitz’ theorem is necessary unless another approach is taken. Remark 3.6.
A similar argument establishes ζ p s q ź p ˆ ´ p s ˙ “ , (3.7)in which the convergence is locally uniform on Re s ą . This directly yields the nonvan-ishing of ζ p s q on Re s ą . However, a separate argument is needed to deduce the locallyuniform convergence of (3.3) from the locally uniform convergence of (3.7). Remark 3.8.
The Euler product formula implies Euclid’s theorem (the infinitude of theprimes). If there were only finitely many primes, then the right-hand side of (3.3) wouldconverge to a finite limit as s Ñ ` . However, the left-hand side of (3.1) diverges as s Ñ ` since its terms tend to those of the harmonic series.4. A NALYTIC C ONTINUATION OF THE Z ETA F UNCTION
We now prove that the Riemann zeta function can be analytically continued to Re s ą , with the exception of s “ , where ζ p s q has a simple pole. Although much morecan be said about this matter, this modest result is sufficient for our purposes. On theother hand, the instructor might wish to supplement this material with some remarks onthe Riemann Hypothesis; see Remark 4.5. Students perk up at the mention of the largemonetary prize associated to the problem. At the very least, they may wish to learn aboutthe most important open problem in mathematics. Theorem 4.1. ζ p s q can be analytically continued to Re s ą except for a simple pole at s “ with residue .Proof. In what follows, t x u denotes the unique integer such that t x u ď x ă t x u ` ; inparticular, ď x ´ t x u ă . For Re s ą , ζ p s q “ ÿ n “ n s “ ÿ n “ n ´ p n ´ q n s “ ÿ n “ nn s ´ ÿ n “ n ´ n s “ ÿ n “ nn s ´ ÿ n “ n p n ` q s “ ÿ n “ n ˆ n s ´ p n ` q s ˙ “ ÿ n “ n ˆ s ż n ` n dxx s ` ˙ “ s ÿ n “ ż n ` n n dxx s ` The assumption Re s ą ensures that both ř n “ nn s and ř n “ n ´ n s converge locally uniformly. STEPHAN RAMON GARCIA “ s ÿ n “ ż n ` n t x u dxx s ` “ s ż t x u dxx s ` . Observe that for Re s ą , ż dxx s “ s ´ ùñ s ´ ` ´ s ż xx s ` dx “ and hence ζ p s q “ s ż t x u dxx s ` “ ˆ s ´ ` ´ s ż xx s ` dx ˙ ` s ż t x u dxx s ` “ s ´ ` ´ s ż x ´ t x u x s ` dx. (4.2)If the integral above defines an analytic function on Re s ą , then ζ p s q can be analyticallycontinued to Re s ą except for a simple pole at s “ with residue . We prove this withtechniques commonly available at the undergraduate-level (see Remark 4.4).For n “ , , . . . , let f n p s q “ ż n ` n x ´ t x u x s ` dx. For any simple closed curve γ in Re s ą , Fubini’s theorem and Cauchy’s theorem imply ż γ f n p s q ds “ ż γ ż n ` n x ´ t x u x s ` dx ds “ ż n ` n p x ´ t x u q ˆż γ dsx s ` ˙ dx “ ż n ` n p x ´ t x u q dx “ . Morera’s theorem ensures that each f n is analytic on Re s ą . If Re s ě σ ą , then ÿ n “ | f n p s q| “ ÿ n “ ˇˇˇˇż n ` n x ´ t x u x s ` dx ˇˇˇˇ ď ÿ n “ ż n ` n ˇˇˇˇ x ´ t x u x s ` ˇˇˇˇ dx ď ÿ n “ ż n ` n dxx Re p s ` q ď ż dxx σ ` “ σ ă 8 . HE PRIME NUMBER THEOREM 9
Consequently, the Weierstrass M -test implies that ÿ n “ f n p s q “ ż x ´ t x u x s ` dx (4.3)converges absolutely and uniformly on Re s ě σ . Since σ ą was arbitrary, it followsthat the series converges locally uniformly on Re s ą . Being the locally uniform limit ofanalytic functions on Re s ą , we conclude that (4.3) is analytic there. (cid:3) Remark 4.4.
The instructor should be aware that many sources, in the interest of brevity,claim without proof that the integral in (4.2) defines an analytic function on Re s ą . Thisis a nontrivial result for an undergraduate course, especially since the domain of integra-tion is infinite. If the instructor has Lebesgue integration at their disposal, the dominatedconvergence theorem, which can be applied to r , , makes the proof significantly easier. Remark 4.5.
It turns out that ζ p s q can be analytically continued to C zt u . The argumentinvolves the introduction of the gamma function Γ p z q “ ş x z ´ e ´ x dx to obtain the functional equation ζ p s q “ s π s ´ sin ´ πs ¯ Γ p ´ s q ζ p ´ s q . (4.6)The extended zeta function has zeros at ´ , ´ , ´ , . . . (the trivial zeros ), along withinfinitely many zeros in the critical strip ă Re s ă (the nontrivial zeros ). To a fewdecimal places, here are the first twenty nontrivial zeros that lie in the upper half plane (thezeros are symmetric with respect to the real axis): . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i, . ` . i. The first nontrivial zeros lie on the critical line Re s “ . The famous RiemannHypothesis asserts that all the zeros in the critical strip lie on the critical line; see Figure 2.This problem was first posed by Riemann in 1859 and remains unsolved. It is consideredthe most important open problem in mathematics because of the impact it would have onthe distribution of the prime numbers.
Remark 4.7.
Students might benefit from learning that the error in the estimate afforded bythe prime number theorem is tied to the zeros of the zeta function. Otherwise the RiemannHypothesis might seem too esoteric and unrelated to the prime number theorem. One canshow that if ζ p s q ‰ for Re s ą σ , then there is a constant C σ such that | π p x q ´ Li p x q| ď C σ x σ log x for all x ě [5, (2.2.6)]. Since it is known that the zeta function has infinitely many zeroson the critical line Re s “ , we must have σ ě .5. T HE LOGARITHM OF ζ p s q In this section we establish a series representation of the logarithm of the zeta function.We use this in Section 6 to establish the nonvanishing of ζ p s q for Re s “ and in Section8 to obtain an analytic continuation of a closely-related function. -2-4-6 1simple polecritical linecritical striptrivial zeros nontrivial zerosoriginal domainanalytic continuation F IGURE
2. Analytic continuation of ζ p s q to C zt u . The nontrivial zeros of the Riemannzeta function lie in the critical strip ă Re s ă . The Riemann Hypothesis asserts thatall of them lie on the critical line Re s “ . Lemma 5.1. If Re s ą , then log ζ p s q “ ÿ n “ c n n s , in which c n ě for n ě .Proof. The open half plane Re s ą is simply connected and ζ p s q does not vanish there(Theorem 3.2). Thus, we may define a branch of log ζ p s q for Re s ą such that log ζ p σ q P R for σ ą . Recall that log ˆ ´ z ˙ “ ÿ k “ z k k (5.2)for | z | ă and observe that Re s ą implies | p ´ s | “ p ´ Re s ă , which permits z “ p ´ s in (5.2). The Euler product formula (3.3), the nonvanishing of ζ p s q for Re s ą , and thecontinuity of the logarithm imply log ζ p s q “ log ź p ˆ ´ p ´ s ˙ “ ÿ p log ˆ ´ p ´ s ˙ (5.3) “ ÿ p ÿ k “ p p ´ s q k k “ ÿ p ÿ k “ { k p p k q s “ ÿ n “ c n n s , in which c n “ $&% k if n “ p k , otherwise . (5.4)The rearrangement of the series above is permissible by absolute convergence. (cid:3) HE PRIME NUMBER THEOREM 11
Lemma 5.5. If s “ σ ` it , in which σ ą and t P R , then log | ζ p s q| “ ÿ n “ c n cos p t log n q n σ , in which the c n are given by (5.4) .Proof. Since σ “ Re s ą , Lemma 5.1 and Euler’s formula provide log | ζ p s q| “ Re ` log ζ p s q ˘ “ Re ÿ n “ c n n σ ` it “ Re ÿ n “ c n e p σ ` it q log n “ Re ÿ n “ c n e ´ it log n e σ log n “ ÿ n “ c n cos p t log n q n σ . (cid:3) Remark 5.6.
The identity (5.3) permits a proof that ř p p ´ diverges; this is Euler’s refine-ment of Euclid’s theorem (the infinitude of the primes). Suppose toward a contradictionthat ř p p ´ converges. For | z | ă , (5.2) implies ˇˇˇˇ log ˆ ´ z ˙ˇˇˇˇ “ ˇˇˇˇˇ ÿ k “ z k k ˇˇˇˇˇ ď ÿ k “ | z | k “ | z | ´ | z | ď | z | . (5.7)For s ą , (5.3) and the previous inequality imply log ζ p s q “ ÿ p log ˆ ´ p ´ s ˙ ă ÿ p p s ď ÿ p p ă 8 . This contradicts the fact that ζ p s q has a pole at s “ . The divergence of ř p p ´ tells usthat the primes are packed tighter in the natural numbers than are the perfect squares since ř n “ n “ ζ p q is finite (in fact, Euler proved that it equals π ).6. N ONVANISHING OF ζ p s q ON Re s “ Theorem 4.1 provides the analytic continuation of ζ p s q to Re s ą . The followingimportant result tells us that the extended zeta function does not vanish on the vertical line Re s “ . One can show that this statement is equivalent to the prime number theorem,although we focus only on deriving the prime number theorem from it. Theorem 6.1. ζ p s q has no zeros on Re s “ .Proof. Recall that ζ p s q extends analytically to Re s ą (Theorem 4.1) except for a simplepole at s “ ; in particular, ζ p s q does not vanish at s “ . Suppose toward a contradictionthat ζ p ` it q “ for some t P R zt u and consider f p s q “ ζ p s q ζ p s ` it q ζ p s ` it q . Observe that(i) ζ p s q has a pole of order three at s “ since ζ p s q has a simple pole at s “ ;(ii) ζ p s ` it q has a zero of order at least four at s “ since ζ p ` it q “ ; and (iii) ζ p s ` it q does not have a pole at s “ since t P R zt u and s “ is the onlypole of ζ p s q on Re s “ .Thus, the singularity of f at s “ is removable and f p q “ . Therefore, lim s Ñ log | f p s q| “ ´8 . (6.2)On the other hand, Lemma 5.5 yields log | f p s q| “ | ζ p s q| ` | ζ p s ` it q| ` log | ζ p s ` it q|“ ÿ n “ c n n σ ` ÿ n “ c n cos p t log n q n σ ` ÿ n “ c n cos p t log n q n σ “ ÿ n “ c n n σ ` ` p t log n q ` cos p t log n q ˘ ě since c n ě for n ě and ` x ` cos 2 x “ p ` cos x q ě , for x P R . Since this contradicts (6.2), we conclude that ζ p s q has no zeros with Re s “ . (cid:3) Remark 6.3.
Since Theorem 3.2 already ensures that ζ p s q ‰ for Re s ą , Theorem 6.1implies ζ p s q does not vanish in the closed half plane Re s ě .7. C HEBYSHEV T HETA F UNCTION
It is often convenient to attack problems related to prime numbers with logarithmicallyweighted sums. Instead of working with π p x q “ ř p ď x directly, we consider ϑ p x q “ ÿ p ď x log p. (7.1)We will derive the prime number theorem from the statement ϑ p x q „ x . Since this asymp-totic equivalence is difficult to establish, we first content ourselves with an upper bound. Theorem 7.2 (Chebyshev’s Lemma) . ϑ p x q ď x .Proof. If n ă p ď n , then p divides ˆ nn ˙ “ p n q ! n ! n ! since p divides the numerator but not the denominator. The binomial theorem implies n “ p ` q n “ n ÿ k “ ˆ nk ˙ k n ´ k ě ˆ nn ˙ ě ź n ă p ď n p “ ź n ă p ď n e log p “ exp ´ ÿ n ă p ď n log p ¯ “ exp ` ϑ p n q ´ ϑ p n q ˘ . HE PRIME NUMBER THEOREM 13
Therefore, ϑ p n q ´ ϑ p n q ď n log 2 . Set n “ k ´ and deduce ϑ p k q ´ ϑ p k ´ q ď k log 2 . Since ϑ p q “ , a telescoping-series argument and the summation formula for a finitegeometric series provide ϑ p k q “ ϑ p k q ´ ϑ p q “ k ÿ i “ ` ϑ p i q ´ ϑ p i ´ q ˘ ď k ÿ i “ i log 2 ă p ` ` ` ¨ ¨ ¨ ` k q log 2 ă k ` log 2 . If x ě , then let k ď x ă k ` ; that is, let k “ t log x log 2 u . Then ϑ p x q ď ϑ p k ` q ď k ` log 2 “ ¨ k log 2 ď x p q ă x since « . ă . (cid:3) Remark 7.3.
The Euler product formula (3.3), which requires the fundamental theorem ofarithmetic, and the opening lines of the proof of Chebyshev’s lemma are the only portionsof our proof of the prime number theorem that explicitly require number theory.
Remark 7.4.
There are many other “theta functions,” some of which arise in the context ofthe Riemann zeta function. For example, the Jacobi theta function θ p z q “ ř m “´8 e ´ πm z ,defined for Re z ą , is often used in proving the functional equation (4.6).8. T HE Φ F UNCTION
Although we have tried to limit the introduction of new functions, we must consider Φ p s q “ ÿ p log pp s , (8.1)whose relevance to the prime numbers is evident from its definition. If Re s ě σ ą , then ÿ p ˇˇˇˇ log pp s ˇˇˇˇ ď ÿ p log pp Re s ď ÿ p log pp σ ă ÿ n “ log nn σ ă 8 by the integral test. The Weierstrass M -test ensures (8.1) converges uniformly on Re s ě σ . Since the summands in (8.1) are analytic on Re s ą and σ ą was arbitrary, theseries (8.1) converges locally uniformly on Re s ą and hence Φ p s q is analytic there. Forthe prime number theorem, we need a little more. Theorem 8.2. Φ p s q ´ s ´ is analytic on an open set containing Re s ě .Proof. For Re s ą , (5.3) tells us log ζ p s q “ log ´ ź p p ´ p ´ s q ´ ¯ “ ´ ÿ p log p ´ p ´ s q . (8.3) The inequality (5.7) implies | ´ p ´ s | ď p Re s , which implies that the convergence in (8.3) is locally uniform on Re s ą . Consequently,we may take the derivative of (8.3) term-by-term and get ´ ζ p s q ζ p s q “ ÿ p p log p q p ´ s ´ p ´ s “ ÿ p p log p q ˆ p s ´ ˙ “ ÿ p p log p q ˆ p s ` p s p p s ´ q ˙ “ ÿ p ˆ log pp s ` log pp s p p s ´ q ˙ “ ÿ p log pp s ` ÿ p log pp s p p s ´ q “ Φ p s q ` ÿ p log pp s p p s ´ q . If Re s ě σ ą , then the limit comparison test and integral test imply ÿ p ˇˇˇˇ log pp s p p s ´ q ˇˇˇˇ ď ÿ n “ log n p n Re s ´ q ď ÿ n “ log n p n σ ´ q ă 8 . The Weierstrass M -test ensures that ÿ p log pp s p p s ´ q converges locally uniformly on Re s ą and is analytic there. Theorem 4.1 implies that Φ p s q “ ´ ζ p s q ζ p s q ´ ÿ p log pp s p p s ´ q extends meromorphically to Re s ą with poles only at s “ and the zeros of ζ p s q .Theorem 4.1 also yields ζ p s q “ p s ´ q ´ Z p s q , Z p q “ , in which Z p s q is analytic near s “ . Consequently, ζ p s q ζ p s q “ ´ p s ´ q ´ Z p s q ` p s ´ q ´ Z p s qp s ´ q ´ Z p s q “ ´ s ´ ` Z p s q Z p s q and hence Φ p s q ´ s ´ “ ´ Z p s q Z p s q ´ ÿ p log pp s p p s ´ q , in which the right-hand side is meromorphic on Re s ą with poles only at the zeros of ζ p s q . Theorem 6.1 ensures that ζ has no zeros on Re s “ , so the right-hand side extendsanalytically to some open neighborhood of Re s ě ; see Remark 8.4. (cid:3) Remark 8.4.
The zeros of a nonconstant analytic function are isolated, so no boundedsequence of zeta zeros can converge to a point on Re s “ . Consequently, it is possibleto extend Φ p s q ´ p s ´ q ´ a little beyond Re s “ in a manner that avoids the zeros of ζ p s q . It may not be possible to do this on a half plane, however. The Riemann Hypothesissuggests that the half plane Re s ą works, but this remains unproven. Compare ř n “ n p n σ ´ q with ř n “ nn σ and observe that ş tt σ dt ă 8 . HE PRIME NUMBER THEOREM 15
9. L
APLACE T RANSFORMS
Laplace transform methods are commonly used to study differential equations and oftenfeature prominently in complex-variables texts. We need only the basic definition and asimple convergence result. The following theorem is not stated in the greatest generalitypossible, but it is sufficient for our purposes.
Theorem 9.1.
Let f : r ,
8q Ñ C be piecewise continuous on r , a s for all a ą and | f p t q| ď Ae Bt , for t ě . Then the
Laplace transform g p z q “ ż f p t q e ´ zt dt (9.2) of f is well defined and analytic on the half plane Re z ą B .Proof. For Re z ą B , the integral (9.2) converges by the comparison test since ż | f p t q e ´ zt | dt ď ż Ae Bt e ´ t p Re z q dt “ A ż e t p B ´ Re z q dt “ A Re z ´ B ă 8 . If γ is a simple closed curve in Re z ą B , then its compactness ensures that there is a σ ą B such that Re z ě σ for all z P γ . Thus, ż | f p t q e ´ zt | dt ď Aσ ´ B is uniformly bounded for z P γ . Fubini’s theorem and Cauchy’s theorem yield ż γ g p z q dz “ ż γ ż f p t q e ´ zt dt dz “ ż f p t q ˆż γ e ´ zt dz ˙ dt “ ż f p t q ¨ dt “ . Morera’s theorem implies that g is analytic on Re z ą B . (cid:3) Theorem 9.3 (Laplace Representation of Φ ) . For Re s ą , Φ p s q s “ ż ϑ p e t q e ´ st dt. (9.4) Proof.
Recall from Theorem 7.2 that ϑ p x q ď x . Thus, for Re s ą ÿ n “ ˇˇˇˇ ϑ p n ´ q n s ˇˇˇˇ ď ÿ n “ ˇˇˇˇ ϑ p n q n s ˇˇˇˇ ď ÿ n “ nn Re s “ ÿ n “ n p Re s q´ ă 8 . (9.5)Consequently, Φ p s q “ ÿ p log pp s p by (8.1) q“ ÿ n “ ϑ p n q ´ ϑ p n ´ q n s p by (7.1) q“ ÿ n “ ϑ p n q n s ´ ÿ n “ ϑ p n ´ q n s p by (9.5) q The interval r , is unbounded and hence the appeal to Fubini’s theorem is more if one uses Riemannintegration; see the proof of Theorem 4.1. “ ÿ n “ ϑ p n q n s ´ ÿ n “ ϑ p n qp n ` q s “ ÿ n “ ϑ p n q ˆ n s ´ p n ` q s ˙ “ ÿ n “ ϑ p n q ˆ s ż n ` n dxx s ` ˙ “ s ÿ n “ ż n ` n ϑ p n q dxx s ` “ s ÿ n “ ż n ` n ϑ p x q dxx s ` p ϑ p x q “ ϑ p n q on r n, n ` qq“ s ż ϑ p x q dxx s ` “ s ż ϑ p e t q e t dte st ` t p x “ e t and dx “ e t dt q“ s ż ϑ p e t q e ´ st dt. (9.6)This establishes the desired identity (9.4) for Re s ą . Since Theorem 7.2 implies ϑ p e t q ď e t , Theorem 9.1 (with A “ and B “ ) ensures that (9.6) is analytic on Re s ą . Onthe other hand, Φ p s q is analytic on Re s ą so the identity principle implies that thedesired representation (9.4) holds for Re s ą . (cid:3)
10. N
EWMAN ’ S T AUBERIAN T HEOREM
The following theorem is a tour-de-force of undergraduate-level complex analysis. Inwhat follows, observe that g is the Laplace transform of f . The hypotheses upon f ensurethat we will be able to apply the theorem to the Chebyshev theta function. Theorem 10.1 (Newman’s Tauberian Theorem) . Let f : r ,
8q Ñ C be a bounded func-tion that is piecewise continuous on r , a s for each a ą . For Re z ą , let g p z q “ ż f p t q e ´ zt dt and suppose g has an analytic continuation to a neighborhood of Re z ě . Then g p q “ lim T Ñ8 ż T f p t q dt. In particular, ş f p t q dt converges.Proof. For each T P p , , let g T p z q “ ż T e ´ zt f p t q dt. (10.2)The proof of Theorem 9.1 ensures that each g T p z q is an entire function (see Remark 10.14for another approach) and g p z q is analytic on Re z ą ; see Remark 10.14. We must show lim T Ñ8 g T p q “ g p q . (10.3) HE PRIME NUMBER THEOREM 17 C R ´ δ R RiR ´ iR F IGURE
3. The contour C R . The imaginary line segment r´ iR, iR s is compact and canbe covered by finitely many open disks (yellow) upon which g is analytic. Thus, there is a δ R ą such that g is analytic on an open region that contains the curve C R . S TEP
1. Let } f } “ sup t ě | f p t q| , which is finite by assumption. For Re z ą , | g p z q ´ g T p z q| “ ˇˇˇˇˇż e ´ zt f p t q dt ´ ż T e ´ zt f p t q dt ˇˇˇˇˇ “ ˇˇˇˇż T e ´ zt f p t q dt ˇˇˇˇ ď ż T e ´ Re p zt q | f p t q| dt ď } f } ż T e ´ t Re z dt “ } f } e ´ T Re z Re z . (10.4)S TEP
2. For Re z ă , | g T p z q| “ ˇˇˇˇˇż T e ´ zt f p t q dt ˇˇˇˇˇ ď ż T e ´ Re p zt q | f p t q| dt ď } f } ż T e ´ t Re z dt ď } f } ż T ´8 e ´ t Re z dt “ } f } e ´ T Re z | Re z | . (10.5)S TEP
3. Suppose that g has an analytic continuation to an open region Ω that containsthe closed half plane Re z ě . Let R ą and let δ R ą be small enough to ensurethat g is analytic on an open region that contains the curve C R (and its interior) formed byintersecting the circle | z | “ R with the vertical line Re z “ ´ δ R ; see Figure 3.S TEP
4. For each R ą , Cauchy’s integral formula implies g T p q ´ g p q “ πi ż C R ` g T p z q ´ g p z q ˘ e zT ˆ ` z R ˙ dzz . (10.6)We examine the contributions to this integral over the two curves C ` R “ C R X t z : Re z ě u and C ´ R “ C R X t z : Re z ď u . S TEP
5. Let us examine the contribution of C ` R to (10.6). For z “ Re it , ˇˇˇˇ z ˆ ` z R ˙ˇˇˇˇ “ ˇˇˇˇ z ` zR ˇˇˇˇ “ ˇˇˇˇ Re it ` Re it R ˇˇˇˇ “ R | Re ´ it ` Re it | “ R | z ` z |“ | Re z | R . (10.7)For z P C , | e zT | “ e T Re z (10.8)and hence (10.4), (10.7), and (10.8) imply ˇˇˇˇˇ πi ż C ` R ` g T p z q ´ g p z q ˘ e zT ˆ ` z R ˙ dzz ˇˇˇˇˇ ď π ˆ } f } e ´ T Re z Re z ˙looooooooomooooooooon by (10.4) p e T Re z q looomooon by (10.8) ˆ | Re z | R ˙looooomooooon by (10.7) p πR q“ } f } R . (10.9)S
TEP A . We examine the contribution of C ´ R to (10.6) in two steps. Since the integrandin the following integral is analytic in Re z ă , we can replace the contour C ´ R with theleft-hand side of the circle | z | “ R in the computation ˇˇˇˇˇ πi ż C ´ R g T p z q e zT ˆ ` z R ˙ dzz ˇˇˇˇˇ (10.10) “ ˇˇˇˇˇˇ πi ż | z |“ R Re z ď g T p z q e zT ˆ ` z R ˙ dzz ˇˇˇˇˇˇ ď π ˆ } f } e ´ T Re z | Re z | ˙looooooooomooooooooon by (10.5) p e T Re z q ˆ | Re z | R ˙looooomooooon by (10.7) p πR q“ } f } R ; (10.11)see Figure 4.S TEP B . Next we focus on the corresponding integral with g in place of g T . Let M “ sup z P C ´ R | g p z q| , which is finite since C ´ R is compact. Since | z | ě δ R for z P C ´ R , ˇˇˇˇ g p z q e zT ˆ ` z R ˙ z ˇˇˇˇ ď M e T Re z δ R . Fix (cid:15) ą and obtain a curve C ´ R p (cid:15) q by removing, from the beginning and end of C ´ R , twoarcs each of length (cid:15)δ R {p M q ; see Figure 5. Then there is a ρ ą such that Re z ă ´ ρ HE PRIME NUMBER THEOREM 19 C R ´ δ R RiR ´ iR F IGURE
4. The integrand in (10.10) is analytic in Re z ă . Cauchy’s theorem ensuresthat the integral over C ´ R equals the integral over the semicircle t z : | z | “ R, Re z ď u . C ` R C ´ R p (cid:15) q RiR Re z “ ´ ρ ´ iR F IGURE C ´ R p (cid:15) q is obtained from C ´ R by removing two segments (red) each of length (cid:15)δ R {p M q . There is a ρ ą such that Re z ă ´ ρ for each z P C ´ R p (cid:15) q . for each z P C ´ R p (cid:15) q . Consequently, lim sup T Ñ8 ˇˇˇˇˇż C ´ R g p z q e zT ˆ ` z R ˙ dzz ˇˇˇˇˇ ď lim sup T Ñ8 ˆ M e ´ ρT δ R ¨ πR looooooomooooooon from C ´ R p (cid:15) q ` Mδ R ¨ (cid:15)δ R M looooomooooon from the two arcs ˙ “ (cid:15). Since (cid:15) ą was arbitrary, lim sup T Ñ8 ˇˇˇˇˇż C ´ R g p z q e zT ˆ ` z R ˙ dzz ˇˇˇˇˇ “ . (10.12) S TEP
7. For each fixed R ą , lim sup T Ñ | g T p q ´ g p q|“ lim sup T Ñ ˇˇˇˇ πi ż C R ` g T p z q ´ g p z q ˘ e zT ˆ ` z R ˙ dzz ˇˇˇˇ p by (10.6) qď } f } R loomoon from C ` R ` ˆ } f } R ` ˙loooooomoooooon from C ´ R p by (10.9), (10.11), (10.12) q“ } f } R .
Since R ą was arbitrary, lim sup T Ñ8 | g T p q ´ g p q| “ that is, lim T Ñ8 g T p q “ g p q . (cid:3) Remark 10.13.
A “Tauberian theorem” is a result in which a convergence result is deducedfrom a weaker convergence result and an additional hypothesis. The phrase originates inthe work of G.H. Hardy and J.E. Littlewood, who coined the term in honor of A. Tauber.
Remark 10.14.
To see that g T p z q entire, first note that since we are integrating over r , T s there are no convergence issues. We may let γ be any simple closed curve in C when wemimic the proof of Theorem 9.1. Another approach is to expand e ´ zt as a power seriesand use the uniform convergence of the series on r , T s to exchange the order of sum andintegral. This yields a power series expansion of g T p z q with infinite radius of convergence.Here are the details. Fix T ą and let M “ sup ď t ď T | f p t q| , which is finite since r , T s is compact and f is piecewise continuous (a piecewise-continuousfunction has at most finitely many discontinuities, all of which are jump discontinuities).Then c n “ ż T f p t q t n dt satisfies | c n | ď M T n ` n ` . Since e z is entire, its power series representation converges uniformly on r , T s . Thus, g T p z q “ ż T f p t q e ´ zt dt “ ż T f p t q ˆ ÿ n “ p´ zt q n n ! ˙ dt “ ÿ n “ p´ q n z n n ! ż T f p t q t n dt “ ÿ n “ p´ q n c n n ! z n defines an entire function since its radius of convergence is the reciprocal of lim sup n Ñ8 ˇˇˇˇ p´ q n c n n ! ˇˇˇˇ n ď lim sup n Ñ8 M n T n ` n p n ` q n p n ! q n “ ¨ T ¨ 8 “ by the Cauchy–Hadamard formula. HE PRIME NUMBER THEOREM 21
Remark 10.15.
Step 6b is more complicated than in most presentations because we areusing the Riemann integral (for the sake of accessibility) instead of the Lebesgue integral.The statement (10.12) follows immediately from the Fatou–Lebesgue theorem in Lebesguetheory; see the proof in [35]. Riemann integration theory cannot prove (10.12) directlysince the integrand does not convergence uniformly to zero on C ´ R .11. A N I MPROPER I NTEGRAL
Things come together in the following lemma. We have done most of the difficult workalready; the proof of Lemma 11.1 amounts to a series of strategic applications of existingresults. It requires Chebyshev’s estimate for ϑ p x q (Theorem 7.2), the analytic continuationof Φ p s q ´ p s ´ q ´ to an open neighborhood of Re s ě (Theorem 8.2), the Laplace-transform representation of Φ p s q (Theorem 9.3), and Newman’s theorem (Theorem 10.1). Lemma 11.1. ż ϑ p x q ´ xx dx converges.Proof. Define f : r ,
8q Ñ C by f p t q “ ϑ p e t q e ´ t ´ and observe that it is piecewise continuous on r , a s for all a ą and | f p t q| ď | ϑ p e t q| e ´ t ` ď for all t ě by Theorem 7.2. Then Theorem 9.1 with A “ and B “ ensures that theLaplace transform of f is analytic on Re z ą . Consequently, for Re z ą ż f p t q e ´ zt dt “ ż ` ϑ p e t q e ´ t ´ ˘ e ´ zt dt “ ż ` ϑ p e t q e ´p z ` q t ´ e ´ zt ˘ dt “ ż ϑ p e t q e ´p z ` q t dt ´ ż e ´ zt dt (11.2) “ ż ϑ p e t q e ´p z ` q t dt ´ z “ Φ p z ` q z ` ´ z p by Theorem 9.3 q . Let z “ s ´ and note that Theorem 8.2 implies that g p z q “ Φ p z ` q z ` ´ z “ Φ p s q s ´ s ´ extends analytically to an open neighborhood of Re s ě ; that is, to an open neighborhoodof the closed half plane Re z ě . Theorem 10.1 ensures that the improper integral ż f p t q dt “ ż ` ϑ p e t q e ´ t ´ ˘ dt “ ż ˆ ϑ p x q x ´ ˙ dxx p x “ e t and dx “ e t dt q“ ż ϑ p x q ´ xx dx converges. (cid:3) Remark 11.3.
Newman’s theorem implies that the improper integral in Lemma 11.1equals g p q although this is not necessary for our purposes. Remark 11.4.
Since | ϑ p e t q| ď e t by Theorem 7.2, the first improper integral (11.2)converges and defines an analytic function on Re z ą by Theorem 9.1 with A “ and B “ . We did not mention this in the proof of Lemma 11.1 because the convergence ofthe integral is already guaranteed by the convergence of ş f p t q e ´ zt dt and ş e ´ zt dt .12. A SYMPTOTIC B EHAVIOR OF ϑ p x q A major ingredient in the proof of the prime number theorem is the following asymptoticstatement. Students must be comfortable with limits superior and inferior after this point;these concepts are used frequently throughout what follows.
Theorem 12.1. ϑ p x q „ x .Proof. Observe that ż ϑ p t q ´ tt dt exists looooooooooooomooooooooooooon by Lemma 11.1 ùñ lim x Ñ8 ż x ϑ p t q ´ tt dt looooooomooooooon I p x q “ . (12.2)S TEP
1. Suppose toward a contradiction that lim sup x Ñ8 ϑ p x q x ą , and let lim sup x Ñ8 ϑ p x q x ą α ą . Then there are arbitrarily large x ą such that ϑ p x q ą αx . For such “bad” x , I p αx q ´ I p x q “ ż αxx ϑ p t q ´ tt dt ě ż αxx αx ´ tt dt p ϑ p x q ą αx and ϑ is increasing q“ ż α αx ´ xux u x du “ ż α α ´ uu du p t “ xu, dt “ x du q“ α ´ ´ log α ą . Since lim inf x Ñ8 x bad ` I p αx q ´ I p x q ˘ ą contradicts (12.2), we conclude lim sup x Ñ8 ϑ p x q x ď . S TEP
2. This is similar to the first step. Suppose toward a contradiction that lim inf x Ñ8 ϑ p x q x ă , and let lim inf x Ñ8 ϑ p x q x ă β ă the limit inferior is nonnegative since ϑ p x q is nonnegative. Then there are arbitrarily large x ą such that ϑ p x q ă βx . For such “bad” x , I p x q ´ I p βx q “ ż xβx ϑ p t q ´ tt dt ď ż xβx βx ´ tt dt p ϑ p x q ă βx and ϑ is increasing q HE PRIME NUMBER THEOREM 23 F IGURE
6. Graph of f p x q “ x ´ ´ log x . “ ż β βx ´ xux u x du “ ż β β ´ uu du p t “ xu, dt “ x du q“ ´ β ` log β ă . Since lim inf x Ñ8 x bad ` I p x q ´ I p βx q ˘ ă contradicts (12.2), we conclude lim inf x Ñ8 ϑ p x q x ě . S TEP
3. Since lim sup x Ñ8 ϑ p x q x ď and lim inf x Ñ8 ϑ p x q x ě , it follows that lim x Ñ8 ϑ p x q{ x “ ; that is, ϑ p x q „ x . (cid:3) Remark 12.3.
Let f p x q “ x ´ ´ log x for x ą . Then f p x q “ ´ { x and f p x q “ { x , so f is strictly positive on p , q and p , ; see Figure 6. This ensures the positivityof α ´ ´ log α for α ą and the negativity of ´ β ` log β for β P p , q . Remark 12.4.
One can show that π p x q „ x { log x implies ϑ p x q „ x , although this is notnecessary for our purposes. In light of Theorem 13.1 below, this implication shows that π p x q „ x { log x is equivalent to ϑ p x q „ x .13. C OMPLETION OF THE P ROOF
At long last we are ready to complete the proof of the prime number theorem. We breakthe conclusion of the proof into three short steps.
Theorem 13.1 (Prime Number Theorem) . π p x q „ x log x .Proof. Recall from Theorem 12.1 that ϑ p x q „ x ; that is, lim x Ñ8 ϑ p x q{ x “ .S TEP
1. Since ϑ p x q “ ÿ p ď x log p ď ÿ p ď x log x “ p log x q ÿ p ď x “ π p x q log x, it follows that “ lim x Ñ8 ϑ p x q x “ lim inf x Ñ8 ϑ p x q x ď lim inf x Ñ8 π p x q log xx . S TEP
2. For any (cid:15) ą , ϑ p x q “ ÿ p ď x log p ě ÿ x ´ (cid:15) ă p ď x log p ě ÿ x ´ (cid:15) ă p ď x log p x ´ (cid:15) q “ log p x ´ (cid:15) q ÿ x ´ (cid:15) ă p ď x “ p ´ (cid:15) qp log x q ˆ ÿ p ď x ´ ÿ p ď x ´ (cid:15) ˙ ě p ´ (cid:15) q ` π p x q ´ x ´ (cid:15) ˘ log x. Therefore, “ lim x Ñ8 ϑ p x q x “ lim sup x Ñ8 ϑ p x q x ě lim sup x Ñ8 ˜ p ´ (cid:15) q ` π p x q ´ x ´ (cid:15) ˘ log xx ¸ “ p ´ (cid:15) q lim sup x Ñ8 ˆ π p x q log xx ´ log xx (cid:15) ˙ “ p ´ (cid:15) q lim sup x Ñ8 π p x q log xx ´ p ´ (cid:15) q lim x Ñ8 log xx (cid:15) “ p ´ (cid:15) q lim sup x Ñ8 π p x q log xx . Since (cid:15) ą was arbitrary, lim sup x Ñ8 π p x q log xx ď . S TEP
3. Since ď lim inf x Ñ8 π p x q x { log x ď lim sup x Ñ8 π p x q x { log x ď , we obtain lim x Ñ8 π p x q x { log x “ . This concludes the proof of the prime number theorem. (cid:3)
It is probably best not to drag things out at this point. Nothing can compete with finish-ing off one of the major theorems in mathematics. After coming this far, the reader shouldbe convinced that the proof of the prime number theorem, as presented here, is largely atheorem of complex analysis (obviously this is a biased perspective based upon our choiceof proof). Nevertheless, we hope that the reader is convinced that a proof of the primenumber theorem can function as an excellent capstone for a course in complex analysis.
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