A Heuristic for the Prime Number Theorem

Transcription

1 A Heuristic for the Prime Number Theorem This article appeared in The Mathematical Intelligencer 8:3 (006) 6 9, and is copyright by Springer- Verlag. Posted with permission. Hugh L. Montgomery, University of Michigan, Ann Arbor, Michigan 4809 hlm@umich.edun (FOOTNOTE Using n-dash: Supported in part by NSF grant DMS ) Stan Wagon, Macalester College, St. Paul, Minnesota 5504 wagon@macalester.edu Why? Why does play such a central role in the distribution of prime numbers? Simply citing the Prime Number Theorem (PNT), which asserts that phl ~ ê ln, is not very illuminating. Here "~" means "is asymptotic to" and phl is the number of primes less than or equal to. So why do natural logs appear, as opposed to another flavor of logarithm? The problem with an attempt at a heuristic eplanation is that the sieve of Eratosthenes does not behave as one might guess from pure probabilistic considerations. One might think that sieving out the composites under using primes up to è!!! would lead to P è!!! p< J - ÅÅÅÅ N as an asymptotic estimate of the count of p numbers remaining (the primes up to ; p always represents a prime). But this quantity turns out to be not asymptotic to ê ln. For F. Mertens proved in 874 that the product is actually asymptotic to -g ê ln, or about. ê ln. Thus the sieve is % (from ê.) more efficient at eliminating composites than one might epect. Commenting on this phenomenon, which one might call the Mertens Parado, Hardy and Wright [5, p. 37] said: "Considerations of this kind eplain why the usual 'probability' arguments lead to the wrong asymptotic value for phl." For more on this theorem of Mertens and related results in prime counting see [3; 5; 6, eer. 8.7; 8]. Yet there ought to be a way to eplain, using only elementary methods, why natural logarithms play a central role in the distribution of primes. A good starting place is two old theorems of Chebyshev (849). Chebyshev's First Theorem. For any, 0.9 ê ln < phl <.7 ê ln. Chebyshev's Second Theorem. If phl ~ ê log c, then c =. A complete proof of Chebyshev's First Theorem (with slightly weaker constants) is not difficult and the reader is encouraged to read the beautiful article by Don Zagier [9] the very first article ever published in

2 this journal (see also [, 4.]). The first theorem tells us that ê log c is a reasonable rough approimation to the growth of phl, but it does not distinguish from other bases. The second theorem can be given a complete proof using only elementary calculus [8]. The result is certainly a partial heuristic for the centrality of since it shows that, if any logarithm works, then the base must be. Further, one can see the eact place in the proof where arises ( Ÿ ê = ln ). But the hypothesis for the second theorem is a strong one; here we will show, by a relatively simple proof, that the same conclusion follows from a much weaker hypothesis. Of course, the PNT eliminates the need for any hypothesis at all, but its proof requires either an understanding of comple analysis or the willingness to read the sophisticated "elementary proof". The first such was found by Erdos and Selberg; a modern approach appears in [7]. Our presentation here was inspired by a discussion in Courant and Robbins []. We show how their heuristic approach can be transformed into a proof of a strong result. So even though our original goal was just to motivate the PNT, we end up with a proved theorem that has a simple statement and quite a simple proof. Theorem. If ê phl is asymptotic to an increasing function, then phl ~ ê ln. Figure shows that ê phl is assuredly not increasing. Yet it does appear to be asymptotic to the piecewise linear function that is the upper part of the conve hull of the graph. Indeed, if we take the conve hull of the full infinite graph, then the piecewise linear function LHL corresponding to the part of the hull above the graph is increasing (see last section). If one could prove that ê phl ~ LHL then, by the theorem, the PNT would follow. In fact, using PNT it is not too hard to prove that LHL is indeed asymptotic to ê phl (such proof is given at the end of this paper). In any case, the hypothesis of the theorem is certainly believable, if not so easy to prove, and so the theorem serves as a heuristic eplanation of the PNT Figure. A graph of ê phl shows that the function is not purely increasing. The upper conve hull of the graph is an increasing piecewise linear function that is a good approimation to ê phl. Nowadays we can look quite far into the prime realm. Zagier's article of 6 years ago was called The First Fifty Million Prime Numbers. Now we can look at the first 700 quintillion prime numbers. Not one at a time, perhaps, but the eact value of ph0 i L is now known for i up to ; the most recent value is due to Gourdon and Sabeh [4] and is ph4 ÿ 0 L = Figure is a log-log plot that shows the error when these stratospheric p values are compared to ê ln and also the much better logarithmic integral estimate lihl (which is Ÿ 0 ê ln t t).

3 Figure. The large dots are the absolute value of the error when ê ln is used to approimate phl, for = 0 i. The smaller dots use lihl as the approimant. Two Lemmas The proof requires two lemmas. The first is a consequence of Chebyshev's First Theorem, but can be given a short and elementary proof; it states that almost all numbers are composite. Lemma. lim Ø phl ê = 0. Proof. First use an idea of Chebyshev to get ph nl - phnl < n ê ln n for integers n. Take log-base-n of both sides of the following to get the needed inequality. 4 n = H + L n > I n n M n<p n p > n = nph nl-phnl n<p n This means that ph nl - phnl Hln 4L n ê ln n. Suppose n is a power of, say k ; then summing over k K, where K is chosen so that K n < K+, gives phnl + k= K k ln 4 ÅÅÅÅÅÅ k Here each term in the sum is at most 3 ê 4 of the net term, so the entire sum is at most 4 times the last term. That is, phnl c n ê ln n, which implies phnl ê n Ø 0. For any > 0, we take n to be the first power of past, and then phl ê phnl ê n, concluding the proof. By keeping careful track of the constants, the preceding proof can be used to show that phl 8. ê ln, yielding one half of Chebyshev's first theorem, albeit with a weaker constant. The second lemma is a type of Tauberian result, and the proof goes just slightly beyond elementary calculus. This lemma is where natural logs come up, well, naturally. For consider the hypothesis with log c in place of ln. Then the constant ln c will cancel, so the conclusion will be unchanged! Lemma. If WHL is decreasing and Ÿ WHtL lnh tl ê t t ~ ln, then WHL ~ ê ln. Proof. Let e be small and positive; let f HtL = lnhtl ê t. The hypothesis implies Ÿ +e WHtL f HtL t ~ e ln (to see this split the integral into two: from to and to +e ). Since WHL is decreasing,

4 4 e ln ~ Ÿ +e WHtL f HtL t WHL Ÿ +e ln t ÅÅÅÅÅÅÅÅ t = e I + ÅÅÅÅ e M WHL Hln L t Thus lim inf Ø WHL ln ë I + ÅÅÅÅ e M. A similar argument starting with Ÿ -e WHtL f HtL t ~ e ln shows that limsup Ø WHL ln ë I - ÅÅÅÅ e M. Since e can be arbitrarily small, we have WHL ln ~. Proof of the Theorem Theorem. If ê phl is asymptotic to an increasing function, then phl ~ ê ln. Proof. Let LHL be the hypothesized increasing function and let WHL = ê LHL, a decreasing function. It suffices to show that the hypothesis of Lemma holds, for then LHL ~ ln. Let f HtL = lnh tl ê t. Note that if ln ~ ghl + hhl where hhl ê ln Ø 0, then ln ~ ghl; we will use this several times in the following sequence, which reaches the desired conclusion by a chain of relations. The notation p k»» n in the third epression means that k is the largest power of p that divides n; the equality that follows the sum comes from considering each p m for m k. ln ~ ÅÅÅÅ ln n n = ÅÅÅÅ n p k»» n k = ÅÅÅÅ n p m» n, m Hcan be done by machine; note L = ÅÅÅÅ p m, m f p m v ~ ÅÅÅÅ p m, m p m Herror is small; note L ~ f HpL + p p m p m ~ f HpL p, m Hgeometric series estimation; note 3L = phdtl f HL - Ÿ phtl f HtL t Hpartial summation; note 4L ~ -Ÿ phtl f HtL t Hbecause phdtl f HL ê ln phl ê Ø 0 by Lemma L ~ -Ÿ t WHtL I ÅÅÅÅÅÅ t - ÅÅÅÅÅÅÅÅ ln t t M t Hbecause phtl ~ t WHtLL ~ Ÿ WHtL ÅÅÅÅÅÅÅÅ ln t t Hl' Hôpital, note 5L Notes: t. It is easy to verify this relation using standard integral test ideas: start with the fact that the sum lies between Ÿ ln t t and ln. But it is intriguing to see that Mathematica can resolve this using symbolic algebra. The sum is just lnhdt!l and Mathematica quickly returns when asked for the limit of ln ê lnh!l as Ø.. error ê ln ÅÅ ln p m ÅÅ ln log p p = ÅÅ ln p ln = phl ê Ø 0 by Lemma 3. The second sum divided by ln approaches 0 because:

5 5 p m, m p m p m= p m Hn-L ê n= Hn-L < = p ÅÅÅÅÅÅ php-l ln n ÅÅÅÅÅÅ n= nhn-l 4. We use partial summation, a technique common in analytic number theory. Write the integral from to as a sum of integrals n + D together with one from dt to and use the fact that phtl is constant on such intervals and jumps by eactly at the primes. More precisely: Ÿ p HtL f HtL t = Ÿ dt p HtL f dt- HtL t + n+ n= Ÿn p HtL f HtL t = p HdtL H f HL - f HdtL L + dt- n= p HnL H f Hn + L - f HnL L = p HdtL f HL - p f HpL 5. L'Hôpital's rule on ln Ÿ t WHtL t ÅÅÅÅÅÅ t yields ÅÅÅÅÅÅ WHLê ê = WHL = ê LHL, which approaches 0 by Lemma. No line in the proof uses anything beyond elementary calculus ecept the call to Lemma. The result shows that if there is any nice function that characterizes the growth of phl then that function must be asymptotic to ê ln. Of course, the PNT shows that this function does indeed do the job. This proof works with no change if base-c logarithms are used throughout. But as noted, Lemma will force the natural log to appear! The reason for this lies in the indefinite integration that takes places in the lemma's proof. Conclusion Might there be a chance of proving in a simple way that ê phl is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. Let LHL be the upper conve hull of the full graph of ê phl (precise definition to follow). The piecewise linear function LHL is increasing because ê phl Ø as Ø. Moreover, using PNT, we can give a proof that LHL is indeed asymptotic to ê phl. But the point of our work in this paper is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the conve hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem. We conclude with the conve hull definition and proof. Let B be the graph of ê phl: the set of all points H, ê phll where. Let C be the conve hull of B: the intersection of all conve sets containing B. The line segment from H, L to any H, ê phll lies in C. As Ø, the slope of this line segment tends to 0 (because phl Ø ). Hence for any positive a and e, the vertical line = a contains points in C of the form Ha, + el. Thus the intersection of the line = a with C is a set of points Ha, yl where < y LHL. The function LHL is piecewise linear and ê phl LHL for all. This function is what we call the upper conve hull of ê phl. Theorem. The upper conve hull of ê phl is asymptotic to phl.

6 6 Proof. For given positive e and 0, define a conve set AH 0 L whose boundary consists of the positive -ais, the line segment from H0, 0L to H0, H + el Hln 0 LL, the line segment from that point to H 0, H + el H + ln 0 LL, and finally the curve H, H + el ln L for 0. The slopes match at 0, so this is indeed conve. The PNT implies that for any e > 0 there is an such that, beyond, H - el ln < ê phl < H + el ln. Now choose 0 > so that ph L < H + el ln 0. It follows that AH 0 L contains B: beyond 0 this is because 0 > ; below the straight part is high enough at = 0 and only increases; and between and 0 this is because the curved part is conve down and so the straight part is above where the curved part would be, and that dominates phl by the choice of. This means that the conve hull of the graph of ê phl is contained in AH 0 L, because AH 0 L is conve. That is, LHL H + el ln for 0. Indeed, H - el ln ê phl LHL H + el ln for all sufficiently large. Hence ê phl ~ LHL. References. D. Bressoud and S. Wagon, A Course in Computational Number Theory, Key College, San Francisco, R. Courant and H. Robbins, What is Mathematics?, Oford Univ. Press, London, J. Friedlander, A. Granville, A. Hildebrand, and H. Maier, Oscillation theorems for primes in arithmetic progressions and for sifting functions, J. Amer. Math. Soc. 4 (99) X. Gourdon and P. Sebah, The phl project, 5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oford Univ. Pr., London, I. Niven, H. S. Zuckerman, and H. L. Montgomery, An Introduction to the Theory of Numbers, nd ed., Wiley, New York, G. Tenenbaum and M. Mendès France, The Prime Numbers and their Distribution, Amer. Math. Soc., Providence, S. Wagon, It's only natural, Math Horizons 3: (005) D. Zagier, The first 50,000,000 prime numbers, The Mathematical Intelligencer 0 (977) 7 9. ü Archive, Code for Figures, and Other Notes