Math 229: Introduction to Analytic Number Theory Elementary approaches II: the Euler product

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1 Math 9: Introduction to Analytic Number Theory Elementary aroaches II: the Euler roduct Euler [Euler 737] achieved the first major advance beyond Euclid s roof by combining his method of generating functions with another highlight of ancient Greek number theory, unique factorization into rimes. Theorem [Euler roduct for the zeta function]. The identity n= n s = rime. () s holds for all s such that the left-hand side converges absolutely. Proof : Here and henceforth we adot the convention: The notation or means a roduct or sum over rime. Every ositive integer n may be written uniquely as c, with each c a nonnegative integer that vanishes for all but finitely many. Thus the formal exansion of the infinite roduct contains each term rime ( cs) () c =0 ( ) s n s = c = cs exactly once. If the sum of the n s converges absolutely, we may rearrange the sum arbitrarily and conclude that it equals the roduct (). On the other hand, each factor in this roduct is a geometric series whose sum equals /( s ). This establishes the identity (). The sum on the left-hand side of () is nowadays called the zeta function ζ(s) = + s + 3 s + = n s ; the formula () is called the Euler roduct for ζ(s). Euler did not actually imose the convergence condition: the rigorous treatment of limits and convergence was not yet available, and Euler either handled such issues intuitively or ignored them. If s is a real number the only case that concerned Euler then it is well known that n= n s converges if and only if s >, by comarison with x s dx (that is, by the Integral Test of elementary calculus). We shall use n=

2 comlex s as well, but the criterion for absolute convergence is still easy: if s has real art σ then n s = ex( s log n) = ex(re( s log n)) = ex( σ log n) = n σ, so the Euler roduct holds in the half-lane σ >. Euler s next ste was to set s = in (). This equates /( ) with the sum n= /n of the harmonic series. Since the sum diverges to +, whereas each factor /( ) is finite, there are infinitely many factors. Therefore, there are infinitely many rimes. This roof does not meet modern standards of rigor, but it is easy enough to fix: instead of setting s equal, let s aroach from above. The next result is an easy estimate on the behavior of ζ(s) for s near. Lemma. The inequalities hold for all s >. s < ζ(s) < s + (3) (More accurate estimates are available using the Euler-Maclaurin formula, but we do not yet need them.) Proof : For all n > 0 we have whence n+ n x s dx = ( n s (n + ) s), s (n + ) s < n s (n + ) s s < n s. Now sum over n =,, 3,.... The sum of (n s (n + ) s )/(s ) telescoes to /(s ). This sum is bounded above by n= n s = ζ(s), and below by n= (n + ) s = ζ(s). This roves the inequalities (3). The lower bound in (3) shows that ζ(s) as s from above (an equivalent notation is as s + ). Since each factor ( s ) in the Euler roduct remains bounded, we have vindicated Euler s argument for the infinitude of rimes. The divergence of /( ) and the behavior of /( s ) as s + give us much more secific information on the distribution of rimes than we could hoe to extract from Euclid s argument. For instance, we cannot have constants C, θ with θ < such that π(x) < Cx θ for all x, because then the Euler roduct would converge for s > θ. To go further along these lines it is convenient to use the logarithm of the Euler roduct: log ζ(s) = log( s ). (4)

3 Euler again took s = and concluded that / diverges. Again we justify his conclusion by letting s aroach from above: Theorem. For any s 0 > there exists M such that s log < M (5) s for all s (, s 0 ]. In articular, / diverges. Proof : By our Lemma, log ζ(s) is between log /(s ) and log s/(s ). Since 0 < log s < s, we conclude that log ζ(s) differs from log /(s ) by less than s < s 0. In the right-hand side of (4), we aroximate each summand log( s ) by s. The error is at most s, so s ( log( s )) < s < ζ(). Hence (5) holds with M = s 0 + ζ(). Letting s we obtain the divergence of /. Interlude on the Big Oh notation O( ). The oint of (5) is that s equals log s within a bounded error, not the secific uer bound M on this error which is why we were content with a bound s 0 + ζ() weaker than what the method can give. Usually in such aroximate formulas we shall be interested only in the existence of constants such as M, not in their exact values. To avoid distractions such as s 0 +ζ(), we henceforth use big Oh notation. In this notation, (5) aears as s = log + O(). (6) s In general, f = O(g) means that f, g are functions on some sace S with g nonnegative, and there exists a constant M such that f(z) Mg(z) for all z S. In articular, O() is a bounded function, so (6) is indeed the same as (5). An equivalent notation, more convenient in some circumstances, is f g (or g f). For instance, a linear ma T between Banach saces is continuous iff T v = O( v ) iff v T v. Each instance of O( ) or or is resumed to carry its own imlicit constant M. If the constant deends on some arameter(s), we may use the arameter(s) as a subscrit to the O or. For instance, we may write O s0 () instead of O() in (6); for any ɛ > 0, we have log x ɛ x ɛ on x [, ). For basic roerties of O( ) and see the Exercises at the end of this section. Back to π(x). The estimate (6) for s does not exlicitly involve π(x). We thus rearrange this sum as follows. Write s as an integral s y s dy, and sum over. Then y occurs in the interval of integration [, ) iff < y, that is, with multilicity π(y). Therefore s = s π(y)y s dy, (7) 3

4 and (6) becomes an estimate for an integral involving π( ). This transformation from the sum in (6) to the integral (7) is an examle of a method we shall use often, known either as artial summation or integration by arts. To exlain the latter name, consider that the sum may be regarded as the Stieltjes integral y s dπ(y), which integrated by arts yields (7); that is how we shall write this transformation from now on. Our eventual aim is the Prime Number Theorem (PNT), which asserts that π(x) is asymtotic to x/ log x as x. Our estimate (6) on the integral (7) does not suffice to rove the PNT, but does rovide suort for it: the estimate holds if we relace π(x) with x/ log x. That is, To rove this, let I(s) = y s log y dy = log + O() ( < s ). s y s log y dy, and differentiate under the integral sign to obtain I (s) = y s dy = s /( s) = /( s) + O(). Thus for < s we have I(s) = I() s I dσ (σ) dσ = + s σ + O() = log s + O() as claimed. While this does not rove the Prime Number Theorem, it does show that, for instance, if c < < C then there are arbitrarily large x, x such that π(x) > cx/ log x and π(x ) < Cx / log x. Remarks Euler s result / = underlies for our exectation that n+ divides + n i= n infinitely often. The residue of n i= n mod n+ should behave like a random element of (Z/ n+ Z), and thus should equal with robability /( ). The exected value of the number of n < N such that n+ divides + n i= n is thus N n= /( ) > N n= / as N. We exect the same behavior for many other roblems of the form how many rimes are factors of f()?, notably f() = (( )! + )/ (the Wilson quotient), f() = (a a)/ (the Fermat quotient with fixed base a > ), and f() = i= /i (the Wolstenholme quotient). We shall soon see that / diverges very slowly: <x / = log log x + O(). Therefore, while we exect infinitely many solutions of f() in each case, we exect that these solutions will be very scarce. Euler s work on the zeta function includes also its evaluation at ositive integers: ζ() = π /6, ζ(4) = π 4 /90, etc. The silliest roof I know of the infinitude We shift the lower limit of integration to y = to avoid the surious singularity of / log y at y =, and suress the factor s because only the behavior as s matters, and multilying by s does not affect it to within O(). We also made the traditional and convenient choice s 0 = ; the value of s 0 does not matter, as long as s 0 >, because we are concerned with the behavior near s =, and by secifying s 0 we can disense with a distracting subscrit in O s0. 4

5 of rimes is to fix one such integer s, and observe that if there were finitely many rimes then ζ(s) = ( s ), and thus also π s, would be rational, contradicting Lindemann s theorem (88) that π is transcendental. It is only a bit less silly to take s = and use the irrationality of π, which though unknown to Euler was roved a few generations later by Legendre (794?). This can actually be used to obtain lower bounds on π(x), but even with modern irrationality measures we can obtain no lower bounds on π(x) better than the log log x bound already available from Euclid s roof. Less frivolously, we note that the integral π(y)y s dy/y aearing in (7) is the Mellin transform of π(y), evaluated at s. The Mellin transform may not be as familiar as the integral transforms of Fourier and Lalace, but the change of variable y = e u yields s dy π(y)y y = 0 π(e u )e su du, which is the Lalace transform of π(e u ), evaluated at s. In general, if f(u) is a nonnegative function whose Lalace transform Lf(s) := f(u)e su du 0 converges for s > s 0, then the behavior of Lf(s) as s s 0 + detects the behavior of f(u) as u. In our case, s 0 =, so we exect that our estimate on π(y)y s dy/y for s near will give us information on the behavior of π(x) for large x. Moreover, inverting the Lalace transform requires a contour integral over comlex s; this suggests that we shall need to consider log ζ(s), and thus the solutions of ζ(s) = 0, in the comlex lane. We shall return to these ideas and the Mellin transform before long. Exercises Concerning the Big Oh (equivalently ) notation:. If f g and g h then f h. If f = O(g ) and f = O(g ) then f f = O(g g ) and f + f = O(g + g ) = O(max(g, g )). Given a ositive function g, the functions f such that f = O(g) constitute a vector sace.. If f g on the interval (a, b) or (a, b] then x a f(y) dy x g(y) dy for all a x in the same interval such that the integrals exist. (We already used this to obtain I(s) = log(/(s )) + O() from I (s) = /( s) + O().) In general differentiation does not commute with (why?). Nevertheless, rove that ζ (s)[= n= n s log n] is /(s ) + O() on s (, ). 3. So far all the imlicit constants in the O( ) or we have seen are effective: we didn t bother to secify them, but we could if we really had to. Moreover the transformations in exercises, reserve effectivity: if the inut constants are effective then so are the outut ones. However, it can haen that we know that f = O(g) without being able to name a constant M such that f Mg. Here is a rototyical examle. Suose x, x, x 3,... is a sequence of ositive reals which we susect are all, but all we can show is that if i j then x i x j < x i + x j. Prove that the x i are bounded, i.e., x i = O(), but that as long as we do not find some x i greater than, we cannot use this to exhibit a 5

6 secific M such that x i < M for all i and indeed if our susicion that every x i is correct then we shall never be able to find M. We shall encounter this sort of unleasant ineffectivity (where it takes at least two outliers to get a contradiction) in Siegel s lower bound on L(, χ); it arises elsewhere too, notably in Faltings roof of the Mordell conjecture, where the number of rational oints on a given curve of genus > can be effectively bounded but their size cannot. Alications of the Euler roduct for ζ(s): 4. Comlete the roof that for each c < there are arbitrarily large x such that π(x) > cx/ log x and for each C > there are arbitrarily large x such that π(x ) < Cx / log x. 5. It is known that there exists a constant M such that π a/b /b M for all ositive integers a, b. Use this together with the Euler roduct for ζ() to rove that π(x) log log x. 6. Prove that there are N/ζ() + O(N / ) squarefree integers in [, N]. Obtain similar estimates for the number of natural numbers < N not divisible by n s for any n > (s = 3, 4, 5,...). NB this and the next few exercises are not quite as easy at they may seem: remember the final exercise for the revious lecture! A hint as to the solution: use the Euler roduct for ζ(s) to obtain a series exansion for /ζ(s). It follows that an integer chosen uniformly at random from [, N] is squarefree with robability aroaching /ζ() = 6/π as N. Informally, a random integer is squarefree with robability 6/π. We shall see that the error estimate O(N / ) can be imroved, and that the asymtotic growth of the error hinges on the Riemann Hyothesis. 7. Prove that there are N /ζ() + O(N log N) ordered airs of relatively rime integers in [, N]. What of relatively rime airs (x, x ) with x < N and x < N? Generalize. Again we may informally deduce that two random integers are corime with robability 6/π. Alternatively, we may regard a corime air (x, x ) with x i N as a ositive rational number x /x of height at most N. Droing the ositivity requirement, we find that there are asymtotically N /ζ() rational numbers of height at most N. This has been generalized to number fields other than Q by Schanuel [979]; a function-field analogue, concerning rational functions of bounded degree on a given algebraic curve over a finite field, was announced by Serre [989,.9] and roved by DiPio [990] and Wan [99] (indeendently but in the same way). The functionfield result was the starting oint of our estimate on the size of the nonlinear codes obtained from rational functions on modular curves [Elkies 00]. Schanuel also obtained asymtotics for rational oints of height at most N in rojective sace of dimension s over a number field K; when K = Q this recovers the asymtotic enumeration of corime s-tules of integers. 8. Prove that as N the number of ordered quadrules (a, b, c, d) of integers in [, N] such that gcd(a, b) = gcd(c, d) is asymtotic to N 4 /5. 6

7 Can this be roved without invoking the values of ζ() or ζ(4)? This can be regarded as a form of a question attributed to Wagstaff in [Guy 98, B48]: Wagstaff asked for an elementary roof (e.g., without using roerties of the Riemann zeta-function) that Q ( + )/( ) = 5/. References [DiPio 990] DiPio, S.A.: Saces of Rational Functions on Curves Over Finite Fields. Ph.D. Thesis, Harvard, 990. [Elkies 00] Elkies, N.D.: Excellent nonlinear codes from modular curves, ages in STOC 0: Proceedings of the 33rd Annual ACM Symosium on Theory of Comuting, Hersonissos, Crete, Greece. Isomorhic with math.nt/0045 at arxiv.org. [Euler 737] Euler, L.: Variae observationes circa series infinitas, Commentarii academiae scientiarum Petroolitanae 9 (744), (resented to the St. Petersburg Academy in 737) = Oera Omnia Ser. 4, [Guy 98] Guy, R.K.: Unsolved Problems in Number Theory. Sringer, 98. [Legendre 794] Legendre, A.-M.: Éléments de Géométrie. Paris: Didot, 794. [Schanuel 979] Schanuel, S.H.: Heights in number fields. France 07, (979). Bull. Soc. Math. [Serre 989] Serre, J.-P.: Lectures on the Mordell-Weil Theorem (trans. M. Brown). F. Vieweg & Sohn, Braunschweig 989. [Wan 99] Wan, D.: Heights and Zeta Functions in Function Fields. In The Arithmetic of Function Fields, ages Berlin: W. de Gruyter, 99. 7