MATH 361: NUMBER THEORY THIRD LECTURE

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1 MATH 36: NUMBER THEORY THIRD LECTURE. Introduction The toic of this lecture is arithmetic functions and Dirichlet series. By way of introduction, consider Euclid s roof that there exist infinitely many rimes: If through n are rime then the number n q = + is not divisible by any i. According to this argument, the next rime after through n could be as large as q. The overestimate is astronomical. Secifically, comute that for n 3, since it follows that i= n + n (7/6) n, i n (7/6) n (7/6) 2 ( n 2 ) 2 (7/6) 4 ( n 3 ) 4 (7/6) 2n 3 ( 2 ) 2n 3 = 7 2n 3 (since 2 = 6) < e 2n 2. So, for examle, the tenth rime 0 satisfies 0 < Since in fact 0 = 29, we see how little Euclid s argument tells us. By contrast, Euler argued that P diverges, and in fact his argument shows more. The argument roceeds as follows. Define ζ(s) = n s, s >. Note that lim ζ(s) =, s + so that also lim log ζ(s) =, s + (The logarithm is natural, of course.) Later we will show that as s aroaches from the right, ζ(s) is asymtotic to /(s ) and so log ζ(s) is asymtotic to log(s ).

2 2 MATH 36: NUMBER THEORY THIRD LECTURE Now, summing over values of n with steadily more rime factors gives n s = (2 s ) e2 = ( 2 s ), n=2 e 2 n=2 e 2 3 e 3 n s =. e 2=0 (2 s ) e2 (3 s ) e3 = ( 2 s ) ( 3 s ), e 2=0 e 3=0 n=2 e 2 e n s = ( 2 s ) ( s ). And so, being very casual about convergence, it is essentially a restatement of unique factorization that the zeta function also has an infinite roduct exression, ζ(s) = n s = ( s ). n Z + P From the general series log( X) = X n /n, X <, we have (again being very casual about convergence) log ζ(s) = log ( s ) = P P = s + ns /n. P P n=2 log( s ) = P ns /n From above, we know that lim s + log ζ(s) =, while one term on the right side of the revious dislay is P s, which we want to understand as s tends to. As for the other term, ns /n < ns = 2s ( s ) = s ( s ), P n=2 P n=2 P P and P s ( s ) < n=2 n s (n s ) < n=2 n(n ) =. Thus the quantity that we want to understand is bounded on both sides by quantities that we do understand, log ζ(s) < P s < log ζ(s). And so and more secifically, lim s =, s + lim s + P s log ζ(s) =.

3 MATH 36: NUMBER THEORY THIRD LECTURE 3 Thus the sum of rime recirocals grows asymtotically as the logarithm of the harmonic series. Recall that the artial sums of the harmonic series themselves grow logarithmically, so that the sum of rime recirocals grows very slowly. Euler s result is much stronger than Euclid s, and it illustrates analytic number theory. 2. Dirichlet Series The zeta function is a articular instance of a Dirichlet series. Definition 2.. An arithmetic function is a comlex-valued function of ositive integers, f : Z + C. Its associated Dirichlet series is a formal series that deends on a arameter s, f(n) F (s) = n s. Using Dirichlet series to discuss arithmetic functions is not really necessary, but I think that the Dirichlet series clarify what is going on. Let F and G be the Dirichlet series associated to the arithmetic functions f and g. Comute that their roduct is F (s)g(s) = f(n) g(m) n s m s = f(n)g(m) (nm) s = de=n f(d)g(e) n s. n m n,m n That is, if we define the convolution (or Dirichlet roduct) of f and g to be f g : Z + C, (f g)(n) = f(d)g(e), de=n then the corresonding roduct of Dirichlet series is (f g)(n) F (s)g(s) = n s. That is, for arithmetic functions f, g, and h, and for Dirichlet series F, G, and H, h = f g H = F G. Since Dirichlet series with nonzero leading term are invertible ( a n n s has inverse bn n s where b = a and b n = a <d n a db n/d for n > ), it follows from the boxed equivalence that the arithmetic functions that do not vanish at form a grou under convolution. 3. Examles, Möbius Inversion With the boxed equivalence in mind, we create a small catalogue of arithmetic functions and their Dirichlet series. The identity arithmetic function is { if n =, i(n) = 0 otherwise. The corresonding Dirichlet series is simly I(s) =.

4 4 MATH 36: NUMBER THEORY THIRD LECTURE Since I(s) is the multilicative identity, i is the convolution identity. The unit arithmetic function is u(n) = for all n. (Ireland and Rosen call this function I.) The corresonding Dirichlet series is the zeta function, U(s) = ζ(s). The recirocal of the zeta function is the Dirichlet series ζ(s) = ( s ) = s + (q) s,q,q,r(qr) s +. The corresonding arithmetic function is the Möbius function, if n =, µ(n) = ( ) k if n = k (distinct rimes), 0 otherwise. Because ζ(s) and ζ(s) are Dirichlet series inverses, u and µ are convolution inverses, µ u = i. That is, since u always returns the value, { if n =, µ(d) = 0 if n >. d n More generally, for any arithmetic functions f and g, because u and µ are inverses, g = f u f = g µ. This equivalence is the Möbius Inversion Formula, g(n) = d n f(d) f(n) = d n g(d)µ(n/d). Often the idea is that f is interesting while g = f u is easy to comute, in which case Möbius inversion roduces f. For examle, consider two arithmetic functions, f(n) = sum of the rimitive nth roots of, g(n) = sum of all nth roots of. Here f is interesting while the finite geometric sum formula shows that g is simly the convolution identity function i. Because f u = g = i, in fact f = µ is the Möbius function. That is, sum of the rimitive nth roots of = µ(n). The reader can enjoy verifying this for n =, 2, 3,... through the first value of n where it is not easy.

5 MATH 36: NUMBER THEORY THIRD LECTURE 5 4. The Euler Totient Function The Euler totient function is an arithmetic function, φ : Z + Z +, φ(n) = #{x {0,..., n } : gcd(x, n) = } = #(Z/nZ). Thus φ() = and φ() = for rime. Now we set u Möbius inversion by counting that n = φ(d) for all n Z +. d n Indeed (noting that (k(n/d), n) = (n/d)(k, d) for the second equality to follow), {0,..., n } = d n{x {0,..., n } : (x, n) = n/d} = d n{k(n/d) : 0 k < d, (k, d) = }, and the desired counting formula follows immediately by definition of the totient function. (For examle, if n = 20 then the disjoint union is {0} {0} {5, 5} {4, 8, 2, 6} {2, 6, 4, 8} {, 3, 7, 9,, 3, 7, 9}, with {2, 6, 4, 8} being the multiles of 20/0 by factors corime to 0, and so on.) Consequently, Möbius inversion gives φ(n) = d n µ(d) n d. That is, φ(n) = n d n µ(d) d = n n +,q n q. The alternating sum-of-sums factors to give the formula for the totient function, φ(n) = n ( ). n (Alternatively, one can observe the formula φ(n) = d n µ(d) n d by inclusion exclusion, leading to the boxed formula for φ(n) and also giving n = d n φ(d) for all n Z + by Möbius inversion.) Some consequences of the totient function formula are φ( e ) = e e for e (this is even if > 2 or e ), φ(mn) = φ(m)φ(n) if (m, n) =, a b = φ(a) φ(b), n 3 = φ(n) is even, n = e e k k = 2 k φ(n) if all i > 2 or 4 n.

6 6 MATH 36: NUMBER THEORY THIRD LECTURE 5. Another Arithmetic Function The sum of divisor kth owers function is σ k : Z + Z +, σ k (n) = d n Esecially, σ 0 counts the divisors of n and σ sums them. Since the Dirichlet series Σ k (s) of σ k is σ k = (kth ower function) u, d k. Σ k (s) = ζ(s k)ζ(s) = ( k s ) ( s ) = ik is js = e c=0 ck es i=0 j=0 = σ k ( e ) es. Also, Möbius inversion gives the formula n k = d n µ(n/d)σ k (d). 6. Multilicative and Totally Multilicative Functions Definition 6.. Let f : Z + C be an arithmetic function. Then f is multilicative if f(nm) = f(n)f(m) for all n and m such that (n, m) =, and f is totally multilicative if Thus: f(nm) = f(n)f(m) for all n and m. For nonzero multilicative functions, f() = and f( e ) = f( e ). And: For totally multilicative functions, furthermore f( e ) = f() e. The corresonding Dirichlet series conditions are f is multilicative F (s) = and f is totally multilicative F (s) = f( e ) es ( f() s ). The first equivalence follows from the formal identity that for any function g of rime owers, g( e ) = g( e ), e =0 e n

7 MATH 36: NUMBER THEORY THIRD LECTURE 7 secialized to g( e ) = f( e )/ es. (The notation e n means that e is the highest ower of that divides n.) The second equivalence follows from the first and from the geometric series formula. Some further facts that are straightforward to check (either directly or by using Dirichlet series) are If f and g are multilicative then so is f g. If f is multilicative and f() 0 then so is the convolution inverse of f. If f is totally multilicative and f() 0 then its convolution inverse is f = µf. To establish the first bullet using Dirichlet series, comute that since f and g are multilicative, their Dirichlet series are F (s) = f( e ) es and G(s) = g( e ) es, and then it follows quickly that F (s)g(s) = (f g)( e ) es, so that f g is again multilicative. As an examle of the first bullet, the observation from earlier that σ k = (kth ower function) u now says that σ k is multilicative, and indeed the calculation of the Dirichlet series Σ k (s) = ζ(s k)ζ(s) has already shown that σ k is determined comletely by the values σ k ( e ). By the finite geometric sum formula, these are σ k ( e ) = k(e+) k. For another examle, because Euler s totient function is a convolution, φ = id µ where id is the identity function id(n) = n and µ is the Mobiüs function, its Dirichlet series is the corresonding roduct Φ(s) = ζ(s )ζ(s) = ( s ) ( s ). (Incidentally we note here that the finite Euler factors s of ζ(s) are exactly as they should be for the Möbius function.) Because id and µ are multilicative, so is φ, and so the corresonding Dirichlet series is Φ(s) = φ( e ) es. Match Euler factors to show that for every rime, φ( e ) es = ( s ) ( s ). Of course this can be verified directly, noting that φ( e ) is for e = 0 and is e ( ) for e.

8 8 MATH 36: NUMBER THEORY THIRD LECTURE To establish the second bullet using Dirichlet series, we need to show that the Dirichlet series H(s) = g( e ) es inverts the Dirichlet series F (s) = e 0 f(e )/ es. Comute for any rime that the roduct ( f() + f() ) ( s + f(2 ) 2s + g() + g() ) s + g(2 ) 2s + has constant term f()g() = (f g)() =, / s -term f()g() + f()g() = (f g)() = 0, / 2s -term f()g( 2 ) + f()g() + f( 2 )g() = (f g)( 2 ) = 0, and so forth. So indeed the roduct is F (s)h(s) is, showing that H(s) is the Dirichlet series G(s) of the convolution inverse g of f, and the form of G(s) shows that g is multilicative. To establish the third bullet using Dirichlet series, we need to show that if f generates the Dirichlet series F (s) then F (s) is the Dirichlet series generated by µf. Comute that indeed since F (s) = ( f() s ), F (s) = ( f() s ) = f() s +,q f(q)(q) s,q,r f(qr)(qr) s + = µ(n)f(n) n s = (µf)(n) n s. Or, to establish the third bullet directly, recall from section 3 that d n µ(d) is for n = and is 0 for n >, and comute that (µf f)(n) = d n µ(d)f(d)f(n/d) = f(n) d n µ(d) = i(n). 7. A Comment on Ireland and Rosen 2.4 Define a rime-counting function, π : R R, The Prime Number Theorem says that lim x π(x) = #{ P : x}. π(x) x/ log x =. The Prime Number Theorem was first roved in 899 by Hadamard and (indeendently) Poussin. An elementary roof was given in the 940 s by Selberg, erhas with a significant contribution from Erdös. Section 2.4 of Ireland and Rosen is showing that easy analytic estimates show that for some constants c and c 2, c x/ log x < π(x) < c 2 x/ log x.

9 MATH 36: NUMBER THEORY THIRD LECTURE 9 For students with background in comlex analysis, see the 997 American Mathematical Monthly article Newman s short roof of the rime number theorem by Zagier.