Dirichlet s Theorem on Arithmetic Progressions

Transcription

1 Dirichlet s Theorem on Arithmetic Progressions Thai Pham Massachusetts Institute of Technology May 2, 202 Abstract In this aer, we derive a roof of Dirichlet s theorem on rimes in arithmetic rogressions. We try to motivate each ste in the roof in a natural way, so that readers can have a sense of how mathematics works. Introduction Number theory is the queen of mathematics, and Dirichlet s theorem on arithmetic rogressions has been considered a gem of that queen. The significance of this theorem lies not only in its simle statement, but also in the beautiful roof given by Dirichlet, which in fact set a background for the study of grou theory and reresentation theory later on. Our main urose in this aer is to discuss this theorem and its roof. Theorem. Dirichlet) Given a, m N with gcda, m) =, there are infinitely many rime numbers in the arithmetic rogression a + km} k N. One can easily show this theorem for m = 4 and a, 3} See Shi & Xie [9]). However, the roof for the general case is much more comlicated and requires many dee algebraic and analytical ideas. Even though Selberg was able to give an elementary roof in 949 See [7]), his roof is rather tedious and unmotivated. In this aer, we will follow Dirichlet s method with some ossible modifications to simlify the original roof. We organize the rest of the aer as follows. Section 2 discusses the motivation for the roof of the theorem. Section 3 reviews background needed for the roof: grou characters, Dirichlet series, and Euler roducts. Section 4 is dedicated to the roof of Dirichlet s theorem. 2 Motivation In this section, we talk about the motivation for Dirichlet s roof of the theorem. Rigorous treatments are resented in the later sections. I am grateful to Sigurdur Helgason and Susan Ruff for their invaluable hel in commenting on the earlier version of the aer in terms of both content and exosition.

2 We first observe that Dirichlet s theorem is in fact an extension of Euclid s theorem, which states that there are infinitely many rime numbers. Secifically, for a =, m = 2) the two theorems are equivalent since all the rimes greater than two are odd. Our urose is to use a similar technique to that in the roof of Euclid s theorem to rove Dirichlet s theorem. Actually, what interests us the most is the stronger version of Euclid s theorem, also known as Euler s theorem on the sum of the recirocals of the rime numbers See Dunham s aer [3]): =. ) Throughout the aer, we write to denote a rime number, unless otherwise secified). Motivated by this theorem, we desire to rove a stronger result than Dirichlet s original theorem: amod m) =. 2) We recall the roof of Euler s theorem to gras the ideas behind it. In his roof, Euler took advantage of the roduct formula: n = s n= for all s C with Res) >. 3) s The left hand side of 3) is known as the Riemann zeta function ζs). Euler roceeded in his roof by writing log ζs) = + gs), 4) s where gs) is bounded as s. The fact that ζs) as s would then imly the result. To rove Dirichlet s theorem, we want the sum to aear in a similar sense. To s amod m) do this, we need a good trick to filter out the rimes congruent to a modulo m from all other rimes. The zeta function needs to be modified, and the Dirichlet series aears naturally; it has the form an)n s, where s, an) C for all n N. A random choice of an) would yield n= nothing. However, when an) is chosen to be a comletely multilicative function i.e. a) = and amn) = am)an) for all m, n N ), we obtain an equation similar to 3), known as the Euler roduct: n= an) n s = for all s C with Res) >. 5) a) s Taking the natural logarithm of both sides of 5), we can write the right hand side in the form a) + hs, a). The choice of an) is not good enough nevertheless, as we have not reached s the sum we desire. We need to fine-tune an) to hel in the filtering rocess. The breakthrough 2

3 oint here is Dirichlet s use of grou characters. With this tool in hand, Dirichlet s theorem is no longer far away. Now, we come to the rigorous treatment for the theorem. 3 Background In this section, we resent the imortant notions which were mentioned earlier and which are necessary for the roof of Dirichlet s theorem. The notions include grou characters, Dirichlet series, and Euler roducts. 3. Grou Characters To understand this art well, readers should have a sufficient knowledge of abstract algebra, secifically grou theory. As a reference, the author recommends Artin s book See [2]) and Serre s book See [8]). About articular grou characters, readers should consult Aostol s book See []). Definition 2. Let G be an abelian grou. A function : G C\0} maing G to the set of non-zero comlex numbers is called a character of G if it is a grou homomorhism, that is g g 2 ) = g )g 2 ) g, g 2 G. We restrict our attention to the finite grou G. In this case, the set of characters i of G forms an abelian grou under multilication j k )g) = j g) k g) g G with rincial character 0 such that 0 g) = g G. This grou is called the dual of G and is usually denoted by Ĝ. We now turn to one imortant roerty of the dual grou. Proosition 3. Any abelian grou G is isomorhic to its dual Ĝ. Proof. For a cyclic grou G, the result is straightforward. Since G can be written as a roduct of cyclic grous, it holds in general. We now consider an imortant roerty of grou characters, which will hel us filter out the rimes we want in the above discussion: the orthogonality roerty of characters. Proosition 4. If Ĝ, then g) = G if = 0, 6) Also, if g G then g) = Ĝ Ĝ if g = the identity element of G), 7) Proof. We can easily see 7) is a corollary of 6) thanks to Proosition 3. So, it is sufficient to rove only 6). 3

4 To this end, we observe that if = 0 then g) = g G. Then, g) = G. If 0, there exists some h G such that h). We have h) g) = gh) = g). 8) Thus g) = 0, and this ends the roof. We observe that, if, ϕ Ĝ then ϕ Ĝ. Moreover, ϕ )g) = g)ϕg). Similarly, if g, h G, then gh G. We have gh ) = g)h). Then by Proosition 4, Ĝ Ĝ we get the following corollary. Corollary 5. If, ϕ Ĝ, then g)ϕg) = G if = ϕ, 9) Also, if g, h G then g)h) = Ĝ Ĝ if g = h, 0) The above result is quite imressive, but it is not the end of the story. We need something more secific, more directly alicable to our theorem. Again, Dirichlet made a big jum in introducing the Dirichlet character. This notion will reaear later on when we talk about Dirichlet series and Euler roducts. Definition 6. A Dirichlet character is any function : Z C which satisfies the following roerties: a) There exists m Z + such that n) = n + m) for all n Z. b) If gcdn, m) >, then n) = 0; if gcdn, m) =, then n) 0. c) nk) = n)k) for all n, k Z. From this definition, we can deduce other roerties. From b) and c), ) =. Combining this with c), we conclude that is a comletely multilicative function. Besides, a) imlies that is eriodic with eriod m. This is why we also call the Dirichlet character modulo m. We make two remarks here. First, according to Definition 2 a character cannot have zero value; however, a Dirichlet character can take the value zero. Second, if gcda, m) = then by Euler s theorem a φm) mod m), where φ is the totient function. Hence, a φm) ) = ) = which imlies that a) φm) =. Thus, a) is a φm)-th root of unity for all a such that gcda, m) =. 4

5 Thanks to these two oints, Dirichlet characters can be viewed in terms of the character grou of the unit grou of the ring Z/mZ. Secifically, let G = Z/mZ) so G = φm)) with the rincial character 0 such that 0 a) = if gcda, m) = and Then, the Dirichlet character modulo m is informally considered the extension of G to Z. We obtain the following imortant formulas. Corollary 7. Let and ϕ be Dirichlet characters modulo m. Then m φm) if = ϕ, g)ϕg) = g=0 ) Similarly, let g and h be integers. Then g)h) = φm) if g h mod m), With this corollary in hand, we can begin to sense how we may obtain the sum amod m) However, the tools we have covered so far are not enough. We now turn to the next critical oint in the roof of the theorem: the Dirichlet series. 3.2 Dirichlet series Our main urose in this section is to review one imortant roerty of the Dirichlet series, and we incororate it into a theorem. a n Theorem 8. Cohen) Let D = n be a Dirichlet series. If D converges for some s = s 0, then s n= it converges uniformly on each comact set of the half-lane Res) > Res 0 ). Moreover, the sum is analytic in this region. The roof of this theorem is quite simle; but as it is not central to the urose of this aer, we refer interested readers to Titchmarsh s book, Chater IX See [0]). We discussed in section 2 that an) in the Dirichlet series being comletely multilicative is not good enough. Moreover, in equations ) and 2) we used the Dirichlet characters and ϕ and mentioned that they will hel in the filtering rocess. Dirichlet was very clever in using such in the Dirichlet series and created the Dirichlet L-series: Ls, ) = n= 2) s. n) n s. 3) The use of Ls, ) is in fact not so imortant when we study the Dirichlet series here or the Euler roducts in the next art; but as it is essential for comleting the roof of Dirichlet s theorem, we bring it along in these discussions. One not-so-direct corollary of theorem 8 is concerned with Ls, ). 5

6 Corollary 9. Let be a Dirichlet character modulo m different from the rincial character. Then Ls, ) converges and is analytic in Res) > 0. Proof. For any a Z, roosition 4 imlies m n + a) = n= m n=0 n) = 0. 4) Let s R +. Let U n = n i= i). By 4), U n} n is bounded. So there is some constant C R + such that U n < C n N. For any M Z +, alying Abel s summation formula See Rudin [6]. 79) we get n=m n) n s = n=m C Since lim = 0, then lim M M s ) U n n s n + ) s C n=m M n=m n s n + ) s = C M. 5) s n) n s = 0 which imlies Ls, ) converges for s R +. By theorem 8, we conclude that Ls, ) converges and is analytic in Res) > 0. We make two remarks here. First, corollary 9 is aarently unnecessary now but it will definitely be useful when we rove Dirichlet s theorem. Second, we defer some roerties of Ls, ) until they become relevant with the context of the roof. Next, we discuss another imortant notion: the Euler roducts. 3.3 Euler roducts This section serves to introduce two imortant results related to Euler roducts. The first one is similar to equation 5). Theorem 0. Ls, ) converges absolutely for Res) >. Moreover in this region, Ls, ) =. 6) ) s Proof. First, notice that is bounded; therefore, Ls, ) converges absolutely for Res) >. Now, for each rime we have: ) s) = ) n ns = n ) ns. This imlies, for n=0 any fixed rime q, that ) = n) s n, where T s q is the set of all natural numbers q n T q whose rime factors are less than or equal to q. Then, for any natural number N we have n=0 N n= n) n s = r ) s n T r,n>n n) n s, 7) 6

7 where r is the largest rime less than or equal to N, and T r is defined in the same way as T q. Letting N aroach infinity, we obtain the desirable result. The second result is concerned only with Ls, 0 ). Proosition. Ls, 0 ) extends to a meromorhic function in Res) > 0 with the only ole at s =. Proof. By theorem 0, we get Ls, 0 ) = m. 8) s We roceed by resenting an imortant result about the Riemann zeta function ζs). Lemma 2. Let ζ be the Riemann zeta function. Then, a) ζs) = for Res) >, and s b) ζs) extends to a holomorhic function in Res) > 0. s We will not rove this lemma, as it is very famous in the literature See e.g. Rudin [6]. 4). Now, we come back to our roosition. By lemma 2a), we can write Ls, 0 ) = ζs) m s ) for Res) >. 9) Note that m s ) is finite. Combining this fact with lemma 2b), we conclude that Ls, 0 ) can extend to a meromorhic function in Res) > 0 and its only ole is at s =. Now, we have enough tools to rove Dirichlet s theorem. 4 Dirichlet s Theorem We want to rove theorem in a way suggested in section 2 of the aer. We start by taking the natural logarithm of the Euler roduct equation 6)): log Ls, ) = [ log ) s )] for Res) >. 20) We will show that the right hand side of 20) can be written in the form ) + hs, ), s where hs, ) is bounded as s. To this end, we fix and use the Taylor s exansion for log x) at x = ) s : log ) s) = ) s + 7 n=2 ) n n ns for Res) >. 2)

8 Moreover, ) = 0 or so ) s s 2. Then for Res) >, we get ) n 2 n ns ) n 2 2 ) s n s ) 2 s 2 2 = ) n 2 s. 22) 2 n=2 n=2 n=2 Let hs, ) = n=2 ) n. Then nns log Ls, ) = hs, ) ) n n ns n=2 ) s + hs, ), where 23) < < for Res) >. 24) 2 n2 Thus, hs, ) is bounded as s as desired. The next ste is the filtering rocess. We recall the orthogonality roerty that given grou G = Z/mZ), φm) if g h mod m), g)h) = 25) We write the sum over to indicate the sum over all Ĝ). By multilying both sides of 23) by a) and summing over all, we obtain a) log Ls, ) = a) n= ) s + a)hs, ). 26) Hence, a) log Ls, ) a)hs, ) = )a) = φm) s amod m) s. 27) The sum we want finally aears. Now notice that Ĝ = φm). Because hs, ) is bounded as s, a)hs, ) is bounded as s. Our goal is to show that the right hand side of 27) diverges to infinity as s. To do this, we need to show a) log Ls, ) as s. By roosition, we know that Ls, 0 ) + as s and thus, so does log Ls, 0 ). Since a) 0 and is bounded, to obtain the desired result it is enough to show log L, ) is bounded below for all 0. This is equivalent to showing L, ) 0 0. Once this statement is roved, we are done with the roof of Dirichlet s theorem. Thus, it is sufficient to rove the following roosition. Proosition 3. For all 0, L, ) 0. 8

9 It is interesting to note that, in the roof of the Prime Number Theorem See Zagier s aer []) the analogous statement for ζs) is also a major ste. Back to our roosition, there are at least two ways to show it. One method is quick but not very illuminating; interested readers can see it in Garrett s aer See [4]). Here, we rovide a much more interesting roof, though it is more comlicated. Another note is that this roof was modified from Dirichlet s original roof. Proof. The key to the roof is the function ζ m s), which is defined as ζ m s) = Ls, ). 28) We observe that for all 0, Ls, ) is analytic in Res) > 0 by corollary 9. Moreover, Ls, 0 ) extends to a holomorhic function in Res) > 0 with the only ole at s = by roosition. Suose that there is some 0 such that L, ) = 0 then ζ m s) would be analytic in Res) > 0 since the zero value at s = of Ls, ) will cancel the ole of Ls, 0 )). We will show that ζ m s) cannot be analytic in Res) > 0 to obtain a contradiction, through which we rove roosition 3. To this end, we denote by ord) the order of the image of in G = Z/mZ) for any rime m. We roceed with the following lemma. Lemma 4. If Res) >, then ζ m s) = m Proof. To rove this lemma, we first note that if m then To see why 28) is true, we start from the identity x ord) = ) φm) ord). 29) ord)s ) ) = ) φm) ord). 30) s ord)s ω U ord) ωx), 3) where U n denotes the set of all n-th roots of unity. Notice that for each ω U ord), there are exactly φm)/ord) Dirichlet characters such that ) = ω. Hence )x) = x ord)) φm) ord). 32) We cannot find any official document saying who was the first to modify Dirichlet s roof; among the ossible contributors is Edmund Landau. 9

10 Let x = s we obtain 28). From here, we have for Res) > ζ m s) = Ls, ) = m ) s = m ) φm) ord), 33) ord)s as desired. So, lemma 4 is roved. The key here is to observe that is a Dirichlet series with non-negative real coefficients. Hence, by lemma 4, ζ m s) is also a Dirichlet series with non-negative real coefficients. To ord)s come u with a contradiction, we need to use the following result, known as Landau s theorem. a n Theorem 5. Landau) Let fs) = n be a Dirichlet series with real coefficients a s n 0. n= Suose that the series defining fs) converges for Res) > s 0 for some real s 0. Suose further that the function f extends to a holomorhic function in a neighborhood of s 0, to say s 0 ɛ, s 0 ) for some ɛ > 0. Then, the series defining fs) converges for Res) > s 0 ɛ. We will not rove this theorem here. Instead, we refer interested readers to Garrett s aer See [4]). Coming back to roosition 3, the roof is now at hand. Our goal is to show that ζ m s) cannot be analytic in Res) > 0. Assume the contrary is true. Recall that ζ m s) is a Dirichlet series whose coefficients are real and non-negative. Moreover, Ls, ) converges for Res) > for all so ζ m s) also converges for Res) >. By Landau s theorem with s 0 = ɛ =, ζ m s) converges for Res) > 0. However, for Res) >, we have ord)s ) φm) ord) which dominates the series + φm)s + 2φm)s + = Then for s >, all the coefficients of ζ m s) = m of =. Hence, φm)s nφm)s m n Z +, gcdn,m)= = + ord)s + 2ord)s + ) φm) ord), 34) ζ m φm) ). φm)s ord)s n Z +, gcdn,m)= ) φm) ord) are greater than those n, 35) which is divergent to infinity. So ζ m s) diverges at s = φm) > 0, which is a contradiction! So roosition 3 is roved, and we are done with the roof of Dirichlet s theorem. 0

11 References []. Aostol, Tom M. 976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Sringer-Verlag. [2]. Artin, Michael, Algebra, Prentice Hall, 99. [3]. Dunham, William 999). Euler: The Master of Us All. The Mathematical Association of America, Dolciani Mathematical Exositions, Vol. 22, 999. [4]. Garrett, Paul, Primes in arithmetic rogressions, 20. htt : // garrett/m/mf ms/notes c/dirichlet.df [5]. Kna, A., Ellitic Curves. Princeton UP, New Jersey, 992. [6]. Rudin, W. Princiles of Mathematical Analysis. 3rd ed. New York, NY: McGraw-Hill, 976. [7]. Selberg, Atle, An Elementary Proof of Dirichlet s Theorem About Primes in an Arithmetic Progression, The Annals of Mathematics, 2nd Series, Vol.50, No.2 Ar., 949), [8]. Serre, J.-P., A Course in Arithmetic, Sringer-Verlag, New York, 973. [9]. Shi, Meiyi & Xie, Tessa, Prime, Modular Arithmetic, and Squares. SWIM 200, Princeton. [0]. Titchmarsh, E. C., The Theory of Functions, Oxford University Press, 932. []. Zagier, D., Newman s short roof of the Prime Number Theorem. The American Mathematical Monthly, 04997),