Dirichlet s Theorem In Arithmetic Progressions

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1 Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205

2 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons, one of the nterestng topcs n number theory. The proof uses complex analyss and s a mlestone for analytc number theory. The theorem was conjectured by Legendre and proved by Drchlet 837. Introducton Drchlet s theorem n arthmetc progressons states that f a and m are postve ntegers wth no common factor except, there are nfntely many prme numbers congruent to a modulo m. That s the seres a, a + m, a + 2m,... contans nfnty many prme numbers. An mmedate observaton of ths theorem by settng a m s Eucld s theorem, that s, there are nfntely many prme numbers. There are several other nterestng problems about prme numbers, one of them beng Landau s fourth problem whch conjectures that there are nfntely many prme numbers of the form n 2 +, where n N. We begn the paper wth some mportant propertes of Drchlet seres, then we defne characters and Drchlet L-functons whch are the man tools used n the proof. Fnally we reformulate the theorem n terms of denstes to reach a deeper result. The general method used n ths paper s based on works of Drchlet, whch can be found n [2]. However, the explanaton of varous concepts and proof of theorems are done by us. Fnally, we would lke to thank Australan Mathematcal Scences Insttute for fundng and makng ths experence possble for us. 2 Drchlet Seres Defnton. Let s be a complex number, {a n } n a sequence of complex numbers and {λ n } n be an ncreasng sequence of real numbers approachng nfnty. Then the general Drchlet seres s defned as n a ne λns. The ordnary Drchlet seres n a nn s s the case where λ n ln n. Theorem 2 Morera s theorem. Let the functon f be contnuous n a smply connected open set U. If C fzdz 0 for all smple closed contours C lyng n U then f s holomorphc on U. Proof. Choose an arbtrary pont z 0 n U and defne functon F : U C by F z z z 0 fτdτ. Note that F s well defned snce ntegral of f over any closed contour s 0 by assumpton and hence the ntegral n F s ndependent of paths from z 0 to z. Now choose z and z n U and let h z z. We have z F z F z z z z 0 fτdτ z z 0 fτdτ z fτdτ h h 2 z

3 and Hence fz h fzh z h fz dτ h F z F z z z z fz h z z z z fzdτ. fτ fzdτ. Snce f s contnuous at z, for any ɛ > 0 there exsts a δ > 0 such that fτ fz < ɛ whenever h < δ. So we have F z F z z z fz z fτ fzdτ h h z z z fτ fz dτ ɛ h z z ɛ. Therefore F s dfferentable wth dervatve f. Snce z was an arbtrary pont n U, thus F s holomorphc and nfnte dfferentable. In partcular F f s holomorphc on U. Theorem 3. Let {f } be a sequence of holomorphc functons defned on an open set U convergng to f : U C unformly. Then f s holomorphc on U. Proof. Let C be a closed contour n U. As f s holomorphc for any, by Cauchy s ntegral theorem we have f zdz 0. By the unform convergence we have fzdz C C C lm f zdz lm f zdz 0. C As C was an arbtrary closed contour n U, hence by Morera s theorem f s holomorphc on U. Lemma 4. Let {a n } and {b n } be two sequences and defne S m,p p a n. Then q a n b n S m,q b q + q S m,n b n b n+. Proof. q q a n b n a m b m + a n b n + q a m b m + S m,n S m,n b n + q q a m b m + S m,n b n S m,n b n + + 3

4 a m b m + S m,m b m + S m,q b q + S m,q b q + q + q + q + S m,n b n q S m,n b n+ S m,n b n b n+ S m,m b m+ S m,n b n b n+ + S m,m b m b m+ S m,q b q + q S m,n b n b n+ Lemma 5. Let a, b, x and y be real numbers wth 0 < a < b and x > 0. If z x + y, then we have e az e bz z e ax e bx. x Proof. We have So z b a e az e bz z e az e bz e zτ dτ z b a z a b a b e zτ dτ z b e zτ dτ z a b a e zτ dτ e zτ dτ e xτ dτ z x e bx e ax z e ax e bx. x Theorem 6. Suppose a general Drchlet seres fs n a ne λns converges for some s 0. Then t converges unformly for any s such that Rs > Rs 0 and args s 0 < α for some α < π 2. Proof. Wthout loss of generalty we can assume s 0 0. Otherwse we apply the transformaton a n a n e λns0 and s s s 0. Then a ne λns a n e λns n converges for s 0. We want to show that fs converges for every s such that Rs > 0 and s args < α. The latter condton s equvalent to sayng M R such that Rs M. Here M cos α. Now by assumng that fs converges for s 0 we know n a n converges. So for any gven ɛ > 0, there exsts an nteger N 0 such that for any m, p N 0 we have S m,p p a n < ɛ. Settng b n e λns n Lemma 4 we have q a n e λns S m,q e λqs + 4 n q S m,n e λns e λn+s.

5 So q a n e λns S m,q e λqs + ɛ + ɛ q q e λ ns e λn+s. S m,n e λns e λn+s Usng Lemma 5 we have e λ ns e λn+s s Rs e λnrs e λn+rs M e λnrs e λn+rs Therefore q a n e λns ɛ + ɛm q fnally snce Rs > 0 we reach e λnrs e λn+rs ɛ + M e λmrs e λqrs. q a n e λns ɛ + M. Hence the unform convergence s shown. Corollary 7. If fs n a ne λns converges for some s 0 C then for any s C such that Rs Rs 0 we have. f converges unformly at s, 2. f s holomorphc at s. Proof. follows from Theorem 6 by choosng α suffcent close to π 2 from below. To prove 2 we defne f s n a ne λns. Clearly f s holomorphc for every and {f } converges to f unformly. Thus by Theorem 3, f s holomorphc at s. Lemma 8. Consder an ordnary Drchlet seres fs n a nn s where {a n } s bounded. Then f converges absolutely for every s C wth Rs >. Proof. There exsts M R such that a n < M for every n. So fs a n n s a n n s M n s M n Rs, n whch clearly converges f Rs >. n Defnton 9. A functon f : N C s sad to be multplcatve where for relatvely prme m, n N we have fmn fmfn. If ths s true for every m, n N then f s sad to be completely multplcatve. n n 5

6 Note: f f 2 f 2 hence f equals 0 or. If f 0 then f s dentcally zero so we requre f to avod the trval case. Theorem 0. Let f : N C be a bounded multplcatve functon and let p, p 2,... be the sequence of prme numbers n ncreasng order. Then fn fp α n s fp α fp α 2 p αs α0 p αs α0 p αs α0 2 n for every s wth Rs >. Proof. Lemma 8 mples that the left sde sum converges absolutely. Let n be a natural number. By the fundamental theorem of arthmetc we know there are unque non-negatve ntegers α, α 2,... only a fnte number of them beng nonzero, such that n pα and so fn fpα. By choosng α α n the th term n the left-hand sde of for, 2, 3,... we produce the term fp α p αs f pα fn pαs n s And we produce each term n the left hand sde of only once by the unqueness of prme factorsaton. Hence there s a bjecton between the nfnte sum and products n. Corollary. If f s bounded by and s completely multplcatve, then for every s C such that Rs > we have fn n s. n fp p s Proof. By Theorem 0 we have n fn n s α0 fp α p αs. Now that f s completely multplcatve we have fp α fp α. Hence fn α fp n s p s n α0 fp p s provded that < for all N. fp p s Defnton 2. Remann zeta functon s defned to be the ordnary Drchlet seres wth a n for all n N, that s, ζs n s, where Rs >. n, 6

7 Usng Corollary we can express Remann zeta functon as an nfnte product where Rs >. Theorem 3. For every s wth Rs >. ζ s holomorphc at s, 2. ζs 0. ζs p s Proof. Suppose Rs > and let r +Rs 2. Snce r >, by Lemma 8, ζr converges. Now by the second part of Corollary 7 and the fact that Rs > Rr r we can conclude that f s holpmorphc at s. To show the second part we note the followng fact: ζs 0 log ζs converges. But log ζs log log p log p s. s p s So t suffces to show that log p converges. Consder the sequence s { } a + log a, where a p. By usng L Hoptal s rule, the lmt of ths sequence as n s s a + log a lm n a 2 lm n a 2 a 2a lm n 2 a 2. Now snce a2 converges, so does a + log a and fnally snce a converges, so does log a log p. s Defnton 4. We defne φs n+ n n s τ s dτ for all s C such that Rs > 0. n Lemma 5. For every s C such that Rs > Proof. If Rs >, then φs + s n n+ n ζs φs + s. n s τ s dτ + τ s dτ n s ζs. n 7

8 From now on we use φs + s Lemma 6. φ s holomorphc on the half-plane Rs > 0. as the defnton of ζ wth the extended doman Rs > 0. Proof. f Rs > 0. sup n τ [n,n+] φs n n+ n n s τ s n s τ s dτ n s n + s n Theorem 7. As s we have Proof. As s log p s n log log ζs log s By the Taylor seres of log about x 0 we have So by applyng settng x p s log s we have j log jp js s. p s x j x j j p s + log. p s jp js. j2 But j2 jp js j p s j2 p s p s p s ps p p. 2 2 Hence as s. p s n log s Corollary 8. There are nfntely many prme numbers. 8

9 3 Characters Let G be a fnte abelan group. Defnton 9. s sad to be a character of G f : G C s a homomorphsm, where C s the group of non zero complex numbers under multplcaton. Denote the set of characters of G by Ĝ. Theorem 20. If n G and Ĝ then we have { n f g 0 f g G Proof. If then g G g g G G. If, ḡ G such that ḡ. Hence we have ḡ g ḡg g. g G g G g G Snce ḡ we can conclude that g G g 0. Corollary 2. For a fnte abelan group G where g G we have { Ĝ f g g 0 f g Ĝ Proof. Ths corollary follows from applyng Theorem 20 to group Ĝ. Let Gm Z/mZ be the set of all ntegers m nvertble modulo m. Defnton 22. s sad to be a character modulo m f : Gm C s a homomorphsm. To extend the doman of a character to all ntegers m we defne g 0 for any g not relatvely prme to m. 4 Drchlet L-functon Defnton 23. Let s be a complex number and be a character modulo m. Then s sad to be a Drchlet L-functon. Ls, n n n s Snce a character s completely multplcatve, by Corollary we have Ls,. p p s 9

10 Example. By settng we have Ls, n coprme to m p s ζs Hence Ls, has a smple pole at s. p m n s p m p m p s p s. p s Lemma 24.. If, then Ls, s bounded on a bounded open neghbourhood of n the half-plane Rs > 0. Proof. Snce, by usng Theorem 20 we have L, < n < φm. n Hence L, s bounded. From Theorem 20 we know f, n nn s converges for s 0. So by the second part of Corollary 7, Ls, s holomorphc on the half-plane Rs > 0. Therefore Ls, remans bounded on a bounded open neghbourhood of n the half-plane Rs > 0. Lemma 25.. If, then L, 0. Proof. The product of L-functons over all the characters modulo m dverges when s φm, where φ s Euler s totent functon [2] page 73. Snce Ls, has a smple pole at s and L, s bounded whenever, we reach the concluson that L, 0 f. Corollary 26. If 0, then log Ls, s bounded on a bounded open neghbourhood of n the half-plane Rs > 0. Proof. Ths s clear from Lemmas 24 and 25. Defnton 27. Let be a character mod m and s be a complex number such that Rs > 0. We defne fs, p p s p m Lemma 28. As s we have fs, log s. 0

11 Proof. Let. Then we have p for every N. By usng Theorem 7 we know that p p s log But fs, dffers from p p s by the fnte sum pp s. p m s. Lemma 29. If then fs, remans bounded as s. Proof. If Rs > log Ls, log The Taylor expanson of log x p p s log p p s about x 0, provded that x < s j x j j x + x2 2 + x By settng x p p s we have x p s <. So log Ls, j j j p p s p p s 2 + j2 j 2 j p 2 But j p j p s j2 j p p s j2 p p p s p s p 2 p s ps p p p. Snce both log Ls, and j2 j subsequently fs,. p p s j are bounded n 2 hence so s p p s Defnton 30. Let a and m be postve ntegers. We defne Q a,m to be the set of prme numbers that are congruent to a modulo m. Theorem 3. Suppose a and m are relatvely prme. Then φm p s p Q a,m fs, a p s and

12 Proof. p m fs, a a p m p p s a p p s a p p s a p p s p m But by Corollary 2 we have a p { φm p m p s p m a p f p a mod m 0 otherwse So fs, a p a mod m p m p s p s a p φm φm p Q a,m p s. Theorem 32 Drchlet s Theorem. If a and m are relatvely prme ntegers, there are nfntely many prme numbers congruent to a modulo m. Proof. From Theorem 3 we have p s φm p Q a,m fs, a. Lemmas 28 and 29 state that as s, fs, s asymptotc to log bounded. Therefore as s p s φm log s. p Q a,m Hence dverges whch shows that Q a,m s nfnte. p Q a,m p s f but otherwse s 2

13 5 Densty Defnton 33. Suppose P s an nfnte set and let Q P. The densty of Q n P s defned to be lm s q Q q s p P p s. Theorem 34. Let P be the set of prme numbers and Q a,m as n Defnton 30. The densty of Q a,m n P s φm for every a. Proof. By usng the results from Theorems 7 and 3, the densty of Q a,m n prme numbers s lm s q Q a,m q s p P p s lm s φm log s log s φm. By settng a and m 0 for nstance, Theorem 34 mples that one quarter of prme numbers end wth a one n the decmal representaton. References [] Tom M. Apostol, Introducton to Analytc Number Theory, Undergraduate Texts n Mathematcs, 976 [2] Jean-Perre Serre, A course n Arthmetc, Graduate Texts n Mathematcs, 973 [3] Dnakar Ramakrshnan, Fourer Analyss on Number Felds, Graduate Texts n Mathematcs, 998 3