A Reformulation of Dirichlet s Primes in Arithmetic Progression

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1 A Reformulatio of Dirichlet s Primes i Arithmetic Progressio Mireille Schitzer Setemer, 2005 Astract This aer resets a reformulatio of Dirichlet s theorem o Primes i Arithmetic Progressios, which states that for a ad relatively rime, the sequece {a+x} x cotais ifiitely may rimes. It is ossile to reduce this well-kow rolem to a simly stated cojecture. Oce assumed, this cojecture allows for comletio of the theorem while avoidig most comlex aalysis. This much more elemetary method would the oly ivolve dual grous ad the comlex character fuctio, simle series aalysis ad some liear algera. I will state the cojecture ad show how the roof follows elegatly from it. Itroductio Let G (Z/Z) so that G refers to the ivertile elemets mod(). Oly itegers of the form a + x where (a, ) have a reresetative cogruecy i G. Therefore, it is clear that the cardiality of G is φ() (where φ() is the Euler hi fuctio). A character o G is a elemet i the dual grou of G. The dual grou of G is defied as G { all homomorhisms χ : G C {0}} This yields multilicative fuctios s.t. χ( m) χ() χ(m). Also otice that this grou icludes the trivial character χ 0 where χ() G A Dirichlet character modulo is a modified character fuctio from the ositive itegers to C such that, for all elemets a Z s.t. (a, ) >, χ(a) 0. Other roerties of dual grous will e discussed further o. The ext tool used is the owerful Euler Product. It is easy to rove the followig equality: rimes <x x-smooth for some x 3, x R

2 Due to this equality ad the multilicativity of the Dirichlet character fuctio, we get: ( ) <x χ() x-smooth χ() for all χ G This sum is true for all fiite values of x, ut sice we desire to show that the left-had side coverges for all (with ay o-trivial χ), it ecomes ecessary to take some sort of limit. Sice we do t yet kow if the right-had side coverges for x or eve if a limit exists, we ca cosider the followig modificatio: χ() x This sum ca e show to have a limit ad coverge coditioally. Note that we caot take ay rearragemet of this sum ad resume a reservatio of covergece. I a attemt to comare this modified summatio with the roduct i equatio ( ), we roceed to slit the summatio as so: <x χ() x χ() x-smooth, >x χ() As log as the series ad roduct are ouded usig x < +, this equality is esured. A method of achievig the desired result requires the followig cojecture o the summatio of the Dirichlet L-fuctio for s over the x-smooth umers greater tha x (for all characters other tha the trivial): x-smooth, >x χ() 0 as x It is ow lai to see that if the cojecture is true, ad this differece gets very small as x gets large, the we will otai: χ() χ() This result will e show to comlete the roof. 2 The Dual of G Here we will show that the character fuctios form a orthoormal asis for the sace of fuctios from G to C. Defiitio 2.: G is the set of all homomorhisms from the grou G to the comlex lae uder multilicatio. 2

3 These fuctios are ituitively easy to costruct ut difficult to geerate while workig with o-cyclic grous. Defiitio 2.2: Let χ e a homomorhism from G C. The corresodig Dirichlet character mod() is defied as the fuctio χ : Z C with the roerties that: for all elemets a i Z such that (a, ) >, χ (a) 0 ad for all elemets c i Z such that (c, ), χ (c) χ(c mod()). These fuctios ehave the same as the characters i the dual of G. Secifically, it is easy to show that a Dirichlet character is also multilicative. Examle 2.: The two Mod(4) Dirichlet characters are the trivial χ 0 ad: 0 0, 2 mod(4) χ() 3 Examle 2.2: The six Mod(7) Dirichlet characters are: 0 0 mod(7) 0 0 mod(7) e 2iπ 3 2 e 4iπ 3 2 χ () e iπ 3 3 χ 2 () e 2iπ 3 3 e 4iπ 3 4 e 2iπ 3 4 e 5iπ 3 5 e 4iπ χ 3 () χ 5 () 0 0 mod(7) mod(7) e 4iπ 3 2 e 5iπ 3 3 e 2iπ 3 4 e iπ χ 4 () χ 0 () 0 0 mod(7) e 2iπ 3 2 e 4iπ 3 3 e 4iπ 3 4 e 2iπ mod(7) Note that there are as may character fuctios as there are elemets i the grou G (Z/Z) (I will rove that this is true i geeral). Also ote that G 7 is cyclic, as is G 7 itself. (As a exlaatio, lease refer to Irelad ad Rose [3] for details o the isomorhism etwee a grou ad its dual.) Thirdly, what is referred to i this last examle as χ 3 is more commoly kow as the 3

4 Legedre symol modulo 7. Proositio 2.: G is a grou uder multilicatio. Proof: Multilicatio of characters is associative, ad χ 0 is the idetity. To show closure ad the existece of iverses, if χ, χ 2 G, defie χ χ 2 as the fuctio that takes a G to χ (a) χ 2 (a). The, we ca defie the iverse χ G as χ (a) : [χ(a)] C. This χ is a homomorhism from the grou to C ad is therefore also i G. Closure is also clear from the fact that the characters are all homomorhisms. Lemma 2.: For χ() χ 0 (), G χ() 0. Ad G χ 0 () φ(). Proof: Choose a ɛ G s.t. χ(a) χ(a) ɛg χ() ɛg χ(a) ɛg χ() ɛg 0 ɛg χ()( χ(a)) 0 ɛg χ() χ(a) ɛg χ() χ() The secod art is ovious as card(g ) φ(). Lemma 2.2: The fuctio from Fuc(G, C) 2 C defied y < χ, ψ > φ() G χ()ψ() where φ() is the Euler hi fuctio is a Hermitia ier roduct o this vector sace. Proof: Clearly, <χ, φ><φ, χ> ad for λ C, λ<χ, ψ><λχ, ψ><χ, λψ>. All that remais to show is ositivity, which follows from: < χ, χ > G χ()χ() φ() G R[χ()]2 + I[χ()]2 φ() > 0 ad is zero R[χ()] 0 ad I[χ()] 0. Lemma 2.3: The χ i are orthoormal i Fuc(G, C) relative to this ier roduct. Proof: Let χ i ad χ j e elemets of G. The < χ i, χ j > G χ i ()χ j () φ() G j χ i () χ (). φ() If χ i χ j the χ i () χ j () for some, sice iverses are uique i a 4

5 grou. Ad sice χ i, χ j ɛ G χ i χ j G ad, y Lemma 2. G χ i ()χ j () φ() 0 If χ i χ j the G χ i ()χ j () φ() G χ 0 () φ() G φ() φ() φ() Lemma 2.4: There exists a atural ijectio etwee G ad its idual G. Proof: G is defied as the dual of the dual grou, or as G Hom(G, C). We ca see this ijectio y defiig the characters (ψ) of the idual with resect to the elemets of G : a G a ψ a (χ) : χ(a) φ is a homomorhism the seds memers of the dual grou to C. Suosig φ is ot ijective would imly that, for some a ad a i G, χ(a) χ(a ) for all χ χ(a (a ) ) for all χ which imlies that a a ad therefore a a. Theorem 2.: The elemets of the dual grou G form a orthoormal asis for the fuctios from G C. Proof: From Lemma 2.3 we get that the χ i are lieraly ideedet ad hece, G G. Additioally, this gives us that G G (sice G is similarly the dual of G ad so the same iequality alies). From Lemma 2.4, we have that G G. Therefore, G G. Sice G dim(fuc(g, C)), the elemets of the dual grou form a orthoormal set with the same cardiality as the dimesio of the sace. Therefore, the χ i s form a orthoormal asis over the sace of fuctios. Corollary 2.: There exists a ijectio etwee G ad G. 3 Euler s Equatio ad the L-Fuctio at s The desirale simlicity of this method comes from side-steig the use of the comlex s-variale i the L-fuctio. After rovig that L(, χ) coverges coditioally for all characters χ χ 0, our cojecture will allow for comletio of the theorem. 5

6 Proositio 3.: The Euler Equality ( ) <x x-smooth where are rime Proof: The exressio i the roduct reresets the sum of a coverget geometric series, which gives us ( ) <x Ad where x is the largest rime less tha x, ( ) ( ) ( + x + ( x ) ) Fially, y the usual rules for multilyig sums, each resultig term will e a uique roduct of oe choice er racket. By N eig a UFD, each roduct will therefore yield a uique term. x-smooth Proositio 3.2: For ay multilicative fuctio f(x), the followig equality holds: f() ( ) <x f() x-smooth Proof: This follows directly from the fact that f( k ) f() k as well as f( q) f() f(q). The it ecomes clear that, f() f( i i i x x ) f( ) i f( 2 ) i 2...f( x ) i x Theorem 3.: The harmoic series of rime umers, rime, diverges. Proof: First, ote that is asymtotic to log( ) for all, meaig that lim log( ). This ca e writte as follows: log( ) A stroger roerty is that the differece etwee the sums <x log( ) <x 6

7 is cotrollale. We get that log( ) <x <x <x 2 <x <x x which coverges y -series. Ad with Pro 3., log x < log x-smooth log( <x ( )) <x log( ) <x (The sums ad roduct are clearly ot equal or covergig to zero, so we ca take cosider their logarithm.) If coverged, so would the log( <x ( )) (y the limit comariso test from asic calculus). If this is so, the the log( ) over the x-smooth umers would e ouded. However, this is a cotradictio, as this series diverges as x. Theorem 3.2: (Dirichlet s Test for covergece) Let a ad e two ifiite series. If { } coverges mootoically to zero, ad the artial sums of a are ouded, the the series a coverges. Proof: Please refer to Gordo [2],.232. Lemma 3.2: The artial sums of χ() (with the Dirichlet character χ) are ouded for χ χ 0. Proof: Let S N N χ() χ() + χ(2) χ(n) 0 Some of these χ(a) are zero ad the rest are i asolute value. There are recisely φ() o-zero χ s, ad so a very rough uer oud o the ifiite series is: χ() φ() Proositio 3.3: L(,χ) χ() is coditioally coverget. Proof: We ve already see that the artial sums of χ() are ouded for 7

8 o-trivial characters. Additioally, { } 0 mootoically. By Theorem 3.2, the series coverges. However, χ() + > + Sice is fiite, L(,χ) diverges i asolute value ad is therefore oly coditioally coverget. over the x-smooth umers greater tha x co- Cojecture: The sum χ() verges to zero as x gets large. Proositio 3.4: Assume the cojecture. The Proof: From Pro 3.2, <x <x <x <x χ() χ() The, y the cojecture, <x χ() χ() x χ() rime x lim χ() χ() x χ() x <x x-smooth + χ() χ(). χ() x-smooth, >x x-smooth, >x 0 as x χ() χ() χ() Proositio 3.5: χ() is coditioally coverget χ χ 0. Proof: Now we have all we eed, sice χ() ( ) 0 ad coverges. It is still true that log( ) χ() <x χ() + <x <x 2 as show i the roof of Theorem 3.. Therefore, usig a exteded logarithm (fuctioig over comlex values), we get that log( ( χ() )) log( χ() 8 ) χ() coverges y LCT

9 Theorem 3.3: (Dirichlet s Theorem o Primes i Arithmetic Progressios) For ay atural umers a ad such that (a, ), the sequece {a + x} x cotais ifiitely may rimes. Proof: Costruct a fuctio ψ such that, for some a (Z/Z) : { a ψ() 0 otherwise ψ() is a fuctio i the sace Fuc(G, C), so it is saed y the χ i s. So, for some c i s (usig t : φ()), ψ() c χ () c t χ t () + c 0 χ 0 (). Sice the characters form a orthoormal asis, we ca determie the values of the coefficiets usig the ier roduct descried i Lemma 2.2. We oly care to oserve that c 0 < ψ(), χ 0 () > φ() t 0 Now, divide the liear comiatio y a rime, ad sum for every. The exressio ecomes: ψ() c χ () c t χ t () Each of the first t sums coverge (y Proositio 3.5) ad the last sum diverges (y Theorem 3.). Ad sice ψ() a we have that a diverges. Therefore, there are ifiitely may rimes cogruet to a mod(). 4 Numerical Aalysis Before advacig the cojecture, I did extesive umerical aalysis (with PARI) o the sum, ad modified the statemet ased o the results. For examle, at first the statemet ivolved two variales x ad x 2 with x 2 x as follows: d(x, x 2 ) χ() <x χ() x 2 + t <x 2, x smooth χ() But whe holdig x costat ad modifyig the smaller x 2, d(x, x 2 ) is smallest whe x x 2. (Please see figure.) The fact that d(x, x 2 ) oly makes sese with x 2 x removed the ossiility of a simle coclusio (if x ad x 2 were free or if x x 2, the we could 9

10 easily have said that, sice χ() x -smooth coverges, the the tail ecomes aritrarily small). I also exerimeted with modifyig the χ fuctio (eve withi a mod grou) ut this choice has little imortace as x gets large. I the aedix, I give a examle summatio usig the mod 7 geerator χ. It is clear that the sum teds drastically to zero amid dimiishig tremors. (Please see figure 2.) Refereces. J. A. Gallia, Cotemorary Astract Algera, fifth ed., Houghto Miffli Comay, New York, R. Gordo, Real Aalysis: A First Course, secod ed., Addiso Wesley, New York, K. Irelad ad M. Rose, A Classical Itroductio to Moder Numer Theory, secod ed., Sriger-Verlag, New York, J.-P. Serre, A Course i Arithmetic, studet ed., Narosa Pulishig House, New Delhi,

11 Figure : A rough idea (with a smoothed curve) of how as(d(x, x2)) varies as x2 is icreased, holdig other variales costat. Here, the mod7 geerator is used ad the x-axis is a log scale.

12 Figure 2: A much more refied icture of the sum as(d(x)) with the mod 7 geerator. Note that the x-axis is o loger a log scale ad that the y-axis is much smaller tha i the revious grah. Also, ulike Grah, here there are 000 samle oits lotted without a smooth curve to give the est ossile idea of the ehaviour of the sum. 2