ANATOLY PREYGEL. def. (1 x k/d ) µ(d),

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1 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Abstract. We resent some of the easier to rove analytic roerties of Dirichlet-Hecke L-functions, including the Dedekind zeta functions. We use Artin recirocity to show that abelian Artin L-functions are Dirichlet-Hecke L-functions, and thus share these roerties. We roceed to show a decomosition formula for the Dedekind zeta function, use this to show non-vanishing at s = of L-series for non-rincial ideal class characters, and to rove the Chebotarev Density Theorem.. Introduction.. Elementary Motivation. We begin with an elementary roof, the extension of which may be viewed as a driving force for this exository aer: Proosition.. Secial Case of Dirichlet s Theorem on Primes). For each fixed ositive integer n, there are infinitely many rimes satisfying mod n). Proof. Assume the contrary, and say that,..., r is the comlete list of such rimes for our fixed n. Let Φ k be the k th cyclotomic olynomial. That is: Φ k def = d k x k/d ) µd), where µd) is 0,, according to whether d is divisible by a square, is the roduct of an even number of rimes, or of an odd number of rimes, resectively. Say q is a rime such that k, q) =. Let K denote the slitting field of f = x k over F q. Note that f = kx k is non-zero for q k, and then f and f have no common roots. Then for d k, x d, x k ) = x d ), and as f = x k has distinct roots, x d x k. Then, counting multilicities in the definition of Φ k shows us that the roots of Φ k are recisely the elements of order exactly k. So, if Φ k has a root over F for k, then this root is an element of order k. By Lagrange s Theorem, this imlies that k F =. So, mod k). Now, let N = n i. As Φ N has only finitely many roots, we may take k Z >0 such that Φ N kn) 0. Then, Φ N kn) has rime factors, and we may let q be any rime dividing it. Note that Φ N 0) =, so Φ N kn) mod k)n, and q is co-rime to kn and so to N. So, q N and kn is a root in F q of Φ N. Thus, q mod N) and so q mod ), but q is not one of the i. This yields a contradiction, establishing our result. We offer an alternative interretation of the idea of the above roof, using some standard results of algebraic number theory. Let K = Qζ n ) be the n th cyclotomic field. Then, O K = Z[ζ n ], and the rimes which ramify in K all divide n. For an unramified rime of K, the ma {ζ n ζ n} is the Frobenius of. Now, the reduction of Φ n in F [x] factors as

2 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL g i= f i e, where e = recall unramified) and deg f i = f is the order of the Frobenius. So, Φ n slits into linear factors if and only if the Frobenius is trivial. This occurs if and only if ζn =, which occurs if and only if mod n). So, our roblem is equivalent to finding many rimes with trivial Frobenius. It turns out that the question of the distribution of rimes with a secified Frobenius is a fruitful generalization, and the goal of this aer will be to develo the techniques to rove the Chebotarev Density Theorem. The above roof is entirely algebraic, and is relatively elementary. However, it seems that such algebraic methods only get us so far. Indeed, in 837 Dirichlet brought in analytic methods to rove the following result generalizing our above roosition: Theorem.. Dirichlet s Theorem on Primes in an Arithmetic Progression). Let a, b N be such that a, b) =. Then, there are infinitely many rimes satisfying a mod b). To this day, results of this nature are handled with analytic machinery. We will begin by develoing sufficient tools to sketch a roof of this result, before extending them to allow us to rove our desired generalization..2. Characters and L-functions. In roving his theorem, Dirichlet introduced a class of analytic objects tied to the rational number field: the Dirichlet L-series. We assume the reader is familiar with basic reresentation theory of finite grous. Throughout this aer, all reresentations will be over C. For an abelian grou G, we will use the word character to refer to a continuous) homomorhism G S C [note that for G finite, these characters are recisely the one-dimensional reresentations under the identification GL C) C, for the image must be a torsion element]. We let HomG, S ) denote the grou of characters of G; we will denote the identity in HomG, S ) by χ 0 which we will call the rincial character. We will rove some of the basic roerties of this construction: Proosition.2.. For G a finite abelian grou written multilicatively here). Then: a) G HomG, S ); b) G = HomHomG, S ), S ) by the evaluation ma; c) { g G χg) = G χ = χ 0 0 otherwise ; d) { χ HomG,S ) χg) = G g = 0 otherwise. e) For any g G, we have χ HomG,S ) χg)t) = G ord t g) ord g. Proof. By the structure theorem for finite abelian grous, we may decomose G into cyclic factors r G Z/n i Z γ with < n i G. 2 i=

3 ANATOLY PREYGEL Let ζ i be a rimitive n th i L-FUNCTIONS AND THE DENSITIES OF PRIMES root of unity, and consider the ma: G HomG, S ) g χ g a) = i ζ γa) iγg) i i We readily check that each χ g is indeed a homomorhism G S for it mas the generator of the i th cyclic summand to ζ γg) i i ). Furthermore, every such homomorhism is of this form, for it must ma this generator to an element of order dividing n i, so to a ower of ζ i. So, this ma is surjective onto HomG, S ). We may exlicitly check that it is a homomorhism. Now, it suffices to rove injectivity. But indeed, χ g = χ 0 imlies that χ g is on the generators of the cyclic summands, which in turn imlies ζ i γg) i ) =, whence γg) i = 0, for each i, and thus g =. This roves a). For b), note that the evaluation ma G g ĝ s.t. ĝχ) = χg) is a homomorhism G HomHomG, S ), S ). For each g G distinct from id, there will be some character with non-trivial value on G by the above characterization of HomG, S ), so this ma is an injection. By a), we have that G = HomHomG, S ), S ), so this ma must be an isomorhism. For c), note that the claim is trivial for χ = χ 0. Then, if χ χ 0, then there exists a g G such that χg ). Then χg ) χg) = χg g) = χg g) = χg), g G g G g g G g G which imlies our desired result. Then, d) follows from b) and c). Now, let H = g. We may take H to be one of our cyclic factors for G. Take it to be our first cyclic factor, and then n = ord g. Then, by the above characterization, we see that: χ HomG,S ) χg)t) = = n n 2 a = a 2 = n a = = t n ) n 2 n r n r a r= ζ a n t ) ζ a n t )) n2 nr = G ord t g) ord g. This shows e). We could also have established this by noting that G = H G/H so HomG, S ) = HomH, S ) HomG/H, S ), and a similar roduct identity. Now, we will introduce Dirichlet s L-series. 3

4 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Definition. Let N > be an integer. Let χ be a character of Z/NZ). We may extend χ to a multilicative function, which we also call χ, on Z by: { χ k) k, N) =, χk) = k = image of k in Z/NZ) 0 otherwise We refer to both of these meanings of χ as the Dirichlet character. There shall be no ambiguity, for it will always suffice to exand to a function on Z. Then, the Dirichlet L-series for the Dirichlet character χ is: Ls, χ) def = n= χn) n s = n,n)= Remark. Not worrying about convergence, the quick reader may immediately notice that the multilicative roerty of χ lets us write the above in a roduct form: ) χ) s. rime Indeed, a Dirichlet series exansion and an Euler roduct exansion are two of the roerties, along with a functional equation, defining L-series..3. Convergence roerties of Dirichlet series. We wish to rove some convergence roerties of the Ls, χ). We will consider a more general class of objects, in order to get results that we can re-use later. Definition. A Dirichlet series is a series of the form a n n s n= with the a n comlex numbers and s a comlex variable. Lemma.3.. If the Dirichlet series a n n n s converges for some s = s 0, then it converges for any s with Res) > Res 0 ), and converges uniformly on any comact subset of this region so, the series reresents an analytic function on that half lane). Moreover, if there exist constants C, σ > 0 such that a a k Ck σ for each k, then the series converges for Res) > σ. Proof. Let P n s) = n i=0 a ii s. For δ > 0, Res) Res 0 ) + δ, i > 0 note that we have: i i+ s s 0 i + ) s s 0 = dx s s 0 ) x s s 0+ Then, for m < n we have n a i P n s) P m s) = i s 0 i=m+ i s s 0 P n s 0 ) = P ms 0 ) n s s 0 m + ) + n s s 0 i i=m+ 4 χn) n s s s 0 i s s 0 Res) Res 0)+ i +δ [ ] P i s 0 ) i s s 0 i + ) s s 0

5 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES and by the above: ) s s0 m + n m + ) s s 0 P ms 0 ) P n s 0 ) n + i=m+ Now, as P n s 0 ) converges, we may take M s.t. M P n s 0 for all n. Then: 2M m + ) δ + M s s 0 i=m+ i +δ P i s 0 ) s s 0 i +δ Now, on any comact set, s s 0 is bounded, and the last sum is convergent by the integral test, so taking m sufficiently large we see that the P n converge uniformly on comact subsets of Res) + δ. Recall that the uniform-on-comacts limit of holomorhic functions on a domain is holomorhic. Alying this to the artial sums of the Dirichlet series, this comletes the first art of the result. This afford us the freedom to be sloy in declaring that convergent Dirichlet series give analytic functions. Now, let S n = a a n. Then, for n > m, δ > 0 and Res) σ + δ: P n s) P m s) = and by the above: A n n s A m m + ) + n s i=m+ C n + C Reσ) s m + ) + n Reσ) s [ ] A k i s i + ) s i=m+ C [ ] i s σ i + ) s σ This roves our result. C n δ + C m + ) + C s σ n δ i=m+ i +δ.4. Sketch of Proof of Theorem... We will use the following analytic results in this roof: a) Convergence of Ls, χ) and ζ K Dedekind zeta) for Res) >, and for Res) > 0 for χ χ 0. b) Existence and convergence of Euler roduct for Ls, χ) and ζ K for Res) >. c) For χ χ 0, Ls, χ) is analytic at s =. Ls, χ 0 ) and ζ K all have simle oles at s =. We rove the first of these here. We rove more general version of the later two in the sequel. Let G = Z/bZ). Then, for χ HomG, S ) not the rincial character, we note that by Pro..2. the sums k n= a n = k n= χk) are cyclic and so bounded. So, we may aly Lemma.3. to Ls, χ) with σ = 0, to get that Ls, χ) is analytic for Res) > 0 and so at s =. Alying Lemma.3. to Ls, χ 0 ) with σ = yields that it is analytic for Res) >. 5

6 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Also, using multilicativity of χ and the Fundamental Theorem of Arithmetic, we may rove the Euler roduct exansion for Ls, χ): Ls, χ) = n= χn) n s = rime χ) s ). For b, let f) denote the order of the image of in G. Then, by Pro..2. we have ) sf) φb)/f). χ HomG,S ) Ls, χ) = b Let K = Qζ b ). Define ζ K = Norm)), where the roduct is over finite rimes of K. Now, note that the only rational rimes ramified in K/Q are those dividing b, by a discriminant argument. So, we may show that ζ K s) = ) sf) φb)/f) = Ls, χ) ) sf) φb)/f). χ HomG,S ) b The last factors are analytic at s =, ζ K and Ls, χ 0 ) have simle oles at s =, and the other factors are analytic there. So, the equality imlies that these analytic factors are non-zero. That is, L, χ) 0 for χ χ 0. Then, letting a be the inverse of a in G, note that by Pro..2. we have a mod b) s = G χ HomG,S ) χ)χa ) s Now, for functions f, g, we write f g if f g may be analytically continued on some neighborhood of s =. Taking logarithms in the Euler roduct for Ls, χ) we get log Ls, χ) = χ) s + m 2 m χm ) sm This last term is absolutely convergent in a neighborhood of s = by comarison to ζ2s), which is analytic on Res) > 2. So, a mod b) a mod b) s G χ HomG,S ) χa ) log Ls, χ) We have that s )Ls, χ 0 ) is analytic and non-vanishing at s = we just kill the simle ole), so log Ls, χ 0 ) log. Also, for χ χ s 0 we have that Ls, χ) is analytic at s =. Then s G χ 0a ) log s = G log s. If there were finitely many rimes a mod b), then this sum would be bounded as s +, but log is not. This yields our desired result. s 6

7 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Remark. There are other ways to establish the crucial fact that L, χ) 0. The above method closely mirrors the roof that we will follow below. Another common aroach is to slit into the case of comlex characters that is, χ 2 χ 0 ) and real characters that is, χ 2 = χ 0 ). Such an aroach may in fact be used to rove the general case that we wish to use here; see for instance [Hei67]..5. Prerequisites and references. We assume that the reader has a reasonable familiarity with basic comlex analysis, the slitting of rimes in number fields, basic commutative algebra incl. localization), and valuations. In addition, we will state and then use without roof the fundamental results of class field theory. In addition, we will occasionally make remarks hinting at the more general theory, secifically Großencharakters, analytic continuation, and Artin L-series. There are many excellent references on this material and the toics surrounding it. For a historical account of class field theory see [Coh85]; for the modern cohomological treatment see one of [Lan94], [Jan96] lacks the idèlic formulation, the closest to the exosition of class field theory given here), [Lan94] no roofs for global theory), [SD0] concise!), [Neu86] s slightly unusual cohomological treatment, without the Brauer grou!), [Neu99] as the revious). For an analytic view on class field theory, including all the theory of L-functions discussed herein and more, see [Gol7]. A full treatment of Dirichlet-Hecke L-functions including a roer treatment of Großencharakters) may be found in [Gol7], [SD0] concise!), or [Tat67] the original source for many of the arguments given in the receding two). These sources also address the question of analytic continuation of these L-functions to all of C, as does [Neu99] with a different aroach, using theta functions, from Hecke). The exosition of the analytic theory in [Neu86] is simle, and is the closest to the exosition given here..6. Notation. Unless otherwise stated, we will use the following notation throughout the remainder of this document: K will denote a number field. By a rime of K we will mean either a non-zero rime ideal of O K a finite rime ), or a real embedding K R a real infinite rime ), or a conjugate air of comlex embeddings K C a comlex infinite rime ). Equivalently, we may regard these as the equivalence classes of valuations on K. We will use ν to denote a not necessarily finite) rime of K, and K ν will denote the comletion of K with resect to the toology induced by the valuation along with the canonical inclusion K K ν written x x ν ). For an extension L/K of number fields, a finite rime of K and P a rime lying above it, we will denote by D P, I P the decomosition and inertia grous of P if our[ extension ] is abelian, we may write in lace of P). For unramified, we will denote by L/K the P ) Frobenius corresonding to P and by L/K the Artin ma of, that is the conjugacy class [ ] { L/K : P } which we will regard as just an element for L/K abelian). If is ramified, P we modify each of these to be I P -cosets. We define the Artin ma on arbitrary ideals by unique factorization into rimes. For functions f, g, we write f g if f g may be analytically continued on some neighborhood of s =. 7

8 L-FUNCTIONS AND THE DENSITIES OF PRIMES 2. Congruence subgrous, class grous, and recirocity ANATOLY PREYGEL In order to generalize the notion of a Dirichlet character and series, we must have an aroriate generalization of the grou Z/nZ). In the context of our roof-sketch above, the field K = Qζ n ) was introduced. Indeed, the Z/nZ) may be viewed as the Galois grou GalK/Q). Let us consider the hrasing of our goal in terms of Frobenius elements of Galois extensions. Observe that for L/F/K a tower of Galois extensions, the restriction ma induces a surjection of decomosition grous D P D, for P a rime of L and = O F. So, to understand the Frobenius elements of F we may look at the Frobenius elements of L. When our base field is Q, the Kronecker-Weber theorem assures us that every abelian extension K/Q is contained in some cyclotomic extension. Now, our characterization of the Frobenius in cyclotomic extensions lets us reduce this study to the study of certain congruences defining the relevant subgrou of the Galois grou of this cyclotomic field. Examle. The above discussion is almost in reverse of how we got to the Frobenius statement in the first lace. Let us bring things back to that context for a quick examle. Say we wanted to look for rimes satisfying ± mod 7). Note that {±} Z/7Z) is a subgrou. Now, we may realize the latter as GalK/Q) for K = Qζ 7 ). Then, let H be the subgrou of L/Q GalK/Q) corresonding to {±}, and) set L = K H = Qζ 7 +ζ7 ). Then, for not ramified ) in K that is, not 7), we have that K/Q = {ζ 7 ζ7}, and ± mod 7) K/Q H ) = id. From this, ± mod 7) slits comletely in L/Q. Finding the minimal olynomial for a rimitive element of L, and showing that the ring of integers is monogenic, we can relate this to olynomials: ± mod 7) x 3 + x 2 2x slits in F [x], a recirocity law! This icture over Q is the rototye for what follows. 2.. Notation. Let I K denote the free abelian grou generated by the finite rimes of K. For a set S of rimes of K, let I S K denote the free abelian grou generated by the finite rimes of K excluding the elements of S. We let ι : K I K be the ma given by a ao K = i e i i I K. Define an modulus as a formal roduct of rimes of K. We write x mod m) to mean that: For each finite rime m we have ord x ) m for m > 0 such that m m; For each real infinite rime ν m we have x ν > 0. Denote I m K Denote def = I S K where S = { finite rime : m}. K m def = ι I m K) = { a b : a, b O K, for each rime S ord a = ord b = 0 } and Finally, denote: P m K def K m def = {x K m : x mod m)}. = ιk m ) I m K and Cl m K 8 def = I m K/P m K.

9 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES We call Cl m K the ray class grou of m. We will often dro the subscrit K from I m K, Pm K, Clm K. If we dro the m, then it is assumed that m = Congruence subgrous. A congruence subgrou defined mod. m) is a subgrou H m of I m K containing Pm K. Whenever we write a suerscrit modulus on a grou, we take that to mean that it is a congruence subgrou defined mod. m. Proosition Say m n, then I m I n. Define H n = H m I n. Then, H n is a congruence subgrou defined mod. n) and: a) H m = H n P m ; b) the inclusion I n I m induces I n /H n = I m /H m. Proof. [Jan96, Ch. V, 6] The general idea of the roof is to use the Chinese Remainder Theorem to show that we may avoid given ideals in a suitable sense. So, if a congruence subgrou is defined mod. m, it is uniquely defined mod. all multiles of m. Then, we may define an equivalence relation on congruence subgrous, with H m H m 2 2 if there is some common multile m of m, m 2 such that they have common restriction to I m that is, if H m I m = H m 2 2 I m ). Call such an equivalence class of congruence subgrous an ideal grou or by abuse of terminology a congruence subgrou. We will write H for an ideal grou, and then H m for its realization mod. m. By the above, the quotients I m i /H m i i for i =, 2 will be isomorhic. So, to each ideal grou we may associate an equivalence class of such quotients, which we will call the congruence class grou. We will sometimes write this as just I/H. Proosition Say H m H m 2 2. Let m be the greatest common divisor of m and m 2. Then, there is a congruence subgrou H m such that H m I m i = H m i i for i =, 2. Proof. [Jan96, Ch. V, 6] Combining Pro and Pro , we see that for any ideal grou H there is a minimal modulus with resect to divisibility) f such that H may be realized mod. f. We call this the conductor of H. Similarly, we have the notion of conductor of a congruence class grou I/H, defined as the conductor of H Finiteness of congruence class grous. Now, we claim that each congruence class grou is finite. If we have P m H m I m, then we may regard I m /H m as a subgrou of I m /P m = Cl m. So, it suffices to show that the ray class grous are finite: Proosition Cl m is finite Proof. Note that [I m : P m K] = [I m : ιk m )] = [I m : ιk m )][ιk m ) : ιk m )]. Note that ιk m ) = I m P. So, by the above, I m /ιk m ) = I P = Cl K. So, the first term is just the class number, which is finite. The second factor is a divisor of [K m : K m ], so it suffices to show that this quantity is finite. Say m = r i= m i where m i = ν n i i with the ν i distinct. Then, consider the reduction ma K m K m 9 r K m i K m i i=.

10 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Note that the kernel of this ma is the set of elements in each K m i, which is recisely K m for the constraints imosed by each m i are indeendent and in total are recisely the constraints imosed by m. So, the ma is injective. By writing out the conditions for something to ma to a rescribed element of the codomain and alying the Weak Aroximation Theorem a weak form of the equivalent of the Chinese Remainder Theorem for valuations), we see that the ma is surjective. So, it suffices to rove that each term in the roduct is finite. If m is a comlex lace, the quotient is trivial; if it a real lace, the quotient has order 2. If m = n is a finite lace, then { a } K m = b : a, b O K, a, b = O K ), where the subscrit denotes localization. Then, K m = + n O K ). So, the quotient is O K ) ) OK ) = = OK + n O K ) n O K ) n Now, O K n is finite, so this last grou is finite. This roves our claim Artin recirocity. In the following sections we will want to invoke class field theory. So, we will briefly review a statement of the imortant results, without roof. Theorem 2.4. Artin Recirocity Theorem). For L/K an abelian extension of number fields, there is a modulus m divisible by all the ramified rimes of L/K and a congruence subgrou H m such that the following is an exact sequence ) H m I m K Exlicitly, we have H m = P m K Norm L/K I m L ). L/K ) GalL/K). When the conditions of the revious theorem hold, we say that L is the class field for K of the congruence class grou I m K /Hm. Furthermore, we say that m is an admissible modulus for L/K and for the corresonding congruence class grou I/H. We define the conductor of L/K to be the minimal with resect to divisibility) modulus f such that f is an admissible modulus for L/K equivalently, the conductor of I/H). We have the following result: Proosition Let f be the conductor of L/K. Then, the rimes dividing f are recisely the ramified rimes of L/K. In addition to the recirocity theorem, there is also a corresondence going the other way: Theorem Existence Theorem). For any congruence subgrou P m K Hm I m K, there is a unique abelian extension L/K such that L is the class field for K of the congruence class grou I m K /Hm. [Equivalently, such that H m = P m K Norm L/K I m L ).] These two theorems rovide a corresondence between objects outside of K, secifically the abelian extensions, and objects inside K, secifically the congruence subgrous. 0

11 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Remark. Let us look at the situation over Q in this context. The abelian extensions over Q corresond to congruence subgrous, which in turn corresond to subgrous of the ray class grous, here Z/mZ). Call the class field corresonding to a ray class grou a ray class field. Then, for any abelian extension K/Q, we have the conductor m. We may assume that m and m = m, in which case we get that the ray class field for m is Qζ m ). So, the notion of admissible modulus and conductor corresond in this setting to finding a cyclotomic extension containing K almost: the case where m leads to the ray class field being the maximal totally real subfield of the cyclotomic extension, to kee the real infinite rime of Q from slitting). 3. Dirichlet-Hecke L-functions 3.. Definitions. Now, we may consider the grou of characters of Cl m K, which is a finite abelian grou by Pro For χ HomCl m K, S ) we may view χ as a function on I m K whose kernel contains Pm K, and then as a multilicative function on I K by setting it to 0 on the ideals not corime to m. We will also denote this ma I K C by χ. As before, we denote the identity element by χ 0 and call it the rincial character. We call these the generalized Dirichlet characters or ideal class characters, and we may define the corresonding Dirichlet-Hecke L-series: Lm, s, χ) = a χa) Norma) s, where the sum is over all integral ideals of O K or equivalently, over those ideals corime to m). Note that we do not indicate an order for the summation. For now, whenever we give a sum over rimes we assume it is taken as a Dirichlet series, that is that the sum if to be taken over ideals in order of increasing norm as we write out in 3.2.). Later we will rove absolute convergence results and this will become largely irrelevant. Examle. Set K = Q, m = m. Note that O K = Z is a PID. Then, each ideal a) I m has two generators, ±a. Maing the ositive generator note m) to its image in Z/mZ) we get a surjective homomorhism, with kernel equal to the set of things congruent to mod m), which is recisely the set of things congruent to mod l ) for l m. So, Cl m K = Z/mZ). So, we get the classical Dirichlet characters and their Dirichlet L-series as secial cases. Examle. Let m = and χ = χ 0 be the rincial character. Then, Lm, s, χ 0 ) = a Norma) s def = ζ K s) This is the so-called Dedekind zeta function. For a modulus m let us define ideal class zeta functions: ζs, c) = a c Norma) s, where c Cl m K is an ideal class viewed as a coset of P m K in Im K ).

12 L-FUNCTIONS AND THE DENSITIES OF PRIMES Note that for an ideal class character χ of modulus m Lm, s, χ) = c Cl m χc)ζs, c) ANATOLY PREYGEL as formal series and on their common domain of convergence which will be the intersections of the domains of convergence of the ideal class zeta functions). Remark. The characters and L-series introduced in this section are generally called generalized Dirichlet characters and L-series. There is a wider class of characters and L-series, the Hecke Großencharakters and their L-series, which maintain all of the interesting roerties of the class described here. They may be thought of as characters of the idèle class grou, or alternatively as generalized Dirichlet characters with an infinite art. There is a discussion of the relationshi between these two classes of characters in [Neu99] and [Hei67] Convergence roerties. Now, we look at the convergence roerties of these L- series. Proosition Let χ be an ideal class character for the modulus m of the field K. Then: a) Lm, s, χ) converges absolutely on Res) >, and uniformly on Res) > + δ for any δ > 0; b) For Res) > we have the convergent Euler roduct identity: Lm, s, χ) = Norm) s χ) ) m Noting that we set χ) = 0 if is not corime to m, we may take the roduct over only those ideals that are. Proof. We write Lm, s, χ) as a Dirichlet series: Lm, s, χ) = a n n s where a n = n= Norma)=n Now, we claim that each ideal class of Cl K can contain at most one ideal of a given norm. Indeed, say I, J are in the same ideal class. Then, I = αj for some α K. Then, NormI) = Normα) NormJ), so NormI) = NormJ) imlies that Normα) =. Then, α is a unit in O K, so I = J. This roves our claim. Then, a n {a : Norma) = n} Cl K. So, for δ > 0 and Res) > + δ we have: a n n s = a n n Res) Cl K n +δ) Then, Lm, s, χ) converges uniformly and absolutely on Res) > + δ by comarison to Cl K n n +δ) which converges by the integral test). This roves i). Now, note that for s > we have the absolutely convergent exansion: s ) = 2 m=0 sm χa)

13 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Then, say,..., r are the rime ideals satisfying Norm) N. By unique factorization of ideals and multilicativity of χ we have Norm) s χ) ) = χa) Norma) s Norm) N So, Norm) N a= k kr r = Norma) N χa) Norma) s + Norm) s χ) ) Lm, s, χ) Norma)>N Norma)>N a= k kr r χa) Norma) s. χa) Norma) s The absolute convergence of Lm, s, χ) on the region in i) thus imly thus the equality in ii). Finally, we ll check that the Euler roduct converges as a roduct that is, its logarithm converges as a sum): Note that log Norm) s χ) ) = m Norm) ms χ m ). This converges absolutely on Res) > by comarison to Lm, s, χ). Exonentiating, we get that Euler roduct converges as an infinite roduct that is, has an non-zero limit) on Res) >. m Now, we look at some simle) analytic continuation roerties. Proosition Let ζ = ζ Q be the Riemann zeta function. Then, ζ may be analytically continued to Res) > 0, excet for a simle ole at s = with residue. Proof. Let ψ r n) = { r r n otherwise Then, define ζ r s) def = ψ r n)n s. n Now, ζ r is a Dirichlet series, with artial sums bounded by r, so by Lemma.3. it has abscissa of convergence at most 0 and so it is analytic on Res) > 0. Now, note that we have the equality of formal series: ζs) = ζ r s) + r r s ζs So, we may continue ζ to Res) > 0 by setting: ζs) = ζ rs) r s The numerator is analytic on Res) > 0 for each r, and the denominator vanishes only when = r s. By the uniqueness of meromorhic continuation, the continuations for different r must agree. So, for s to be a ole, we must have = r s for each r = 2, 3,.... 3

14 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL This imlies s =. This gives our desired continuation. Note that ζ 2 ) = ln 2, so the order of the ole at s = is the order of vanishing of the denominator. Exanding out the Taylor series, we see that it is of order. So, s )ζs) is analytic at s =, and to comute res s= ζs) = lim s s )ζs) we can comute the limit for s + real. But, for s > a ositive real: s = So, the residue is indeed. dt t ζs) + dt s t = + s s Remark. We do not need, thus do not rove, the following functional equation for ζ: ζ s) = 22π) s cos 2 sπ)γs)ζs) Along with the above, this lets us analytically continue ζ to the whole comlex lane excet for the ole at s =. In the receding we used only the finiteness of the class grou. We can get better convergence results by using the above and looking at the ideal class zeta functions by estimating the distribution of norms in the ideal classes. We state without roof a result of this nature: Proosition Let c Cl m K be an ideal class. Let Sc, n) = {a c : : Norma) n}. Then, Sc, n) = κ m n + On N ) where N = [K : Q] and κ m = 2 r+s regm)π s ω m Normm) K 2 where r, s are the number of real and comlex rimes of K, ω m the number of roots of unity in O K Km, regm) the regulator of m, K the discriminant of O K. Proof. See [Jan96, Ch. IV, 2] or [Lan94, Ch. VI, 3]. This allows us to get: Proosition Let K, m, χ be as in Pro. 3.2., c, κ m be as in Pro , and let N = [K : Q]. Then: a) ζs, c) may be analytically continued to the region Res) > excet for the simle N ole at s = with residue κ m. b) If χ = χ 0, then Lm, s, χ) may be analytically continued to the region Res) > N excet for the simle ole at s = with residue Cl m K κ m. c) ζ K s) may be analytically continued to the region Res) > excet for the simle N ole at s = with residue Cl K κ d) If χ χ 0, then Lm, s, χ) converges on Res) >, and is analytic in that half N lane. 4

15 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Proof. Let fs) = ζs, c) κ m ζ Q s) be an equality of Dirichlet series. Then, f is given by a Dirichlet series with coefficients a i such that k a i = Sc, k) κ m k, i= which by Pro is Ok N ). By Lemma.3., f is analytic on Res) >. By Pro , ζ N Q may be analytically continued to Res) > 0 excet for a simle ole at s = with residue. So, setting ζs, c) = fs) + κ m ζ Q s) gives a continuation of ζs, c) to Res) >. Furthermore, N res s= ζs, c) = res s= fs) + κ m res s= ζ Q s) = κ m This roves a). Now, note that Lm, s, χ 0 ) = ζs, c). c Cl m With a), this gives the desired continuation. Also, the residue is just the sum of the residues, which is Cl m K κ m. Setting m = we get Lm, s, χ 0 ) = ζ K s), giving us c). Now, recall that we have Lm, s, χ) = χc)ζs, c) c Cl m K This is a Dirichlet series with coefficients a i such that k a i = χc)sc, k) i= c Cl m = ) κ m k + Ok N ) c Cl m χc) = Ok N + kκm for χ χ 0, we may use Pro..2. to write this as: c Cl m χc) So, Lemma.3. imlies d). = Ok N ) 4. L-series and Galois grous 4.. Conductors of characters. For a modulus m and a character χ HomCl m K, S ), let the conductor f χ of χ be the conductor of ker χ viewed as a subgrou of I m K containing Pm K, or equivalently the conductor of I m K / ker χ. Proosition 4... Let m, χ, f χ be as above. Then, f χ is equal to the least modulus n such that χ factors through Cl n K. Also, there is a unique χ HomCl fχ K, S ) such that χ factors through χ. 5

16 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Proof. Let H m = ker χ I m K ; note that it is a congruence subgrou. As Pfχ K Hfχ and by Pro we have the mas which induce I fχ K /Pfχ K I fχ K /Hfχ I m K/H m HomCl fχ K, S ) HomI fχ K /Hfχ, S ) HomI m K/H m, S ). Now, χ factors through I m K /Hm, and so under this ma it factors through Cl fχ K. Furthermore, χ is defined by this, giving uniqueness. Say χ factors through Cl n K for some n m: Cl m K χ χ S Cl n K Then, ker χ = I n K ker χ. Set H n = ker χ, we see that H n realizes ker χ mod. n, and so f χ n by the minimality of the conductor and Pro So, f χ is minimal with this roerty. We say that a character χ HomCl m, S ) is rimitive if m = f χ. To every character χ there corresonds a unique rimitive character, the χ of Pro. 4..; we will continue to use χ to refer to the rimitive character corresonding to χ. Examle. For any modulus m and χ 0 HomCl m K, S ) the rincial character, we have f χ0 = and χ 0 is the rincial character in HomCl K, S ). Let us comare the Dirichlet-Hecke series of χ and χ. Proosition ) Say χ HomCl m, S ) with conductor f. Let F be the fixed field in L of L/K that is, the image of ker χ under the Artin ma). Then, for Res) > : ker χ Lf, s, χ) = Lm, s, χ) ) χ) Norm) s m f = ) χ) Norm) s unr. in F/K Proof. The first equality follows at once from writing out the Euler roducts on both sides. Let L/K be the class field of Cl m. By Pro. 4.. the conductor of χ is equal to the conductor of I m / ker χ, so by Pro the rimes dividing f χ are recisely the ramified rimes of the class field of I m / ker χ. Artin Recirocity together with the Galois corresondence imly that F is the class field of I m / ker χ. So, the rimes dividing f χ are recisely the ramified rimes of F. The second equality follows. Using Artin Recirocity, we may treat a generalized Dirichlet character as a character on a certain abelian Galois grou and vice versa. The class of Artin characters and L-functions further generalizes this notion. 6

17 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES 4.2. Artin L-functions. Let L/K be a Galois extension of number fields, with Galois grou G = GalL/K). Then, we may consider an irreducible finite dimension comlex reresentation of G, ρ : G AutV ). [ ] Let be a finite rime of L/K. Then, for a rime P lying over, L/K gives a coset in G P [ ] of the inertia grou I P. Then, ρ L/K is a linear ma on V I P P. Note that the characteristic olynomial of this ma is unchanged by conjugation, and so is unchanged by relacing P with any other rime τpτ lying over. Then, the quantity [ ] ) L/K det id ρ P is also indeendent of the choice of rime P lying over. So, we may define the Artin L-function for ρ by the Euler roduct: LL/K, s, ρ) = [ ] ) L/K det id ρ P V I P [ ] As ρ L/K has finite order we may conclude that it is diagonalizable, with root of unity P eigenvalues. Taking logarithms, and writing this in terms of the eigenvalues, we may exlicitly establish convergence roerties. We will not do this, because the only class which we are interested in will be shown in Corollary 4.2. to be equivalent to certain Dirichlet-Hecke L-functions, whose convergence roerties we have already considered. Note that sometimes the Artin L-functions are defined without the factors for the ramified rimes e.g. in [Hei67]). However, it seems better to include these factors. Various functorial roerties of these L-functions with resect to change of reresentation are true in this form, for examle how the L-function behaves under induction of characters. Also, this form allows the Artin L-functions to actually generalize the Dirichlet L-functions, as we shall see: Proosition Let L/K be an abelian extension of number fields, and ρ an irreducible, so one dimensional, reresentation of G = GalL/K). As ρ is one dimensional, view it as an element of HomG, S ) under the identification S C GL C). Let F be the fixed field in L of ker ρ. Then: LL/K, s, ρ) = )) L/K ρ unr. in F/K That each term in the roduct is well defined is art of the conclusion.) Proof. Galois theory gives us the short exact sequence V I P GalL/F ) GalL/K) res GalF/K). So, for F = L ker ρ we have that ρ factors through GalF/K). Note that V = C for our reresentation, so for a rime P of L lying over we have { V I P C ρi P ) = {id} = 0 otherwise 7

18 L-FUNCTIONS AND THE DENSITIES OF PRIMES Then, we may write det id ρ [ ] L/K P V I P ) = ) L/K I P ker ρ otherwise { ρ ANATOLY PREYGEL Note that the I P are conjugate, and ker ρ is a normal subgrou. So, this quantity is indeed well defined with resect to different choices of rimes lying over. In fact, we knew this a riori, for the LHS is well defined with resect to different choices of rimes lying over.) Also, we have I P ker ρ F = L ker ρ L I P unr. in F/K. This roves our result. Corollary Let L/K, ρ, F be as in Pro Under the Artin ma, ρ induces a character χ HomCl m K, S ) for some m. Let f χ be the conductor of χ, and χ the corresonding rimitive character. Then: LL/K, s, ρ) = Lf χ, s, χ) Proof. By Theorem 2.4. the Artin ma induces GalL/K) = I m K /Hm for some m and congruence subgrou H m. So, under the Artin ma ρ induces a character χ HomCl m K, S ). Then, by Pro. 4..2, noting that the F there matches the F here, we have that Lf χ, s, χ) = χ)). By Pro we have that LL/K, s, χ) = unr. in F/K unr. in F/K ρ )) L/K. Note that ker ρ GalL/F ) by construction so we may define ρ HomGalF/K), S ) such that ρ restricts to ρ on GalL/K). Now, note that under the Artin ma ρ induces χ HomCl fχ K, S ) for f χ is the conductor of L/F by comments in the roof of Pro. 4..2). Note that χ is defined by ) F/K χ ) = ρ = ρ L/K Then, by the uniqueness claim from Pro. 4.. we must have χ = χ. This roves our result Extension of zeta functions, non-vanishing of L-functions. We are now ready to move forward towards the crucial art of our roof, showing how zeta functions extend under extension of field and using this to show the non-vanishing of the L-series corresonding to non-rincial characters at s =. 8 ).

19 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Proosition Let L/K be an abelian extension of number fields. Then on their common domain of definition ζ L s) = ρ LL/K, s, ρ) = Lf χ, s, χ) χ = ζ K s) χ χ 0 Lf χ, s, χ) where the first roduct is over all irreducible reresentations equivalently characters) of G = GalL/K), the second is over the corresonding in the sense of Corollary 4.2.) Dirichlet characters χ, and the final is over all such characters which are not the the rincial character equivalently, which do not come from the trivial reresentation). Proof. Let, P denote rimes of K, L resectively. Let e), f), g) be such that = P P g) ) e) so then, f) is the inertial degree of any Pi over ), and note that their roduct is N = [L : K]. Then: ζ L s) = P NormP) s ) = Norm) sf) ) g) Now, by Pro we have LL/K, s, ρ) = ρ ρ = unr. in L ker ρ /K ρi P )=0 ρ ρ ) ) L/K Norm) s ) ) L/K Norm) s Now, we note that the irreducible reresentations of G are one dimensional, and are thus just the characters on G. Furthermore, the characters ) satisfying χi P ) = 0 corresond exactly to characters of G/I P. The image of in this quotient will have order equal to the order L/K of the Frobenius in the residue field extension, that is f). Noting that G/I P = f)g), and alying Pro..2. we get that our revious roduct is equal to = Norm) sf) ) g). This roves the first equality in the statement. The second equality follows by Corollary The final equality follows by noting that for χ equal to the rincial character, we have f χ =, and Lf χ, s, χ) = ζ K s). Proosition Let K, m be as above. Then Lm,, χ) 0 for χ χ 0. 9

20 L-FUNCTIONS AND THE DENSITIES OF PRIMES Proof. Let L/K be the ray class field mod. m. Then, by Pro ζ L s) = ζ K s) χ χ 0 Lf χ, s, χ). ANATOLY PREYGEL By Pro , all quantities may be continued to a neighborhood of s =, so this equality must hold there. Also, by Pro , ζ L and ζ K have simle oles at s =, so the roduct must have neither a ole nor a zero at s =. But, by Pro the Lf χ, s, χ) are analytic there. So, no terms of the roduct have a ole there and so none can vanish there. Remark. Note that we could have develoed all of the results we need without introducing Artin L-functions, sticking to Dirichlet-Hecke L-functions arising from rimitive characters. Class field theory would still be crucial to the argument, arising in Pro The advantage lies in exosing ties to more general theory. From Corollary 4.2. we get that all abelian Artin L-functions are Dirichlet-Hecke L-functions, and so share their analytical roerties. Hecke and Tate showed that these extend analytically to all of C, excet for a ole at s = for the rincial character which corresonds to the trivial reresentation). Then, we may develo a general theory for the functorial roerties of Artin L-functions under change of reresentation, such as direct sum and inducing a reresentation from a subgrou which would yield Pro by letting the subgrou be {id} GalL/K)). Then, a result of Brauer shows that the character of any Galois reresentation is a Z-linear combination of the characters of abelian characters, giving an arbitrary L-function as a quotient of roducts of abelian L-series, which imlies that it can be meromorhically extended to C. Of course, Artin s Conjecture, that they may in fact be analytically extended excet for s = if it contains the trivial reresentation), remains quite oen. 5.. Density. 5. Dirichlet density and the Chebotarev Density Theorem Definition. Let S mseco K ) be a set of finite rimes of K. Then, the Dirichlet density of S is given by S δs) def = lim Norm) s s, + Norm) s where the sum in the denominator is taken over all of mseco K ). We note that the denominator is just ζ K s), which converges for Res) > by Pro. 3.2., and the numerator converges by comarison to it. So, our limit has a chance of being meaningful. Note that we take the limit as s + to avoid having to worry about continuation to to the left of Res) = although by Pro we can do this for the denominator). We can in fact say more: Proosition 5... For S any set of finite rimes of K, if δs) exists then: a) 0 δs) ; b) δs) = lim s + S Norm) s ; logs ) 20

21 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES Proof. Note that the fraction in our limit is always non-negative and is bounded above by. This yields a). Using the Euler roduct for ζ K, we have that for Res) > : log ζ K s) = log Norm) s ) Exanding this latter in a ower series, and re-grouing terms: = Norm) s + Now, we may bound m from below by 2 to get m Norm) sm m 2 m 2 m Norm) sm. by, sum the result geometric series, and bounding the Norm Norm) 2 Res) ζ K 2 Res)). Then, as ζ K s) converges for Res), we have that our left hand term does so for Res) > /2. Viewing it as a Dirichlet series, we see that it gives an analytic function there, and in articular in a neighborhood around s =. So, Norm) s log ζ K s) log s. The last relation follows for ζ K s) has a simle ole at s, so s )ζ K s) is analytic and non-zero around s =. Then, logs )ζ K s) = logs ) + log ζ K s) is analytic at s =, and thus bounded near there. Noting that lim log s + s log s = + so lim s = + Norm) s we get b): δs) = lim s + S Norm) s. logs ) Now, as the denominator gets arbitrary large as s +, changing S in ways which changes the numerator by a bounded amount does not affect δs). More recisely: Proosition Let S, S be two sets of finite) rimes of K. Say S, S, s) = S Norm) s S Norm) s is bounded for s, + ɛ) for some ɛ > 0. Then, δs) = δs ). This is true, in articular in the following cases: a) If S and S differ by a finite number of elements; b) If S and S = S A, where A is the set of rimes having inertial) degree. 2

22 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL Proof. The first statement follows by the comments before the statement of this roosition. Now, if S and S differ by a finite number of elements, then S, S, s) = Norm) s Norm) s S\S S \S has only finite many terms and so is analytic at s = and bounded there, so we may aly the first art of our claim. Also, S, S, s) / A Norm) s. Now, for each rational rime, there are at most [K : Q] rimes lying over it and so at most that many of them not in A. Furthermore, each of those that are not in A have norm at least 2, so: 2s [K : Q]ζ2s) / A Norm) s [K : Q] Now, ζ2s) is analytic on Res) > /2 and in articular at s =, and so bounded in a neighborhood around. So, we may aly the first art of our claim Comarison to natural density. Now, we called the above concet a density. However, there is a more intuitive notion of density: Definition. Let S be as above. For a set T of rimes of K, define π K T, n) def = #{ T : Norm) n}. If T is omitted, then we assume T = mseco K ). Then, the natural density of S is given by δ nat S) def π K S, n) = lim n π K n). While we will not need to use any of this, it is worthwhile to consider the relationshi between these two notions of density. The following justifies calling the δs) a density : Proosition Let S be as above. If δ nat S) exists, then δs) exists and the two are equal. Proof. See [Gol7, 4-]. Note that the other direction is not true. Let K = Q and S = { rime : has leading decimal digit }. Then, using the Prime Number Theorem, it is ossible to show that δs) = log 0 2 while δ nat S) fails to exist as mentioned in [Ser73]). Remark. Note that all the results which we rove below for Dirichlet density are also true for natural density. The roofs require sharer analytic information and generalize the rime number theorem. See [Gol7]. 22

23 ANATOLY PREYGEL L-FUNCTIONS AND THE DENSITIES OF PRIMES 5.3. Density of Frobenius in abelian extensions. We roceed to rove a generalized version of the argument we gave for Theorem.., with the crucial ingredient of the nonvanishing of L, χ) for χ χ 0 rovided by Pro Proosition Let K be a number field, m a modulus, and H m a congruence subgrou grou. If c is a coset of I m /H m, then the set Sc) of rime ideals in c has density δsc)) = [I m :H m ]. Proof. Let G = I m /H m. Note that by Pro..2. Norm) s = G c = G χ HomG,S ) χ HomG,S ) χc ) χ)χc ) Norm) s χ) Norm) s. Then, taking logarithms in the Euler roduct formula of Pro. 3.2., and exanding the ower series for log, we have, for Res) > : t log Lm, s, χ) = χ) Norm) s + m χm ) Norm) sm. We have that m χm ) Norm) sm m 2 m 2 m 2 m Norm) Res)m [K : Q]ζ Q 2 Res)) and as ζ Q 2s) converges on Res) > /2 + δ, our LHS does as well and so defines an analytic function there. So: log Lm, s, χ) χ) Norm) s. Then, we note that for χ χ 0 we have that Lm, s, χ) is analytic at s = and Lm,, χ) 0 Pro , Pro ), so log Lm, s, χ) is analytic in a neighborhood of s =. So, Norm) s χc ) log Lm, s, χ) G G Lm, s, χ 0). c χ HomG,S ) Now, we note that Lm, s, χ 0 ) has a simle ole at s = Pro ), and so Then, we have c log Lm, s, χ 0 ) log for g bounded in a neighborhood of s =. Then, c δs) = lim Norm) s s + logs ) s. Norm) s logs ) = + gs) G = lim s + 23 G ) gs) = logs ) G,

24 L-FUNCTIONS AND THE DENSITIES OF PRIMES ANATOLY PREYGEL where the last equality follows as g is analytic at s = thus bounded in a neighborhood of it, while logs ) gets arbitrarily large. Corollary Let L/K be an abelian extension of number fields, with Galois grou G = GalL/K). Let σ G. Let ) L/K S = { unramified finite rime of K : = σ}. Then, δs) = G. Proof. We know that L/K is the class field for some congruence divisor class grou I/H = G with conductor f. Then, Theorem 2.4. and Pro tells us that S is the set of rimes in some coset c I f /H f, u to a finite number of rimes dividing f. Then, by Pro we have that δs) = [I f : H f ] = I/H = G Reduction of Chebotarev Density Theorem. Now, we may reduce the Chebotarev Density Theorem to the abelian case. In conjunction with the revious section, this will rove it. We follow a roof essentially from [Mac68] though now widesread e.g. in [Gol7, Neu86, Jan96, Neu99]). Theorem 5.4. Chebotarev Density Theorem). Let L/K be a Galois extension of number fields, with Galois grou G = GalL/K). Let σ G, and c σ be the conjugacy class of σ. Let ) L/K S = { finite rime of K : = c σ }. Then, δs) = cσ G. Proof. Now, let H = σ. Let L H be the fixed field of H. Let S = { S : unramified in L/K}. ) Let T = {P rime of L H over S : L/L H = σ, f L H /KP ) = e L H /KP ) = } L/L H is P abelian, so the Artin ma will indeed yield [ a] single element). Let U = {P rime of L over S : L/K = σ}. P Consider the ma U T given by P P O L H. We readily see that its image does lie in T. for P must be unramified and the inertial degrees for P and P O L H will both be H. Furthermore, as H is the decomosition grou of P, we have no slitting from L H to L, so it is injective. Finally, we note that it is surjective, for given a rime P of L H, there will be a rime P of L lying over it; the only things that could revent P from being in U are ramification from K to L H, or change of Frobenius the first is revented by requiring e L H /KP ) =, the second is revented by requiring f L H /KP ) =. Then, by Pro we may ignore the finitely many ramified rimes and the rimes of degree > over Q, so certainly over K), Pro gives us that δt ) =. H Now, consider the ma U S given by P P O K. It is of course surjective. Now, how many elements ma to the same S? Say P is one re-image, then τp mas to 24