MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS ROBERT J. LEMKE OLIVER

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1 MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS ROBERT J. LEMKE OLIVER Abstract. Granville and Soundararajan have recently suggested that a general study of multilicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called retentious view of analytic number theory. Here we study multilicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/Q, we construct a natural class S K of comletely multilicative functions whose values are dictated by Artin symbols, and we show that the only functions in S K whose artial sums exhibit greater than exected cancellation are Dirichlet characters. 1. Introduction and statement of results In a recent series of aers ([1], [3], [4], [5], [6] as a few examles), Granville and Soundararajan have introduced the notion of retentiousness in analytic number theory. The idea of retentiousness is to study generic comlex-valued multilicative functions of modulus bounded by 1 as an alternative to focusing on the zeros of L-functions. In this sense, this hilosohy can be viewed as establishing a framework for the elementary roof of the rime number theorem due to Erdős and Selberg. Indeed, the theory has advanced well beyond that oint there are now retentious roofs of many dee theorems in analytic number theory, including versions of the large sieve inequality [], Linnik s theorem [], and, due to Koukoulooulos [10], a quantitative version of the rime number theorem for rimes in arithmetic rogressions of the same quality as can be obtained by traditional methods and yet retentiousness is still in its nascence. That is, even though they are essentially very basic objects, we still have much to learn about multilicative functions. Arguably the most striking multilicative functions are Dirichlet characters. These are comletely multilicative functions defined on the rimes via congruence relations, yet they also exhibit a strong additive structure eriodicity. One way in which this additional structure manifests itself is by the startling amount of cancellation exhibited by the summatory function. If f(n) is any multilicative function, let S f (X) := n X f(n). Generically, if f() 1, the best cancellation we can exect is what we would find in a random model, which would yield S f (X) = O(X 1/+ɛ ). However, if (n) is a non-rincial Dirichlet character modulo q, we have the much stronger statement that S (X) = O q (1) as X, where the subscrit q indicates that the imlied constant may deend on q. Given this startling, albeit elementary, cancellation exhibited by Dirichlet characters, we may ask the following question. Question 1. Given a comletely multilicative function f(n) such that f() 1 for almost all rimes, if S f (X) X 1/ δ for some fixed δ > 0, must f(n) necessarily come from a Dirichlet character? 1

2 ROBERT J. LEMKE OLIVER Of course, this question is hoelessly vague, but a more recise version can be found in the author s work with J. Jung [8]. Even with a more recise formulation requiring both that f() 1, so that f(n) fits into the classical domain of retentiousness, and, say, that n X f(n) X, so that f(n) is not too small Question 1 aears to be intractable at resent. Not all is lost, however, as we are able to rovide an answer for a certain, natural class of functions, which moreover seems like a not unreasonable lace to look for consiracies akin to the eriodicity of Dirichlet characters. This class of functions will be defined via the arithmetic of number fields, with Dirichlet characters arising from cyclotomic) extensions. Thus, let K/Q be a finite Galois extension, not necessarily abelian, and let rime unramified in K, ( K/Q ( K/Q denote the Artin symbol, defined so that for each rational ) is the conjugacy class in Gal(K/Q) of elements acting like Frobenius modulo for some rime of K dividing ; recall that by the Chebotarev density theorem, each class occurs for a ositive roortion of rimes. We let S K denote the class of comlex-valued comletely multilicative functions f(n) satisfying the following two roerties. First, we require that f() 1 for all rimes, with equality holding if slits comletely, so that f both fits into the context of retentiousness and is of the same size as a Dirichlet character in absolute value. Secondly, generalizing the deendence of () only on the residue class of (mod q), we require f() to deend only on the Artin symbol 1 and are any two unramified rimes such that ( ) ( ) K/Q K/Q =, 1 ( K/Q ). That is, if we must have that f( 1 ) = f( ). We note that if K = Q(ζ m ), the m-th cyclotomic extension, S K includes all Dirichlet characters modulo m, and by taking K to be a nonabelian extension, we can obtain other functions of arithmetic interest which are intrinsically different from Dirichlet characters; see the examles following Theorem 1.1. We are now interested in the following reformulation of Question 1 to the class of functions in S K. Question. Suose f S K is such that S f (X) = O f (X 1/ δ ) as X for some fixed δ > 0. Must f(n) coincide with a Dirichlet character? That is, must f() = () for all but finitely many rimes? Modifying techniques of Soundararajan [1] that were develoed to show that degree 1 elements of the Selberg class arise from Dirichlet L-functions, we are able to answer this question in the affirmative. Theorem 1.1. If f S K is such that S f (X) = O f (X 1/ δ ), then f() = () for all unramified rimes, where is a Dirichlet character of conductor dividing the discriminant of K. Two remarks: First, as mentioned above, the author and Jung [8] recently asked a more general version of Question 1 in their work on retentiously detecting ower cancellation in the artial sums of multilicative functions. It is unfortunate that while Question 1 fits nicely into the retentious hilosohy, the roof of Theorem 1.1 is highly non-retentious, as it relies critically on L-function arguments. However, we still consider this roof to be of merit, as it highlights the interface between retentious questions and techniques relying on L-functions.

3 MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS 3 Second, as the roof will show, the conditions on the class S K are not otimal. We have chosen the definition of S K that we did for aesthetic uroses, but in fact, the class can be exanded in a few ways. First, we do not actually require that f(n) is comletely multilicative, only that f( ) is also determined by the Artin symbol ( K/Q ) and that f( k ), k 3, can be suitably bounded indeendent of. We also need to assume nothing about the size of f() for rimes that do not slit comletely in K. In articular, the condition that f() is determined by the Artin symbol imlies that square root cancellation is still the exected random order, so that the question remains interesting. It would also be nice to comletely remove the restriction on those rimes which slit comletely, but this would likely require greater understanding of a articular extension of the Selberg class. At resent, following techniques used in the roof of Theorem 1.1, it would likely be ossible to allow f() 1 for rimes that slit comletely. However, we consider within this setting the case of f() = 1 to be the most interesting, as this is the only case in which there are f(n) with S f (X) X 1/ δ ; that is, if f() is dictated by Artin symbols and 0 < f() < 1 for rimes that slit comletely in K, then S f (X) X 1/ δ. We also note that, if we do not bound f() at all, then coefficients of Artin L-functions associated with K are also ermitted, and, assuming they are automorhic, they would also likely exhibit more than square root cancellation. We conclude this section with three examles of functions in some S K which we believe to be of arithmetic interest. Examle 1. Let F (x) = x 3 + x x + 1, and let K be the slitting field of F (x), which has Galois grou G = S 3 and discriminant 196 = Let ρ() denote the number of inequivalent solutions to the congruence F (x) 0 (mod ), and define the function f S K by f() = 1 if and ρ() = 0, 0 if or ρ() = 1, 1 if and ρ() = 3. There is a unique Dirichlet character ( ) in S K, which corresonds to the alternating character 11 of S 3, and is given by () =. Alternatively, we can write () in terms of ρ() by 1 if 11 and ρ() = 1, or =, () = 1 if and ρ() = 0 or 3, 0 if = 11. Since f() () for those rimes such that ρ() = 0 or 1 and since such rimes occur a ositive roortion of the time by the Chebotarev density theorem, we should not exect to see more than square root cancellation in the artial sums of f(n) by Theorem 1.1, and indeed, we find the following. X S f (X) S f (X) / X S f (X) /(X/ log 7/6 X)

4 4 ROBERT J. LEMKE OLIVER In the next examle, we discuss the aarent convergence in the fourth column. Examle. The astute reader may object to the above examle by noting that f(n) as constructed has mean 1/6 on the rimes as a consequence of the Chebotarev density theorem. The Selberg-Delange method [13, Chater II.5] redicts that, for a multilicative function g(n) of mean z on the rimes, the summatory function S g (X) should be of the order X(log X) z 1. Γ(z) In articular, for f(n) as in the revious examle, we would exect that S f (X) should have order X/(log X) 7/6, which indeed matches the data more closely (and exlains the fourth column). Notice, however, that the redicted main term is 0 when the mean on the rimes is 0 or 1 (or any non-ositive integer, but recall that we are inside the unit disc). The latter ossibility essentially corresonds to the Möbius function µ(n), so the most interesting case occurs when the mean on the rimes is 0. It is a simle exercise to see that any such function g S K (where K is as in Examle 1) arises as the twist of (n) we must have that g() = ω() for all rimes and some ω satisfying ω = 1. Taking g() = i() for all rimes, we comute the following. X S g (X) S g (X) / X i i i i i i Here, the fact that S g (X) is not O(X 1/ δ ) for some δ > 0 is less aarent than was the case in Examle 1 (there is even more fluctuation than is visible in the limited information above for examle, S g (810000)/ ), but nevertheless, since g(n) does not coincide with a Dirichlet character, Theorem 1.1 guarantees that the artial sums are not O(X 1/ δ ). Examle 3. Let F (x) = x 4 +3x+3, and let K be the slitting field of F (x), which has Galois grou G = D 4 and discriminant = There are five conjugacy classes of G, three of which, each of order two, can be determined by exloiting the quadratic subfields Q( 3) and Q( 7). To distinguish the remaining two conjugacy classes, each of which consists of a single element, we exloit the factorization of F (x) modulo. As in the revious examles, let ρ() denote the number of inequivalent solutions to the congruence F (x) 0 (( (mod ), and additionally define l() to be the air defined by 3 f() = ζ 3, if l() = ( 1, 1), ζ3, if l() = (1, 1), 0, if 1. ), ( 7 1, if l() = (1, 1) and ρ() = 4, 1, if l() = (1, 1) and ρ() = 0, 1, if l() = ( 1, 1), )). We now consider f S K

5 MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS 5 We note that f(n) is neither a Dirichlet character nor its twist each of the three Dirichlet characters ( ) ( 3, 7 ) ( ), and 1 has the same value on the singleton conjugacy classes, and these are the unique characters in S K yet it has mean 0 on the rimes. We find the following. X S f (X) S f (X) / X i i i i i 0.16 As with Examle, we find the fact that S f (X) is not O(X 1/ δ ) to be not entirely clear, yet it is guaranteed to be so. In this case, there is even more fluctuation in the values of S f (X) / X. For examle, when X = , we have that S f (X) / X 0.01, yet when X = , we have that S f (X) / X Thus, without knowledge of Theorem 1.1, it would be difficult to guess the correct order of S f (X), although if one were forced to seculate, O(X 1/ ) would robably be the most reasonable guess. In fact, under the generalized Riemann hyothesis, we have that S f (X) = O ɛ (X 1/+ɛ ) for all ɛ > 0, so that by Theorem 1.1, square root cancellation is the truth in this case. Acknowledgements The author would very much like to thank the anonymous referees, one of whom suggested that an unconditional roof of the theorem may roceed along the lines below.. Proof of Theorem 1.1 We begin by recalling the setu in which we are working. K/Q is a finite Galois extension with Galois grou G := Gal(K/Q), and f S K if and only if f() 1 for all rimes, with equality holding if slits comletely in K, and f( 1 ) = f( ) for all unramified rimes 1 and such that ( ) ( ) K/Q K/Q =. 1 Looking only at unramified rimes, we can therefore regard f as a class function of G, and as such, it can be decomosed [11,. 50] as (.1) f = a, Irr(G) where Irr(G) denotes the set of characters associated to the irreducible reresentations of G and each a C. We remark that, because we have disregarded any information coming from the ramified rimes, we lose all control over the values f assumes on such rimes. In the roof to come, we will show that f() = () for all unramified rimes. To do so, that is, to establish Theorem 1.1, we will show that a = 1 for some one-dimensional and that a = 0 for all other. We do so incrementally, first establishing that each a is, in fact, rational, then, using techniques due to Soundararajan [1] develoed to study elements of the Selberg class of degree 1, we rove the result.

6 6 ROBERT J. LEMKE OLIVER Let L(s, f) denote the Dirichlet series associated to f(n), so that we have f(n) L(s, f) := = ( 1 f() ) 1, n s s n=1 recalling that f(n) is comletely multilicative. By matching the coefficients of s in each Euler factor, the decomosition (.1) then guarantees the Euler roduct factorization L(s, f) = ( L(s, ) a 1 + O( s ) ), valid in the region of absolute convergence R(s) > 1, and where L(s, ) is the Artin L- function associated to the reresentation attached to. In fact, we will need to go further with this factorization. The coefficient of s in the Euler roduct is again essentially a class function of G, so it can be decomosed as a linear combination of the characters, and we obtain (.) L(s, f) = L(s, ) a L(s, ) b A(s), where A(s) is analytic and non-zero in the region R(s) > 1/3. Recall that Artin L-functions factor as roducts of integral owers of Hecke L-functions, so that for each non-trivial, the function L(s, ) is analytic and non-zero in some neighborhood of the region R(s) 1/, and for the trivial character 0, we have that L(s, 0 ) = ζ(s) is again analytic and non-zero in a neighborhood of R(s) 1/, excet at s = 1/. Thus, the roduct L(s, ) b A(s) is analytic and non-zero in some neighborhood of R(s) 1/, excet ossibly at s = 1/. Now, the condition that f(n) = O(X 1/ δ ) n X guarantees that L(s, f) is analytic in the region R(s) > 1/ δ, so by the above, we must have that L(s, ) a is analytic in a neighborhood of R(s) 1/, excet ossibly at s = 1/. In articular, we must have that (.3) ord s=s0 L(s, f) = a ord s=s0 L(s, ) for any s 1/ with R(s) 1/. Recall now that we wish to show that each a is rational. This would follow from (.3) above if there are n := #Irr(G) choices s 1,..., s n such that the matrix M n (s 1,..., s n ) := (ord s=si L(s, j )), 1 i, j n, is invertible over Q. If this is not the case, then there must be integers n, not all zero, such that, for all s 0 1/ with R(s 0 ) 1/, we have n ord s=s0 L(s, ) = 0.

7 MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS 7 To see this, induct on n, the number of characters, noting that the result is obviously true if n = 1. If n > 1, then either there is a choice of {s 1,..., s n 1 } such that the matrix associated to the characters 1,..., n 1 is invertible, or there are such integers n with n, and we may obviously take n n = 0. Assuming that we are in the former case, the matrix (ord s=si L(s, j )), 1 i n 1, 1 j n, has a kernel of dimension one, and if there is no choice of s n such that M n (s 1,..., s n ) is invertible, any linear relation between the columns must lie in this kernel, which yields the desired n. Comleting the L-functions and using the functional equation, the roduct, Λ(s) := Λ(s, ) n, will be analytic and non-vanishing on C, excet ossibly at s = 1/. We note that this is, essentially, an L-function of degree d := n dim and conductor q := qn, where each q is the conductor of L(s, ). Where this breaks from standard definitions of L-functions is in the gamma factor, G(s) := Γ(s, ) n, where Γ(s, ) is the gamma factor for L(s, ), as tyically one does not allow negative exonents. Nevertheless, much of the formalism carries through, and in articular, it is ossible to show, for examle by following Iwaniec and Kowalski [7, Theorem 5.8], that the number of zeros of height u to T is N(T ) = T π log qt d + O(log QT ), (πe) Q d := q n. Since Λ(s) is non-vanishing excet ossibly at s = 1/, there can be no main term, and so we see that d = 0 and q = 1. Moreover, since Λ(s) is entire (excet ossibly at s = 1/) and is of order one, we have the factorization Λ(s) = (s 1/) m e A+Bs, where m Z and A, B C. Since d = 0 and q = 1, by considering the functional equation, first as s with s R, we see that R(B) = 0, and by s = 1/ + it, t, that B = 0. Writing G(s) as ( s G(s) = Γ(s) a Γ ) ( ) b c s + 1 Γ and alying Stirling s formula, we see that log G(s) = a + b + c ( ( 1 + log s log s a + (b + c) ( a + b )) s ) log s + (a + b + c) log π + b log + O( s 1 ). Because Λ(s) = C (s 1/) m, the coefficients of both s log s and s must be zero. The fact that a + b + c = 0 is a restatement of the fact that the degree is zero, but the condition that ( ) 1 + log a + (b + c) = 0

8 8 ROBERT J. LEMKE OLIVER imlies that a = 0 and b = c, since each of a, b, and c is an integer. We thus have that whence G(s) = (s/) b/ (1 + O( s 1 )), L(s, ) n = (s 1/) m (s/) b/ (1 + O( s 1 )). Uon taking the limit as s, we find that b = m = 0, so, in fact, L(s, )n = 1. But this, due to the linear indeendence of characters, cannot haen, so no such n exist, and each of the quantities a must be rational. At this stage, we are now able to rove the theorem. The advantage gained by knowing that each a is rational, is that the function F (s) := L(s, ) a enjoys nice analytic roerties. In articular, aart from a ossible branch along the ray (, 1/], it will be holomorhic. To see this, let k be the denominator of the a, and note that, from (.3), we must have Λ(s, ) ka = (s 1/) m h(s) k for some entire function h(s). Ignoring the branch, F (s) essentially behaves as an L-function of degree a dim. However, we note that this is also the evaluation of f() at a rime that slits comletely in K or, equivalently, at the identity of Gal(K/Q) by (.1). Thus, it is a rational number of absolute value 1, and so is either 1 or 1. However, there are no holomorhic L-functions of negative degree, as can be seen, for examle, by a zero counting argument (which can be modified simly to account for the ossible branch), and so the degree must be 1. Moreover, it is known that a degree 1 element of the Selberg class must come from a Dirichlet L-function, a fact which is originally due to Kaczorowksi and Perelli [9] and was reroved by Soundararajan [1]. However, as before, F (s) does not satisfy the axioms of the Selberg class, as its gamma factor may have negative exonents, so we must modify Soundararajan s roof to our situation. There are only two key comonents in Soundararajan s roof an aroximate functional equation for F (s) and control of the gamma factors on the line R(s) = 1/. The roof of the aroximate functional equation naturally requires the analytic roerties of F (s), and as it may have a branch in our situation, we must modify the roof slightly; we do so in Lemma.1 below. On the other hand, the control over the gamma factors is rovided from our assumtion on the degree, so in articular, the same estimates hold. We state these estimates in (.4) and we give the idea of Soundararajan s roof below, after the roof of Lemma.1. Lemma.1. For any t R such that t and any X > 1, we have that a(n) F (1/ + it) = n 1/+it e n/x + O ( X 1+ɛ (1 + t ) 1+ɛ + X 1/+ɛ e t ). n=1 Proof. Consider the integral I := (1) F (1/ + it + w)x w Γ(w)dw.

9 MULTIPLICATIVE FUNCTIONS DICTATED BY ARTIN SYMBOLS 9 On one hand, relacing F by its Dirichlet series and directly comuting, we find that I is given by a(n) I = n 1/+it e n/x. n=1 On the other hand, moving the line of integration to the left and letting F (w) denote the integrand, we find that 1+ɛ (t+1/)i 1+ɛ+i I = F (1/ + it) + F (w)dw + F (w)dw + F (w)dw, 1+ɛ i 1+ɛ (t 1/)i where C denotes the contour formed by the union of the segments [ 1 + ɛ (t 1/)i, 1/ + ɛ (t + 1/)i], [1/ + ɛ (t + 1/)i, 1/ + ɛ (t 1/)i], and [1/ + ɛ (t 1/)i, 1 + ɛ (t 1/)i]. Notice that the contour C is bounded away from the branch cut of F (s), whence its contribution can be bounded as O(X 1/+ɛ e t ). The contribution from the two vertical contours is handled exactly as in Soundararajan s roof, and yields a contribution of O(X 1+ɛ (1 + t ) 1+ɛ ). As mentioned above, we must also have some control over the gamma factors on the line R(s) = 1/. This is straightforward, as Stirling s formula yields, if G(s) = Γ(s, )a, that there are constants B, C R such that (.4) G(1/ it) G(1/ + it) = e it log ( t πi +ib+ e 4 C it 1 + O(t 1 ) ). (For the reader closely following Soundararajan, we do not have a t ia term because, for each, all µ,j are real.) This is not the most natural reresentation, but it roves to be convenient for the roof. Now, the idea of Soundararajan s roof is to consider, for any real α > 0, the quantities F(α, T ) := 1 α αt αt t it log F (1/ + it)e πeα πi 4 dt and 1 F(α) := lim F(α, T ). T T Armed with Lemma.1, one can evalute F(α) in two ways, either using the functional equation or not. The first method shows that F(α) = 0 unless παcq Z, where q = qa and C is as in (.4), in which case it is, essentially, the coefficient a(παcq ). The second method, on the other hand, shows that F(α) is eriodic with eriod 1, whence πcq Z and the coefficients a(n) are eriodic modulo πcq =: q 0 Z. As remarked above, the roof of this follows Soundararajan s [1] exactly, with the only modification necessary being the relacement of his aroximate functional equation with ours, Lemma.1. In fact, this argument extends to show that, if we modify the definition of the Selberg class to allow rational exonents on the gamma factors and for there to be finitely many lases of holomorhicity, then still, the only degree 1 elements are those coming from the traditional Selberg class. To conclude the roof of Theorem 1.1, we note that since the coefficients F (s) are eriodic modulo q 0 and are also multilicative, we must have that, away from rimes dividing q 0, that C

10 10 ROBERT J. LEMKE OLIVER they coincide with a Dirichlet character q0 (mod q 0 ). Thus, we have that. L(s, ) a = L(s, q0 ), where. = means that equality holds u to a finite roduct over rimes. By linear indeendence of characters, the only way this can haen is if = q0 and a = 1 for some and a = 0 for all others. Moreover, this shows that the Euler factors absorbed into the. = come only from the ramified rimes. This is exactly what we wished to show, so the result follows. References [1] A. Granville. Pretentiousness in analytic number theory. J. Théor. Nombres Bordeaux, 1(1): , 009. [] A. Granville and K. Soundararajan. Multilicative number theory. in rearation. [3] A. Granville and K. Soundararajan. The sectrum of multilicative functions. Ann. of Math. (), 153(): , 001. [4] A. Granville and K. Soundararajan. Decay of mean values of multilicative functions. Canad. J. Math., 55(6): , 003. [5] A. Granville and K. Soundararajan. Large character sums: retentious characters and the Pólya- Vinogradov theorem. J. Amer. Math. Soc., 0(): (electronic), 007. [6] A. Granville and K. Soundararajan. Pretentious multilicative functions and an inequality for the zetafunction. In Anatomy of integers, volume 46 of CRM Proc. Lecture Notes, ages Amer. Math. Soc., Providence, RI, 008. [7] H. Iwaniec and E. Kowalski. Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 004. [8] J. Jung and R. J. Lemke Oliver. Pretentiously detecting ower cancellation. Math. Proc. Camb. Phil. Soc., 154(3): , 013. [9] J. Kaczorowski and A. Perelli. On the structure of the Selberg class. I. 0 d 1. Acta Math., 18():07 41, [10] D. Koukoulooulos. Pretentious multilicative functions and the rime number theorem for arithmetic rogressions. Comos. Math., to aear. [11] J. Neukirch. Algebraic number theory, volume 3 of Grundlehren der Mathematischen Wissenschaften [Fundamental Princiles of Mathematical Sciences]. Sringer-Verlag, Berlin, Translated from the 199 German original and with a note by Norbert Schaacher, With a foreword by G. Harder. [1] K. Soundararajan. Degree 1 elements of the Selberg class. Exo. Math., 3(1):65 70, 005. [13] G. Tenenbaum. Introduction to analytic and robabilistic number theory, volume 46 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Translated from the second French edition (1995) by C. B. Thomas. Deartment of Mathematics and Comuter Science, Emory University, 400 Dowman Dr, Atlanta, GA address: rlemkeo@emory.edu