Multiplicative Toeplitz matrices and the Riemann zeta function
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Hilberdink, T. (2015) Multiplicative Toeplitz matrices and the Riemann zeta function. In: Bringmann, K., Bugeaud, Y., Hilberdink, T. and Sander, J. (eds.) Four faces of number theory. European Mathematical Society, Zurich, pp. 77-122. ISBN 9783037191422 doi: https://doi.org/10.4171/142 Available at http://centaur.reading.ac.uk/50981/
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Reading’s research outputs online Chapter 3 Multiplicative Toeplitz matrices and the Riemann zeta function
Titus Hilberdink
Contents 1 Toeplitz matrices and operators – a brief overview ...... 80 1.1 C.T/, W.T/, and winding number ...... 82 1.2 Invertibility and fredholmness ...... 82 1.3 Hankel matrices ...... 83 1.4 Wiener–Hopf factorization ...... 84 2 Bounded multiplicative Toeplitz matrices and Operators ...... 85 2.1 Criterion for boundedness on l2 ...... 85
2.2 Viewing 'f as an operator on function spaces; the Besicovitch space ...... 89 2.3 Factorization and invertibility of multiplicative Toeplitz operators ...... 95 3 Unbounded multiplicative Toeplitz operators and matrices ...... 99 3.1 First measure – the function ˆ˛.N / ...... 99 2 3.2 Second measure – '˛ on M and the function M˛.T / ...... 109 2 4 Connections to matrices of the form .f .ij=.i; j / //i;j N ...... 117 Bibliography ...... 120
Introduction
In this short course, we aim to highlight connections between a certain class of ma- trices and Dirichlet series, in particular the Riemann zeta function. The matrices we study are of the form
f.1/ f.1 /f.1 /f.1 / 2 3 4 f.2/ f.1/ f.2 /f.1 / 0 3 2 1 f.3/ f.3 /f.1/f.3 / ; ( ) B 2 4 C B f.4/ f.2/ f.4 /f.1/ C B 3 C B : : : : : C B : : : : :: C B C @ A i.e., with entries aij f.i=j/for some function f Q C. They are a multiplica- D W C ! tive version of Toeplitz matrices which have entries of the form aij ai j . For this reason we call them Multiplicative Toeplitz Matrices. D 78 Titus Hilberdink
Toeplitz matrices (and operators) have been studied in great detail by many au- thors. They are most naturally studied by associating with them a function (or ‘sym- bol’) whose Fourier coefficients make up the matrix. With aij ai j , this ‘symbol’ is D 1 n a.t/ ant :tT D n 2 D1X Then properties of the matrix (or rather the operator induced by the matrix) imply properties of the symbol and vice versa. For example, the boundedness of the opera- tor is essentially related to the boundedness of the symbol, while invertibility of the operator is closely related to a.t/ not vanishing on the unit circle. For matrices of the form ( ) we associate, by analogy, the (formal) series f.q/qit; q QC X2 where q ranges over the positive rationals. Note, in particular, that if f is supported on the natural numbers, this becomes the Dirichlet series
f.n/nit: n N X2 ˛ In the special case where f.n/ n , the symbol becomes .˛ it/. It is quite natural then to ask to what extentD properties of these Multiplicative Toeplitz Matrices are related to properties of the associated symbol. Rather surprisingly perhaps, these type of matrices do not appear to have been studied much at all – at least not in this respect. Finite truncations of them have appeared on occasions, notably Redheffer’s matrix [27], the determinant of which is related to the Riemann Hypothesis. Denoting by An.f / the n n matrix with entries f.i=j/if j i and zero otherwise, it is easy to see that j
An.f /An.g/ An.f g/; where f g is Dirichlet convolution; D since the ij th entry on the left product is
n i r i=i TS3 An.f /irAn.g/rj f g f g.d/ D r j D d r 1 j r i d i=j XD Xj j Xj if j i by putting r jd, and zero otherwise. With 1 and denoting the constant j D 1andtheM¨obius functions, respectively, it follows that An.1/An./ In –the D identity matrix. Note also that det An.1/ det An./ 1. Redheffer’s matrix is D D 011 1 000 0 Rn An.1/ An.1/ En; D C 0 1 D C 000 0 B C @ A TS3 Is this intended to be i=j? 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 79 say, where the matrix En has only 1s on the topmost row from the 2nd column on- n wards. Then, with M.n/ r 1 .r/, D D P M.n/ M.n/ 1 00 0 01 0 RnAn./ In 0 : 1 ; D C 0 1 D : : 00 0 B C B C B 00 1 C @ A @ A so that det Rn M.n/. The well-known connection between the Riemann Hypothe- D 1 " sis (RH) and M.n/ therefore implies that RH holds if and only if det Rn O.n2 / D C for every ">0.(Seealso[18] for estimates of the largest eigenvalue of Rn). Briefly then, the course is designed as follows: In ÷1, we recall some basic aspects of the theory of Toeplitz operators, in particular their boundedness and invertibility. In ÷2, we study bounded multiplicative Toeplitz operators. This is partly based on some of Toeplitz’s own work [32], [33] and recent results from [15]and[16], but we also present new results, mainly in ÷2. Thus Theorem 2.1 is new, generalising Theorem 1.1 of [16], which in turn is now contained in Corollary 2.2. Also Subsection 2.2 and parts of 2.3 are new.
Preliminaries and Notation
p (a) The sequence spaces l .1 p< ) consist of sequences .an/ for which p 1 n1 1 an converges. They are Banach spaces with the norm D j j 1=p P 1 p .an/ p an : k k D j j n 1 XD The space l 1 is the space of all bounded sequences, equipped with the norm p .an/ supn N an . We shall also use l .QC/, which is the space of se- k k1 D 2 j j p quences aq where q ranges over the positive rationals such that q aq < , with analogous norms and also for p . j j 1 2 2 D1 P l and l .QC/ are Hilbert spaces with the inner products
1 a; b anbn and a; b aqbq; h iD h iD n 1 q QC XD X2 respectively. (b) Let T z C z 1 – the unit circle. We denote by L2.T/ the space of square-integrableDf 2 Wj functionsjD g on T. L2.T/ is a Hilbert space with the inner product and corresponding norm given by
1 2 f; g f g f.ei/g.ei/d; f f 2: h iD D 2 k kD j j ZT Z0 sZT 80 Titus Hilberdink
The space L1.T/ consists of the essentially bounded functions on T with norm 2 f denoting the essential supremum of f . (Strictly speaking, L and L1 consistk k1 of equivalence classes of functions satisfying the appropriate conditions, with two functions belonging to the same class if they differ on a set of measure zero.) n 2 Let n.t/ t for n Z.Then. n/n Z is an orthonormal basis in L .T/ 2 D 2 22 and L .T/ is isometrically isomorphic to l .Z/ via the mapping f .fn/n Z, 7! 2 where fn are the Fourier coefficients of f , i.e.,
fn f; n f n: Dh iD ZT (c) A linear operator ' on a Banach space X is bounded if 'x C x for all x X. In this case the operator norm of ' is defined to bek k k k 2 'x ' sup k k sup 'x : k kDx X;x 0 x D x 1 k k 2 ¤ k k k kD The algebra of bounded linear operators on X is denoted by B.X/.
(d) An infinite matrix A .aij / induces a bounded operator on a Hilbert space H if there exists ' B.H/Dsuch that 2 aij 'ej ;ei ; Dh i where .ei / is an orthonormal basis of H. Note that not every infinite matrix induces a bounded operator, and it may be difficult to tell when it does. (e) For the later sections we require the usual O; o; ; notation. Given f; g de- fined on neighbourhods of with g eventually positive, we write f.x/ 1 D o.g.x// (or simply f o.g/) to mean limx f.x/=g.x/ 0, f.x/ O.g.x// to mean f.x/D Ag.x/ for some constant!1 A and allDx sufficientlyD j j large, and f.x/ g.x/ to mean limx f.x/=g.x/ 1. !1 D The notation f g means the same as f O.g/, while f . g means f.x/ .1 o.1//g.x/ . D C
1 Toeplitz matrices and operators – a brief overview
Toeplitz matrices are matrices of the form
a0 a 1 a 2 a 3 a1 a0 a 1 a 2 T 0 a2 a1 a0 a 1 1 ; (3.1) D a a a a B 3 2 1 0 C B : : : : : C B : : : : :: C B : : : : C @ A 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 81 i.e., T .tij /,wheretij ai j . They are characterised by being constant on diagonals.D D For a Toeplitz matrix, the question of boundedness of T was solved by Toeplitz.
Theorem 1.1 (Toeplitz [32]). The matrix T induces a bounded operator on l 2 if and only if there exists a L1.T/ whose Fourier coefficients are an .n Z/.Ifthisis the case, then T 2 a . 2 k kDk k1 We refer to the function a as the ‘symbol’ of the matrix T , and we write T.a/. Sketch of Proof. For a L2.T/, the multiplication operator 2 M.a/ L2.T/ L2.T/; f af W ! 7! is bounded if and only if a L1.T/. If bounded, then M.a/ a . The matrix 2 k kDk k1 representation of M.a/ with respect to . n/n Z is given by 2
M.a/ j ; i a j ; i a j i a i j ai j ; h iDh iD D D ZT ZT i.e., by the so-called Laurent matrix
a0 a 1 a 2 a 3 a 4 0 a1 a0 a 1 a 2 a 3 1 L.a/ : (3.2) WD B a2 a1 a0 a 1 a 2 C B C B a3 a2 a1 a0 a 1 C B C B C @ A The matrix for T is just the lower right quarter of L.a/. We can therefore think of T as the compression PL.a/P ,whereP is the projection of l 2.Z/ onto l 2 l 2.N/. D An easy argument shows that T is bounded if and only if a L1,andthen T L.a/ a . 2 k kD k kDk k1 Hardy space. Let H 2.T/ denote the subspace of L2.T/ of functions f whose Fourier coefficients fn vanish for n<0.LetP be the orthogonal projection of 2 2 L onto H , i.e., P. n Z fn n/ n 0 fn n. The operator f P.af/ has 2 D 2 7! matrix representation (3.1). For, with j 0 (so that j H .T/), P P 2
T.a/ j ; i P.a j / i P an n j i an j n i ai j h iD D C D D ZT ZT n Z n 0 ZT X2 X if i 0, and zero otherwise. Hence, we can equivalently view T.a/as the operator T.a/ H 2.T/ H 2.T/; f P.af/: W ! 7! 82 Titus Hilberdink
1.1 C.T/, W.T/, and winding number Let C.T/ denote the space of con- tinuous functions on T.Fora C.T/ such that a.t/ 0 for all t T,wede- note by wind.a; 0/ the winding2 number of a with respect¤ to zero. More2 generally, wind.a; / wind.a ;0/ denotes the winding number with respect C.For D 2 example, wind. n;0/ n. The Wiener AlgebraDis the set of absolutely convergent Fourier series:
1 1 W.T/ an n an < : D ( W j j 1) X1 X1 Some properties: (i) W.T/ forms a Banach algebra under pointwise multiplication, with norm
1 a W an : k k WD j j X1 (ii) (Wiener’s Theorem) If a W and a.t/ 0 for all t T,thena 1 W . 2 ¤ 2 2 (iii) If a W.T/ has no zeros and wind.a; 0/ 0,thena eb for some b W.T/. 2 D D 2 We have 2 W.T/ C.T/ L1.T/ L .T/: 1.2 Invertibility and fredholmness Let A be a bounded operator on a Banach space X. TS2 (i) A is invertible if there exists a bounded operator B on X such that AB BA 1 D D I . As such, B is the unique inverse of A, and we write B A .Thespectrum of A is the set D .A/ C I A is not invertible in X : Df 2 W g The kernel and image of A are defined by Ker A x X Ax 0 ; Im A Ax x X : Df 2 W D g Df W 2 g (ii) The operator A is Fredholm if ImA is a closed subspace of X and both KerA and X=ImA are finite-dimensional. As such, the index of A is defined to be Ind A dim Ker A dim .X=Im A/: D For example, T. n/ is Fredholm with Ind T. n/ n. D Equivalently, A is Fredholm if it is invertible modulo compact operators; that is, there exists bounded operator B on X such that AB I and BA I are both compact. The essential spectrum of A is the set
ess.A/ C I A is not Fredholm in X : Df 2 W g Clearly ess.A/ .A/. Note that A invertible implies A is Fredholm of index zero. For Toeplitz operators, the converse actually holds (see [3], p. 12).
TS2 Please check the range of the following enumeration. 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 83
1.3 Hankel matrices These are matrices of the form
a1 a2 a3 a4
a2 a3 a4 a5
H 0 a3 a4 a5 a6 1 ; (3.3) D a a a a B 4 5 6 7 C B : : : : : C B : : : : :: C B : : : : C @ A i.e., H .hij /,wherehij ai j 1. They are characterised by being constant on cross diagonals.D The boundednessD C of H was solved by Nehari, and the compactness of H by Hartman.
Theorem 1.2 ([26], [12]). The matrix H generates a bounded operator on l 2 if and only if there exists b L1.T/.with Fourier coefficients bn/ such that bn an for n 1. Furthermore,2 the operator H is compact if and only if b C.T/. D 2 We refer to the function a as the ‘symbol’ of the matrix H, and we write H.a/. Given a function a defined on T,leta be the function Q a.t/ a.1=t/ .t T/: Q D 2 Proposition 1.3. For a; b L .T/, 2 1 T.ab/ T.a/T.b/ H.a/H.b/ D C Q H.ab/ H.a/T.b/ T.a/H.b/: D Q C Proof. The matrix L.a/ in (3.2) is of the form
T.a/ H.a/ L.a/ Q : D H.a/ T.a/ Since L.ab/ L.a/L.b/, the result follows by multiplying the 2 2 matrices. D As a special case, we see that T.abc/ T.a/T.b/T.c/ for a H ;b D 2 1 2 L1;c H 1. The space H 1 is defined analogously to L1.(T.a/ is upper- triangular2 and T.c/is lower-triangular.) By Theorem 1.2,ifa; b C.T/,thenH.a/H.b/ is compact, so that T.ab/ 2 Q 1 T.a/T.b/is compact. In particular, if a has no zeros on T, we can take b a C.T/.ThenT.ab/ T.1/ I ,soT.a/ is invertible modulo compact operatorsD 2 D D 1 (i.e., Fredholm) with ‘inverse’ T.a /. This type of reasoning leads to:
Theorem 1.4 (Gohberg [7]). Let a C.T/.ThenT.a/is Fredholm if and only if a has no zeros on T,inwhichcase 2
Ind T.a/ wind.a; 0/: D 84 Titus Hilberdink
Hence T.a/ is invertible if and only if a has no zeros on T and wind.a; 0/ 0. Equivalently, since T. a/ I T.a/for C, we have D D 2 ess.T .a// a.T/; D .T.a// a.T/ C a.T/ wind.a; / 0 : D [f 2 n W ¤ g Sketch of Proof. We have seen that a 0 on T implies T.a/ is Fredholm. In this ¤ case, let wind.a; 0/ k.Thena is homotopic to k , and (since the index varies continuously) D Ind T.a/ Ind T. / k wind .a; 0/: D k D D For the converse, suppose T.a/ is Fredholm with index k,buta has zeros on T. Then a can be slightly perturbed to produce two functions b; c W.T/ without zeros such that a b and a c are as small as we please,2 but wind.b; 0/ and wind.c; 0/kdiffer byk1 one. Ask the indexk1 is stable under small perturbations, T.b/and T.c/are Fredholm with equal index. But Ind T.b/ wind.b; 0/ and Ind T.c/ wind.c; 0/ (by above), so wind.b; 0/ wind.c; 0/ D0 — a contradiction. D D 1.4 Wiener–Hopf factorization Since W.T/ C.T/, Theorem 1.4 applies to W.T/. However, for Wiener symbols we can obtain a quite explicit form for the inverse when it exists. This is because Wiener functions can be factorized. Denote by W and W the subspaces of W consisting of functions C
1 n 1 n ant and ant t T 2 n 0 n 0 XD XD respectively, where an < . j j 1 Theorem 1.5 (Wiener–HopfP factorization). Let a W.T/ such that a has no zeros, and let wind.a; 0/ k. Then there exist a W 2and a W such that D 2 C 2 C a k a a : D C b Proof. We have wind.a k;0/ 0.Soa k e for some b W .Butb b b ,whereb W and bD W . Hence, D writing a eb2and a ebDC gives C C 2 C 2 C D C D b bC a k e e a a : D D C Theorem 1.6 (Krein [21]). Let a W.T/.ThenT.a/ is Fredholm if and only if a has no zeros on T,inwhichcase 2 Ind T.a/ wind.a; 0/: D In particular, T.a/is invertible if and only if a has no zeros on T and wind.a; 0/ 0. In this case D 1 1 1 T.a/ T.a /T .a /; D C where a a a is the Wiener–Hopf factorization of a. D C 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 85
Proof of second part. Note that if a W ,thenH.a/ 0, while if a W ,then H.a/ 0. Suppose a has no zeros on2 Tand wind.a; 0/D 0.Thena factorizes2 C as a Q aDa with a W . Applying Proposition 1.3 withDa and a in turn gives D C ˙ 2 ˙ C 1 1 1 1 T.a /T .a / T.a a / I T.a a / T.a /T .a /; D D D D 1 1 1 1 T.a /T .a / T.a a / I T.a a / T.a /T .a /; C C D C C D D C C D C C 1 1 so that T.a / are invertible with T.a / T.a /.ButalsoT.a/ T.a a / ˙ ˙ D ˙ 1 D1 C1 D T.a /T .a / (by Proposition 1.3). Hence T.a/ T.a / T.a / 1 C 1 D C D T.a /T .a /. C
2 Bounded multiplicative Toeplitz matrices and Operators
2.1 Criterion for boundedness on l 2 Now we consider the linear operators induced by matrices of the form . /, regarding them as operators on sequence spaces, in particular l 2. For a function f Q C on the positive rationals, we define W C ! m f.q/ lim f ; whenever this limit exists: D N n q QC !1 m; n N X2 .m;X n/ D 1 We shall sometimes abbreviate the left-hand sum by f.q/. We say that f q 2 l 1.Q / if C P f.q/ j j q QC X2 converges. In this case, the function
F.t/ f.q/qit t R D 2 q QC X2 exists and is uniformly continuous on R. Note that, for >0, TS3
T 1 it f.q/; if q QC; lim F.t/ dt D 2 (3.4) T 2T T D 0; otherwise. !1 Z 1 Theorem 2.1. Let f l .Q / and let ' denote the mapping .an/ .bn/ where 2 C f 7!
1 n bn f am: D m m 1 XD TS3 Please check: due to reasons of a coherent design of all papers, we had to add the commas here as well as in similar formulas. 86 Titus Hilberdink
2 Then 'f is bounded on l with operator norm
it 'f sup f.q/q F : k kDt R Dk k1 2 ˇq QC ˇ ˇ X2 ˇ ˇ ˇ ˇ ˇ 2 Proof. We shall first prove that 'f is bounded on l , showing 'f F in the process, and then show that F is also a lower bound. k kk k1 For q Q and N N klet k1 2 C 2 N .N / q 1 q b f am and bq f am: q D m D m m 1 m 1 XD XD .N / Note that bq bq as N for every q QC, whenever an is bounded. We have the following! formulae:!1 2
T N 2 N 1 it .N / lim F.t/ ann dt anbn (3.5) T 2T T D !1 Z ˇn 1 ˇ n 1 ˇXD ˇ XD T ˇ N ˇ2 1 ˇ itˇ .N / 2 lim F.t/ ann dt bq : (3.6) T 2T T D j j !1 Z ˇ n 1 ˇ q QC ˇ XD ˇ X2 ˇ ˇ (These hold for an bounded.)ˇ To prove theseˇ expand the integrand in a Dirichlet series. For the first formula we have N T 2 T it 1 it 1 n F.t/ ann dt aman F.t/ dt 2T D 2T m Z T ˇn 1 ˇ m;n N Z T ˇXD ˇ X ˇ ˇ N ˇ ˇ n .N / amanf anb ! m D n m;n N n 1 X XD as T . For the second formula, note first that !1 N it it F.t/ ann f.q/an.qn/ D n 1 q QC;n N XD 2 X r it .N / it f an r b r ; D n D r r QCn N r QC X2 X X2 the series converging absolutely. Thus
N 1 T 2 1 T q it F.t/ a nit dt b.N /b.N / 1 dt b.N / 2 2T n q1 q2 2T q q T D C T 2 ! Cj j Z ˇ n 1 ˇ q1;q2 Q Z q Q ˇ XD ˇ X2 X2 ˇ ˇ as T ˇ . ˇ !1 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 87
Since F.t/ F ,wehave j jk k1 T N 2 N .N / 2 2 1 it 2 2 bq F lim ann dt F an : j j k k1 T 2T T Dk k1 j j q QC !1 Z ˇn 1 ˇ n 1 X2 ˇXD ˇ XD ˇ ˇ 2 .Nˇ / 2 ˇ 2 2 Thus if a .an/ l ,wehave n1 1 bn F a for every N . Letting D 2 2 D j j k2 k1k k N shows that .bn/ l too (indeed .bq/ l .QC/), and so 'f is bounded on l 2,!1 with 2 P 2 'f F : k kk k1 Now we need a lower bound. By Cauchy–Schwarz, 2 1 1 2 1 2 anbn an bn : j j j j ˇn 1 ˇ n 1 n 1 ˇXD ˇ XD XD ˇ ˇ 2 Thus 'f n1 1 aˇnbn for everyˇ a .an/ l with a 1. Choose an as follows:k letkjN N (toD be determinedj later)D and put2 k kD 2P it n an for n N , and zero otherwise. D d.N/ j 2 Here d.N/ is the numberp of divisors of N . Thus .an/ l and a 1. With this choice, 2 k kD 1 n it .N / bn f m . b / D m D n d.N/ m N Xj and so p it 1 1 it n it 1 n n anbn n f m f D d.N/ m D d.N/ m m n 1 n N m N m;n N XD Xj Xj Xj 1 it f.q/q Sq.N /; D d.N/ q QC X2 where Sq.N / 1: D m; njN n mXD q k n k Put q l ,where.k; l/ 1.Thenm l if and only if ln km.Since.k; l/ 1, this forcesD k n and l m. So,D for a contributionD to the sum, weD need k; l N , i.e., klDN . Suppose thereforej thatj kl N .Then j j j Sq.N / 1 1mrl;n rkTS4 with r N D D D D 2 m; njN rk;rl N lnXD km Xj N 1 d : D D kl r N Xj kl
TS4 Please check if the replacement of m rk;n rl with m rl;n rk is correct. D D D D 88 Titus Hilberdink
Writing q kl whenever q k in its lowest terms, gives j jD D l
1 it d.N= q / anbn f.q/q j j : (3.7) D d.N/ n 1 q 2 QC XD XjqjjN The idea is now to choose N in such a way that it has all ‘small’ divisors while d.N= q / j j is close to 1 for all such small divisors q .TakeN of the form d.N/ j j
˛p log P N p ; where ˛p . D D log p p P Y h i Thus every natural number up to P is a divisor of N .Everyq such that q N is of ˇp j jj the form q p P p (0 ˇp ˛p), so that j jD Q d.N= q / ˇp j j 1 : d.N/ D ˛p 1 p P Y C If we take q log P ,thenpˇp log P for every prime divisor p of q . j j log log P j j Hence, for such p, ˇp and ˇp 0 if p> log P . Thus p 2 log p Dp d.N= q / ˇp p log log P j j 1 1 d.N/ D ˛p 1 2 log P p plog P C p plog P Y Y log log P .plog P/ 1 ; D 2 log P where .x/ is the number of primes up to x.Since.x/ O. x /, it follows that D log x for all P sufficiently large, the RHS above is at least A 1 log P for some constant A. p n2 Let ">0. Then there exists n such that f.q/ <". Choose P e 0 0 q >n0 j j j j so that log P n . Then the modulus of the sum in (3.7) can be made as close to 0 P F as we please by increasing P , for it is at least k k1 p A f.q/qit f.q/ f.q/ log P j j j j ˇ q plog P ˇ q plog P q >plog P ˇj jX ˇ j jX j j X ˇ ˇ pA ˇ f.q/qit ˇ f.q/ 2 f.q/ log P j j j j ˇq QC ˇ q QC q >plog P ˇ X2 ˇ X2 j j X ˇ A ˇ p > ˇF.t/ 0ˇ 2"; j j log P p 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 89 where " can be made as small as we please by making P large. Since this holds for any t, we can choose t to make F.t/ as close as we like to F . Hence 'f F and so we must have equality. k k1 k k k k1 In the special case where f 0, the supremum of F.t/ is attained when t 0, j j D in which case F F.0/ f 1;QC . Thus: k k1 D Dk k 2 Corollary 2.2. Let f QC C such that f 0.Then'f is bounded on l if and 1 W ! only if f l .Q /,inwhichcase ' f C . 2 C k f kDk k1;Q Examples. Take f.n/ n ˛ for n N, and zero otherwise. Then F.t/ .˛ it/. D 2 D We shall denote 'f by '˛ in this case. Applying Corollary 2.2, we see that '˛ is bounded on l 2 if and only if ˛>1, and the norm is .˛/.
2.2 Viewing 'f as an operator on function spaces; the Besicovitch space We can view 'f as an operator on functions rather than sequences. For this we need to construct the appropriate spaces. Let A denote the set of trigonometric polynomials; i.e., the elements of A are all finite sums of the form n ik t ake ; k 1 XD where a C and R. The space A has an inner product and a norm given by k 2 k 2 1 T 1 T f; g lim f g and f f; f lim f 2: h iDT 2T T k kD h iDsT 2T T j j !1 Z p !1 Z Now let B2 (Besicovitch space) denote the closure of A with respect to this inner 2 2 product; i.e., f B if f fn 0 as n for some fn A.WeturnB into a Hilbert space by2 identifyingk f k!and g whenever!1f g 0.(See[2 2], Chapter II.) k kD Now write .t/ it (>0;t R)andletf./denote the Fourier coefficient D 2 O T T 1 1 it f./ lim f lim f.t/ dt where it exists: O D T 2T T D T 2T T !1 Z !1 Z F R C Denote by the space of locally integrable f such that f./O exists for all >0. W ! (a) Fourier coefficients and series For f B2, the Fourier coefficients f./ex- 2 O ist and f./O is non-zero on at most a countable set, say n n N. The function int f g 2 f has the .formal/ Fourier series n 1 f.n/e . O (b) Uniqueness f; g B2 have the same Fourier series if and only if f g 0. 2 P k kD 90 Titus Hilberdink
(c) Parseval For f B2, 2 1 T f lim f 2 f./2; (3.8) k kDT s2T T j j D j O j !1 Z sX and, more generally, for f; g B2, 2 1 T f; g lim f g f./g./: h iDT 2T T D O O !1 Z X 2 2 (d) Riesz–Fischer Theorem Given n R and an l , there exists f B int 2 2 2 such that f.t/ ane . n 1 2 F F (e) Criterion for membershipP in B : With as before, if f and Parseval’s identity (3.8) holds, then f B2. 2 2 Indeed, the set of for which f./O 0 is necessarily countable and we may ¤ 2 2 write this as 1;2;::: with k1 1 f.O k/ f TS5 .Letfn.t/ fik t g D j j Dk k D k n f.k/e .Then O P 2 2 2 2 P f fn f fn f./ 0 as n . k k Dk k k k D j O j ! !1 k>nX
2 2 The analogues of the Hardy and Wiener spaces: , , C , . BQC BN WQ WN (a) Let B2 denote the subspace of B2 of functions with exponents log q QC D for some q Q . Alternatively, start with the subset of A consisting of finite 2 C trigonometric polynomials of the form aq q,whereq ranges over a finite subset of QC, and take its closure. 2 2 P (b) Let BN denote the subspace of B of functions with exponents log n for some n N. This is the analogue of the Hardy space. D 2 2 C (c) Let WQ denote the set of all absolutely convergent Fourier series in BQC ;i.e.
W C c.q/ q c.q/ < : Q D W j j 1 q QC q QC n X2 X2 o This is the analogue of the Wiener algebra. As we saw earlier, if TS6
f c.q/ q W C ; D 2 Q q QC X2
then f.q/ c.q/. With pointwise addition and multiplication, W C becomes O D Q an algebra. Further, WQC becomes a Banach algebra with respect to the norm
f W f.q/: k k D j O j q QC X2 it Analogously, let WN denote the set of absolutely convergent series n1 1 ann . D TS5 Please check if the replacement of f 2 with f 2 is correct. k kB k k P TS6 The following formula had to be detached from the text because of typesetting reasons. Please check. 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 91
2 2 BQC has . q/q QC as an orthonormal basis while BN has . n/n N as an orthonor- 2 2 2 2 2 Q 2 mal basis. The spaces BQC and BN are isometrically isomorphic to l . C/ and l , respectively, via the mappings
f f.q/ q QC and f f.n/ n 1: 7! f O g 2 7! f O g
2 2 The operator M.f /. Let P be the projection from BQC to BN;thatis,
P c.q/qit c.n/nit: D q QC n N X2 X2 2 2 For f W C , we define the operator M.f / B B by g P.fg/. 2 Q W N ! N 7! The matrix representation of M.f / w.r.t. n n N is the multiplicative Toeplitz f g 2 matrix .f.i=j//i;j 1. Indeed, if f q f.q/ q ,then O D O P P.f j / P f.q/ O q j P f.q/ O qj D q D q X X 1 P f.q=j/ q f.n=j/ n: D O D O q n 1 X XD Hence
1 i M.f / j ; i P.f j /; i f.n=j/ n; i f : h iDh iD O h iD O j n 1 XD In terms of the operator ', 1 M.f / 'f : D O Interlude on .s/. Since this work concerns connections to Dirichlet series and the Riemann zeta function in particular, we recall a few relevant facts regarding .s/. The Riemann zeta function is defined for s>1by the Dirichlet series < 1 1 .s/ : D ns n 1 XD In this half-plane .s/ is holomorphic and there is an analytic continuation to the whole of C except for a simple pole at s 1 with residue 1. Furthermore, .s/ satisfies the functional equation D .s/ .s/.1 s/ D where 1 s s 1 . / s s 1 s 2 2 2 .s/ 2 .1 s/ sin s : D 2 D .2 / 92 Titus Hilberdink
The connection to prime numbers comes from Euler’s remarkable product formula 1 .s/ s>1 1 p s D p < Y The order of .s/. Considering . it/ as a function of the real variable t for fixed (but arbitrary) , it is known thatC for t large j j def A .t/ . it/ O. t / for some A: D C D j j The infimum of such A is the order of and is called the Lindelof¨ function; i.e., ./ inf A . it/ O. t A/ : D f W C D j j g From the general theory of functions it is known that the Lindel¨of function is convex and decreasing. Since is bounded for >1,but 0, it follows that ./ 0 for 1. By the functional equation and continuity of6! we then have D 0; if 1; ./ 1 (Þ) D ; if 0. 2 For 0<<1,thevalueof./ is not known, but it is conjectured that the two 1 line segments in (Þ) above extend to 2 . This is the Lindelof¨ Hypothesis.Itis equivalent to the statement that D .1 it/ O.t"/ for every ">0: 2 C D The Lindel¨of Hypothesis is a major open problem TS7 and is a consequence of the Riemann Hypothesis, which states that .s/ 0 for >1 . ¤ 2
Upper and lower bounds for . Let
Z .T / max . it/ : D 1 t T j C j j j (The restriction t 1 is only added to avoid problems for the case 1.) The following resultsj holdj for large T . D
(a) Z .T / ./ for >1. ! 2 (b) For 1, unconditionally it is known that Z1.T / O..log T/3 /, while on RH D D
Z1.T / . 2e log log T: On the other hand, Granville and Soundararajan [11] proved that
Z1.T / e .log log T log log log T log log log log T/; C for some arbitrarily large T . They further conjectured that it equals the above with an O.1/ term instead of the quadruple log-term.
TS7 Note from publisher: Paper by Albeverio and Cheng (proof of the Lindelf Hypothesis)? 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 93
1 a (c) For 2 <<1, unconditionally one has Z .T / O.T / for various a>0, while on RH D 2 2 .log T/ log Z .T / A .1 /log log T for some constant A. Montgomery [25] showed that
1 1=2 .log T/ log Z .T / p 20 .log log T/ and, using a heuristic argument, he conjectured that this is (apart from the con- stant) the correct order of log Z .T /. In a recent paper (see [22]), Lamzouri suggests that 1 .log T/ log Z .T / C./ .log log T/ 2 1 with C./ G1./ .1 / ,where D 2n 1 1 .u=2/ du G1.x/ log : D .nŠ/2 u1 1=x Z0 n 0 ! C XD
1 32 c 0:156:: (d) For unconditionally one has Z 1 .T / O.T 205 .log T/ / O.T / D 2 2 D D (see [17]), while on RH log T log Z 1 .T / A 2 log log T for some constant A. On the other hand, it is known that
log T log Z 1 .T / c 2 slog log T (see [1], [29]). Using a heuristic argument, Farmer et al. ([6]) conjectured that
1 log Z 1 .T / log T log log T: 2 r2
1 (e) For <2 , the functional equation for .s/ reduces the problem to the case 1 >2 . So, for example,
1 T 2 Z .T / .1 / for <0. 2 94 Titus Hilberdink
Mean values. For . 1 ;1/, the mean-value formula 2 2 T 1 2 1 1 lim . it/ dt 2 .2/; T 2T T j j D n D !1 Z n 1 XD is well-known (see [31]). Furthermore,
T 2 4 1 4 1 d.n/ .2/ lim . it/ dt 2 : T 2T T j j D n D .4/ !1 Z n 1 XD For higher powers, however, present knowledge is very patchy. It is expected that the above formulas extend to all higher moments, i.e.,
T 2 1 2k 1 dk.n/ lim . it/ dt 2 : T 2T T j j D n !1 Z n 1 XD This is equivalent to the Lindel¨of Hypothesis. Examples. 2 2 2 1 (a) The mean values for and imply that ; BN for . 2 ;1/. Note that 2 2 2 2 this also implies B . j j 2 QC For higher powers, however, only partial results are known. For example, it is k 2 1 known that BN if .1 k ;1/. Slightly better bounds are available, especially for particular2 values2 ofk, but it is expected that much more holds, k 2 1 namely: BN for every k N and all . 2 ;1/. This is (equivalent to) the Lindel¨of Hypothesis.2 2 2 an bn (b) Let g.s/ n1 1 ns and h.s/ n1 1 ns be two Dirichlet series which con- D D D D verge absolutely for s>0 and s>1, respectively. Let ˛>0 and ˇ>1 < < and put f.t/P g.˛ it/h.ˇ it/P.Thenf W C with D C 2 Q 1 1 a b f.q/ md nd for q m with .m; n/ 1. O D m˛nˇ d ˛ ˇ D n D d 1 C XD We can prove this by multiplying out the series for g.˛ it/ and h.ˇ it/.We have C
1 a 1 b a b m it f.t/ m n m n D m˛ it nˇ it D m˛nˇ n m 1 n 1 C m;n 1 XD XD X 1 ambn m it 1 1 amd bnd m it D m˛nˇ n D d ˛ ˇ m˛nˇ n d 1 m; n 1 d 1 C m; n 1 XD .m;X n/ D d XD .m;X n/ D 1
1 1 a b m it md nd : D m˛nˇ d ˛ ˇ n m; n 1 d 1 C .m;X n/ D 1 XD 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 95
Further Properties of WQC and WN. Notation. For a unital Banach algebra A, denote by GA the group of invertible elements of A, and by G0A, the connected component of GA which contains the identity. If A is commutative, then
b a G0A a e for some b A. 2 ” D 2
(a) Wiener’s Theorem for WQC and WN (see [13], Theorems 1 and 2):
Let f W C .Then1=f W C if and only if infR f >0; i.e., GW C 2 Q 2 Q j j Q D f W C infR f >0: f 2 Q W j j g it an Let f.t/ n1 1 ann WN and put F.s/ n1 1 ns for s 0.Then D D 2 D D < 1=f WN if and only if there exists ı>0such that F.s/ ı for all s 0. 2 P Pj j < The example f.t/ 2it shows that the condition f.t/ ı>0for all t R is D j j 2 not sufficient for 1=f WN. 2 1 (b) Let f GW C . Then the average winding number w.f /,definedby 2 Q arg f.T/ arg f. T/ w.f / lim D T 2T !1 exists, and w.f / log q for some q Q (see [19], Theorem 27). D 2 C It is easy to see that (i) w.fg/ w.f / w.g/, and (ii) w. q/ log q. D C D g (c) G0WN GWN;i.e.forf WN,wehave1=f WN if and only if f e for D 2 2 D some g WN 2 2.3 Factorization and invertibility of multiplicative Toeplitz operators The analogue of the factorization T.abc/ T.a/T.b/T.c/ for a H 1;b L ;c H for Toeplitz matrices holds forD Multiplicative Toeplitz matrices.2 2 1 2 1
Theorem 2.3. Let f WN, g W C , and h WN.Then 2 2 Q 2 M.fgh/ M.f /M.g/M.h/: D
Proof. We show the matrix entries agree. By Proposition 2.1, fgh W C and 2 Q
M.f /M.g/M.h/ M.f /ikM.g/klM.h/lj ij D k;l 1 X i k l f g h D O k O l O j k; l 1 ijkXand j jl 1 mi n f g h : D O m O nj O 1 m;n 1 X
1Also known as mean motion. 96 Titus Hilberdink
On the other hand, since all QC-series converge absolutely we have
it f.t/g.t/h.t/ f.qO 1/g.q2/h.qO 3/.q1q2q3/ D C O q1;q2;q3 Q X2 q it f.q1/g h.q3/ q D O O q1q3 O q QCq1;q3 X2 X fgh.q/qb it: D q QC X2 Hence b i i M.fgh/ij fgh f.q1/g h.q3/ M.f /M.g/M.h/ ; j O jq q O ij D D q ;q O 1 3 D X1 3 since 1=q1 and q3 must range over N.
In view of Theorem 2.3, it is of interest to know when a given f W C factorises 2 Q as f gh with g WN and h WN.ForthenM.f / M.g/M.h/ and the invertibilityD of M.f /2follows from2 knowing when M.g/ andD M.h/ are invertible. Thus, if h is invertible in WN,then 1 1 1 1 M.h/M.h / M.hh / M.1/ I M.1/ M.h h/ M.h /M.h/; D D D D D D 1 1 1 so that M.h/ M.h /. Similarly, if g is invertible in WN,thenM.g/ M.g 1/. It followsD then that M.f / 1 M.h/ 1M.g/ 1. D D Let FWQC denote the set of functions in WQC which factorise as
f f qf (3.9) D C where f GWN, f GWN,andq QC. 2 C 2 2 1 1 Note that with f as above, then 1=f f .1=q/f ,so1=f FWQC .In D C 2 particular, FW C GW C . Note that M. q/ is invertible if and only if q 1. Q Q D
Theorem 2.4. Let f FWQC .ThenM.f / is invertible if and only if w.f / 0.If 2 1 1 1 D this is the case, then M.f / M.f /M.f /, with f as in (3.9). D C ˙
Proof. Write f f qf as in (3.9). Then M.f / M.f /M. q/M.f /.Now D C D1 1 C M.f / and M.f / are invertible, with inverses M.f / and M.f /, respectively. C C Thus M.f / is invertible if and only if M. q/ is invertible. But this happens if and only if q 1. D Since w.f / w.f / w. q/ w.f / w. q/ log q, we see that M.f / is invertible if andD only if w.fC / 0C. C D D Now suppose w.f / 0. ThenD the above gives D 1 1 1 1 1 1 M.f / M.f /M.f / M.f / M.f / M.f /M.f /: D C D C D C 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 97
Multiplicative coefficients and Euler products.
Definition. (a) A function a Q C is multiplicative if a.1/ 1 and W C ! D a.pa1 pak / a.pa1 / a.pak /; 1 k D 1 k
for all distinct primes pi and all ai Z.Wesaya is completely multiplicative if, in addition to the above, 2
k k k 1 k a.p / a.p/ and a.p / a.p / ; D D for all primes p and k N. 2 F MF (b) For a subset S of ,let denote the set of f S for which f.O / is multi- plicative. 2 F k (c) Let f and p prime. Suppose that k Z f.p / converges. Then define 2 2 j O j P k fp f.p / k : D O p k Z X2 2 ] Note that fp is periodic with period .Definefp T C by log p W !
] i k ki f .e / fp.= log p/ f.p /e for 0 <2. p D D O k Z X2
Further, denote by WQC;p the set of those f WQC whose QC-coefficients are k 2 supported on p k Z . (Thus fp W C by definition.) f W 2 g 2 Q ;p Note that, for fixed p, there is a one-to-one correspondence between WQC;p and W.T/ via the mapping ].
In [14], the Euler product formulas
f f.q/ q fp and M.f / M.fp/; D O D D q QC p p X2 Y Y were shown to hold whenever f MW C . 2 Q The analogue of the Wiener–Hopf factorization holds for MWQC -functions with- out zeros.
Theorem 2.5. Let f MW C such that f has no zeros. Then f FW C . 2 Q 2 Q
Proof. We have f p fp,wherefp WQC;p and each is non-zero. Hence ] D 2 ] fp W.T/ and has no zeros. Let kp wind .fp ;0/. Note that kp 0 for all 2 Q D D 98 Titus Hilberdink sufficiently large p,since
f ].t/ 1 f.pm/ f.q/ 0 as p . j p j j O j j O j! !1 m 0 q p X¤ jXj
kp Let q p p (a finite product). ByD1.1(iii) TS8 ,wehave Q ] f ] ] egp ; p D pkp ] for some gp W.T/. Hence 2 gp fp kp e ; D p with gp W C . Thus for P so large that kp 0 for p>P,wehave 2 Q ;p D
fp q exp gp : D p P p P Y X
Now fp.t/ 1 as p uniformly in t, so can choose gp so that gp.t/ 0 as ! !1 ! p (uniformly in t). Hence, for all sufficiently large p (and all t), fp 1 g!1p 1 j jD e 1 gp ,sothat j j 2 j j m gp 2 fp 1 2 f.p / : j j j j j O j m 0 X¤ .n/ .n/ Let g p n gp.Then g is a Cauchy sequence in WQC :forn>m D f g .n/ P.m/ m g g gp 2 f.p / 2 f.k/ 0 j j j j j O j j O j! m
m X X X¤ X .n/ .n/ as m . Thus g g W C . But each g is of the form hn kn with !1 ! 2 Q C hn WN and kn WN (since gp WQC;p). Thus g h k with h WN, k WN. 2 2 h k 2 D C 2 2 It follows that f qe e , which is of the required form. D Note that, as such,
kp ] w.f / w. p / kpw. p/ wind.fp ;0/log p; D p D p D p X X X where the sum is finite.
Corollary 2.6. Let f MWQC such that f has no zeros and w.f / 0.Then M.f / is invertible. 2 D
1 Example. M.˛/ is invertible for ˛>1with M.˛/ M.1=˛/. D TS8 What does 1.1 refer to? Theorem 1.1 does not include an enumeration. Is it meant to be Subsection 1.1? 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 99 3 Unbounded multiplicative Toeplitz operators and matrices
The last section has, in various ways, been a relatively straightforward extension of the theory of bounded Toeplitz operators to the multiplicative setting. The theory of unbounded Toeplitz operators is rather less satisfactory and not easy to generalise. Our particular concern is in fact the operator M.˛/ for ˛ 1, since we expect a connection with the Riemann zeta function. The hope is that if a satisfactory theory is developed for this case, it can be generalised to other symbols. In this section, we shall therefore concentrate on the particular case when f.n/ ˛ D n , when the symbol is .˛ it/. Throughout we write '˛ (equivalently, M.˛/) for 'f . From ÷2 we see that for ˛ 1, '˛ is unbounded. It is interesting to see to what extent properties of '˛ are related to properties of ˛. The above theory is only valid for absolutely convergent Dirichlet series, when the symbols are bounded. But .˛ it/ is unbounded for ˛ 1. How to measure unboundedness? We shall investigate two different measures. The first case can be considered as restricting the range, while in the second case we shall restrict the domain.
'˛ 3.1 First measure – the function ˆ˛.N / With bn defined by a .an/ ˛ D 7! .bn/, i.e., bn d n d an=d ,let D j P N 1=2 2 ˆ˛.N / sup bn : D j j a 1n 1 k kD XD Theorem 3.1. We have the following asymptotic formulae for large N :
ˆ1.N / e log log N O.1/ .˛ 1/ D C D 1 ˛ .log N/ 1 log ˆ˛.N / <˛<1 log log N 2 1 log N 2 1 log ˆ 1 .N / : ˛ 2 log log N D 2 Sketch of proof. .For the proof see [15]/. We start with upper bounds. First we note that for any positive arithmetical function g.n/,
2 g.n/ 1 ˆ˛.N / max : (3.10) n˛ n N g.d/d˛ n N d n X Xj This is because 2 2 2 1 g.d/an=d 1 g.d/ an=d bn ˛ ˛ ˛ j ˛ j ; j j D g.d/d 2 d 2 g.d/d d ˇ d n p ˇ d n d n ˇXj ˇ Xj Xj ˇ ˇ ˇ p ˇ 100 Titus Hilberdink
1 ˛ by Cauchy–Schwarz. Writing G.n/ d n g.d/ d ,wehave D j 2 2 g.d/ an=dP g.d/ 2 bn G.n/ j j max G.n/ an : j j d ˛ n N d ˛ j j n N n N d n d N n N=d X X Xj X X Taking a 2 1 yields (3.10). The idea is to choose g appropriately, so that the RHS ofk (3.10k )isassmallaspossible.D For 1 <˛ 1, choose g.n/ to be the following multiplicative function: for 2 aprimepowerpk let 1; if pk M; g.pk / . M /ˇ ; if pk >M. D ( pk Here M .2˛ 1/ log N and ˇ is a constant satisfying 1 ˛<ˇ<˛. Note that g.pk / Dg.p/ for every k N and p prime. We estimate the expressions2 in (3.10) separately. First
k g.n/ 1 g.p / g.p/ g.p/ 1 1 exp n˛ C pk˛ C p˛ 1 p˛ 1 n N p k 1 p n p o X Y XD Y X (3.11) and, after some manipulations using the Prime Number Theorem, one finds for the case ˛<1 g.n/ ˇM 1 ˛ log . : (3.12) n˛ .1 ˛/.˛ ˇ 1/ log M n N X C Now consider G.n/, which is multiplicative because g.n/ is. At the prime powers we have k 1 1 1 1 G.pk / D p˛rg.pr / D p˛r C M ˇ p.˛ ˇ/r r 0 r 0 r k XD pXr M pXr >M 1 1 1 : C p˛ 1 C M ˛.1 pˇ ˛ / (Here we require ˇ<˛.) Note that this is independent of k. It follows that 1 1 1 G.n/ exp : p˛ 1 C M ˛ 1 pˇ ˛ p n p n Xj Xj The right-hand side is maximised when n is as large as possible (i.e. N )andN is of the form N 2:3:::P. For such a choice, log N .P/ P , so that (using the prime numberD theorem) D 1 1 .log N/1 ˛ log N log max G.n/ . 1 : n N p˛ 1 C M ˛ .1 ˛/ log log N C M ˛ log log N p P p P X X (3.13) 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 101
From (3.12)and(3.13) it follows that M should be of order log N for optimality. So taking M log N (with >0), (3.10), (3.12)and(3.13)thenimply D 1 ˛ 1 ˛ ˇ 1 1 .log N/ log ˆ˛.N / . 2.1 ˛/.˛ ˇ 1/ C 2.1 ˛/ C 2˛ log log N C ˇ for every ˇ .1 ˛; ˛/ and >0.Since˛ ˇ 1 decreases with ˇ, the optimal 2 C choice is to take ˇ arbitrarily close to ˛. Hence we require inf>0 h./,where ˛1 ˛ 1 1 h./ : D .1 ˛/.2˛ 1/ C .1 ˛/ C ˛ ˛ Since h0./ ˛C1 . 2˛ 1 1/, we see that the optimal choice is indeed 2˛ 1. SubstitutingD this value of gives D .1 .2˛ 1/ ˛/ .log N/1 ˛ log ˆ .N / . C : ˛ 2.1 ˛/ log log N For ˛ 1,weusethesamefunctiong.n/ as before (though with possibly differ- ent valuesD of M and ˇ). From (3.11) it follows that
g.n/ 1 M ˇ 1 : n 1 1 C pˇ .p 1/ n N p M p p>M X Y Y ˇ 1 ˇ ByMertens’Theorem, thefirstproductise log M O.1/, while M p>M p O.1=log M/,andso C D P g.n/ e log M O.1/ exp O.1=log M/ e log M O.1/: (3.14) n C f gD C n N X For the G.n/ term we have, as for the ˛<1case, 1 1 G.pk / : 1 1 C M.1 pˇ 1/ p Thus, with N 2:3:::P, D 1 1 1=p G.N / 1 1 1 C M.1 pˇ 1/ p P p Y P e log P O.1/ 1 O : D C C M log P Taking M log N and noting that P log N , the right-hand side is e log log N O.1/. CombiningD with (3.14)showsthat C
ˆ1.N / e log log N O.1/: C 102 Titus Hilberdink
1 1 The case ˛ 2 . The function g as chosen for ˛ . 2 ;1 is not suitable for D 1 1 2 an upper bound as we would require 2 <ˇ< 2 ! Instead we take g to be the multiplicative function following: for a prime power pk ,let
1 M 2 g.pk / min 1; : D pk .log p/2 n o Here M>0is independent of p and k and will be determined later. Thus g.pk / 1 if and only if pk .log p/2 M . Note that g.pk/ g.p/ 1 for all k 1 andD all primes p. Thus (3.11) holds with ˛ 1 and (using the prime number theorem) D 2 g.n/ 1 1 pM log . pM : (3.15) pn pp 1 C p log p log M n N p. M p& M X X.log M/2 X.log M/2
(The first sum is of order pM=.log M/2 and the main contribution comes from the second term.) Regarding G.n/,thistimewehave2
k 1 1 k G.n/ G.pk / 1 log p ; D C pr=2 C pM pk n pk n r 1 r 1 Yk Yk XD XD so that 1 1 log n 1 log G.n/ k log p : pp 1 C pM pM C pp 1 p n pk n p n Xj Xk Xj The right-hand side above is maximal when n N 2:3:::P, hence D D log N 1 log N 2 log N log max G.n/ . : n N pM C pp pM C log log N p P p X Combining with (3.15), (3.10)thengives
pM log N log N log ˆ 1 .N / . : 2 2 log M C 2pM C logp log N The optimal choice for M is easily seen to be M log N log log N , and this gives the upper bound in (iii). D Now we proceed to give lower bounds. For a fixed n N,let 2 1 ad if d n, and zero otherwise. D d.n/ j p 2Here as usual, pk n means pk n but pkC1 n. jj j 6 j 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 103
Then a 2 1, while for d n k k D j 1 1 ˛.d/ b : d D c˛ D d.n/ c d d.n/ Xj Hence for N n, p p 2 2 1 2 bk bd ˛.d/ ˛.n/: j j D d.n/ DW k N d n d n X Xj Xj
With this notation ˆ˛.N / maxn N ˛.n/, and the lower bounds follow from 1 the maximal order of ˛.n/.For <˛<1this can be found easily. Since ˛.p/ 2 p D 1 p ˛ 1 p 2˛ for p prime, we find for n 2:3:::P (so that log n P )that C C 2 D 1 1 1 ˛.n/ 1 O exp .1 o.1// D C p˛ C p2˛ D C p˛ p P p P Y X .1 o.1//P 1 ˛ .1 o.1//.log n/1 ˛ exp C exp C : D .1 ˛/ log P D .1 ˛/ log log n th Now, if t is the k number of the form 2:3 P (i.e., t p1 p ), then log t k k D k k k log k log tk 1. Hence for tk N For ˛ 1, we have to be a little subtler to obtain maxn N 1.n/ e log log N O.1/.D We omit the details, which can be found in [15]. D C 1 p For the case ˛ 2 , the above choice doesn’t give the correct order and we lose a power of log log ND. Instead, we follow an idea of Soundararajan [29]. Let f be the multiplicative function supported on the squarefree numbers whose values at primes p is . M /1=2 1 ; for M p R; f.p/ p log p D 0; otherwise. Here M log N.log log N/as before and log R .log M/2. D 1=2 D 2 Now take an f.n/F.N/ ,whereF.N/ n N f.n/ so that 2 D D n N an 1.ThenbyH¨older’s inequality D P P N 1=2 N N 2 1 f.n/ b anbn pdf.d/ n D F.N/ pn n 1 n 1 n 1 d n XD XD XD Xj 1 f.n/ f.d/2: (3.16) D F.N/ pn n N d N=n X .n;X d/ D 1 104 Titus Hilberdink Now using ‘Rankin’s trick’3 we have, for any ˇ>0 f.n/ f.n/ f.d/2 f.d/2 f.d/2 n1=2 D n1=2 n N d N=n n N d 1 d>N=n X .n;X d/ D 1 X .n;X d/ D 1 .n;X d/ D 1 f.n/ n ˇ 1 f.p/2 O 1 pˇ f.p/2 : D n1=2 C C N C n N p− n p − n X Y Y (3.17) The O-term in (3.17) is at most a constant times 1 ˇ 1=2 ˇ 2 1 ˇ 2 ˇ 1=2 f.n/n 1 p f.p/ 1 p f.p/ p f.p/ ; N ˇ C N ˇ C C n N p− n p X Y Y while the main term in (3.17) is (using Rankin’s trick again) f.p/ 1 1 f.p/2 O 1 f.p/2 pˇ 1=2f.p/ : 1=2 ˇ p C C p C N p C C Y Y Hence (3.16) implies N 1=2 1 f.p/ b2 1 f.p/2 n F.N/ C C p1=2 n 1 p XD Y 1 O 1 pˇ f.p/2 pˇ 1=2f.p/ : ˇ C N p C C Y The ratio of the O-term to the main term on the right is less than f.p/ exp ˇ log N .pˇ 1/ f.p/2 ; C C p1=2 M p R X which equals M M 1=2 exp ˇ log N .pˇ 1/ : C p.log p/2 C p.log p/ M p R X 3 1=2 1=2 " Take ˇ .log M/ . The term involving M is at most .log N/ C for every ">0, whileD the remaining terms in the exponent are (by the prime number theorem 3 ˇ 1 ˇ If cn >0, then for any ˇ>0, n>x cn x nD1 n cn. P P 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 105 in the form .x/ li.x/ O.x.log x/ A/ for all A) D C R t ˇ 1 log N ˇ log N M dt O C t.log t/3 C .log log N/A ZM R dt R dt ˇ log N ˇM O ˇ2M D C t.log t/2 C t log t ZM ZM log N log log log N ˇ ; log log N after some calculations. Finally, since F.N/ .1 f.p/2/, this implies p C Q 1 f.p/ ˆ .N / 1 ; 1=2 2 C p1=2.1 f.p/2/ M p R Y C for all N sufficiently large. Hence 1 log N 1=2 log ˆ .N / & M 1=2 ; 1=2 p.log p/ log log N M p R X as required. 1 Remark. The result for 2 <˛<1is .log N/1 ˛ .1 .2˛ 1/ ˛/ .log N/1 ˛ . log ˆ .N / . C : 2.1 ˛/ log log N ˛ 2.1 ˛/ log log N It would be nice to obtain an asymptotic formula for log ˆ˛.N /. Indeed, it is possible to improve the lower bound at the cost of more work by using the method for the case ˛ 1 , but we have not been able to obtain the same upper and lower limits. D 2 Connections between ˆ˛.N / and the order of j.˛ C it/j. The lower bounds 1 obtained for ˆ˛.N / for 2 <˛ 1 can be used to obtain information regarding the maximum order of .s/ on the line s ˛. < D ˛ Proposition 3.2. With bn d n d an=d we have, for any ˛, D j P 2˛ 2 aman.m; n/ 1 bn : j j D m˛n˛ k2˛ n N m;n N k N X X XŒm;n Proof. We have 2 1 ˛ ˛ 1 ˛ ˛ bn bnbn c d aca c d aca ; j j D D n2˛ d D n2˛ d c n;d n Œc;d n jXj Xj 106 Titus Hilberdink since c n; d n if and only if Œc; d n. Hence j j j ˛ ˛ 2 ˛ ˛ 1 c d acad 1 bn c d aca ; j j D d n2˛ D Œc; d2˛ k2˛ n N c;d N n N;Œc;d n c;d N k N X X X j X XŒc;d by writing n Œc; dk.Since.c; d/Œc; d cd, the result follows. D D 1 It follows that if an 0 for all n and ˛> ,then 2 2˛ 2 aman.m; n/ bn .2˛/ : (3.18) j j .mn/˛ n N m;n N X X 1 2 Theorem 3.3. Let 2 <˛ 1 and let a l with a 2 1.LetAN .t/ N it 2 2 1 k k D D n 1 ann .LetN T ,where0<< 3 .˛ 2 /. Then for some >0, D P T 2˛ 1 2 2 aman.m; n/ .˛ it/ AN .t/ dt .2˛/ O.T /: (3.19) T j C j j j D .mn/˛ C Z1 m;n N X 1 Proof. We shall assume 2 <˛<1, adjusting the proof for the case ˛ 1 after- wards. For ˛ 1, we can integrate from 0 to T since the error involvedD is at most ¤ O.N=T/ O.T /. D ˛ it ˛ Starting from the approximation .˛ it/ n t n O.t /,wehave C D C 2P 2 1 1 2˛ .˛ it/ O.t /: j C j D n˛ it C ˇn t C ˇ ˇX ˇ ˇ ˇ Let k; l N such that .k; l/ 1.LetˇM maxˇ k; l T 1 km it 1 T km it dt dt: .mn/˛ ln D .mn/˛ ln Z0 m;n t m;n T Zmax m;n X X f g The terms with km ln (which implies m rl;n rk with r integral) contribute D D D 1 T rM .2˛/ M 2˛ 1T 2 2˛ T O : .kl/˛ r2˛ D .kl/˛ C .kl/˛ r T=M X 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 107 The remaining terms contribute at most 1 1 2 2M 2˛ .mn/˛ log km .kmln/˛ log km m; n T j ln j m; n T j ln j kmXln kmXln ¤ ¤ 1 2M 2˛ ˛ m1 .m1n1/ log m1 kT;n1 lT n1 j j m1Xn1 ¤ 1 2M 2˛ ˛ m1 .m1n1/ log m1;n1 MT n1 j j m1Xn1 ¤ 2˛ 2 2˛ O.M .MT / log.MT // D 2 2 2˛ O.M T log T/; D using Lemma 7.2 from [31]. Hence T it 2 k .2˛/ 2 2 2˛ .˛ it/ dt ˛ T O.M T log T/: 0 j C j l D .kl/ C Z It follows that for any positive integers m; n < T , T it 2˛ 2 m .2˛/.m; n/ 2 2 2˛ .˛ it/ dt ˛ T O.max m; n T log T/: 0 j C j n D .mn/ C f g Z N it Thus, with AN .t/ n 1 ann , D D T P T it 2 2 2 m .˛ it/ AN .t/ dt aman .˛ it/ dt j C j j j D j C j n Z0 m;n N Z0 X a a .m; n/2˛ .2˛/T m n D .mn/˛ m;n N X 2 2˛ 2 O T log T max m; n aman : C f g j j m;n N X 2 2 3 The sum in the O-term is at most N . n N an / N , using Cauchy–Schwarz. Hence j j P T 2˛ 3 1 2 2 aman.m; n/ N log T .˛ it/ AN .t/ dt .2˛/ O : T j C j j j D .mn/˛ C T 2˛ 1 Z0 m;n N X Since N 3 T 3 and 3 < 2˛ 1, the error term is O.T / for some >0. 108 Titus Hilberdink If ˛ 1 we integrate from 1 to T instead and the O-term above will contain an D extra log T factor, but this is still O.T /. We note that with more care the N 3 could be turned into an N 2, so that we can take <˛ 1 in the theorem. This is however not too important for us. 2 Corollary 3.4. Let 1 <˛ 1. Then for every ">0and N sufficiently large, 2 2 .˛ 1 / " max .˛ it/ ˆ˛.N 3 2 / O.N / (3.20) t N j C j C for some >0. 2 1 Proof. Let an 0 be such that a 2 1,andtakeN T with < 3 .˛ 2 /.By (3.18)and(3.19), k k D D T 2 1 2 2 bn .˛ it/ AN .t/ dt O.T / j j T j C j j j C n N Z0 X T 2 1 2 max .˛ it/ AN .t/ dt O.T / t T j C j T 0 j j C Z 2 2 max .˛ it/ an .1 O.N=T// O.T / D t T j C j j j C C n N X using the Montgomery and Vaughan mean value theorem. The implied constants in the O-terms depend only on T and not on the sequence an . Taking the supremum over all such a,thisgives f g 2 2 2 ˆ˛.N / sup bn max .˛ it/ O.T /; D a 1 j j t T j C j C 2 n N k k D X for some >0,and(3.20) follows. In particular, this gives the (known) lower bounds c.log T/1 ˛ max .˛ it/ exp t T j C j log log T n o 1 for 2 <˛<1and maxt T .1 it/ e log log T O.1/ (obtained by Levinson in [23]). j C j C Morever, we can say more about how often .˛ it/ is as large as this. For A R and c>0,let j C j 2 FA.T / t Œ1; T .1 it/ e log log T A : (3.21) D 2 Wj C j 1 ˛ c.log T/ F˛;c.T / t Œ0; T .˛ it/ exp : (3.21 ) D 2 Wj C j log log T 0 n o 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 109 1 We outline the argument in the case 2 <˛<1.Wehave 2 1 2 2 bn .˛ it/ AN .t/ dt O.T /: j j T F˛;c.T / C Œ0;T F˛;c.T / j C j j j C n N Z Z n X (3.22) The second integral on the right is at most 1 ˛ T 1 ˛ 2c.log T/ 1 2 2c.log T/ exp AN .t/ dt O exp ; log log T T 0 j j D log log T n o Z n o 1=2 while, by choosing an d.N/ for n N and zero otherwise, the LHS of (3.22)is D =3 j at least ˛.N /. Every interval ŒT ;T contains an N of the form 2:3:5:::P.As c0.log T/1˛ such, ˛.N / exp for some c >0. Hence, for 2c < c , log log T 0 0 n o 1 ˛ 1 2 2 c0.log T/ .˛ it/ AN .t/ dt exp : T F˛;c.T / j C j j j 2 log log T Z n o 2 " We have .˛ it/ O.T / for some and AN .t/ d.N/ O.T /,so j C jD j j D 1 2 2 2 1 " .˛ it/ AN .t/ dt T C .F˛;c.T //: T j C j j j ZF˛;c.T / 1 2 " Thus .F˛;c.T // T for all c sufficiently small. 1 ˛ In particular, since < 3 ,wehave: Theorem 3.5. For all A sufficiently large .and positive/, log T .FA.T // T exp a log log T .1 2˛/=3 for some a>0, and for all c sufficiently small, .F˛;c.T // T C for all sufficiently large T . Furthermore, under the Lindelof¨ Hypothesis, the exponent can be replaced by 1 ". 2 3.2 Second measure – '˛ on M and the function M˛.T / For ˛ 1, 2 2 2 4 '˛ is unbounded on l and so '˛.l / l (by the closed graph theorem –see[30], p. 183). However, if we restrict the domain6 to M2, the set of multiplicative elements of l 2,wefindthat'.M2/ l 2. More generally, if f is multiplicative then, as we shall see, ' .M2/ l 2 in many cases (and hence ' .M2/ M2). f f 4 2 2 Being a ‘matrix’ mapping, '˛ is necessarily a closed operator, and so '˛ .l / l implies '˛ is bounded. 110 Titus Hilberdink M2 M2 2 Notation. Let and c denote the subsets of l of multiplicative and completely multiplicative functions, respectively. Further, write M2 for the non-negative mem- M2 M2 C bers of , and similarly for c . M2 M2C M2 Let 0 denote the set of functions f for which f g whenever g M2;thatis, 2 2 M2 f M2 g M2 f g M2 : 0 Df 2 W 2 H) 2 g M2 M2 M2 Thus for f 0, 'f . / . 2 M 2 M2 It was shown in [16]that is not closed under Dirichlet convolution, so 0 M2. A criterion for f M2 to be in M2 was given, namely: ¤ 2 0 M2 k Proposition 3.6. Let f be such that k1 1 f.p / converges for every prime k 2 D j j M2 p and that k1 1 f.p / A for some constant A independent of p.Thenf 0. D j j M2 P 2 k On the other hand, if f with f 0 and for some prime p0, f.p0 / P 2 k M2 decreases with k and k1 1 f.p0 / diverges, then f 0. D 62 The proof is basedP on the following necessary and sufficient condition: Let f; g M2 be non-negative. Then f g M2 if and only if 2 2 1 m n m k n k f.p /g.p /f .p C /g.p C / converges: p m;n 1 k 0 X X XD This can be proven in a direct manner. M2 M2 M2 Thus, in particular, c 0.Forf c if and only if f.p/ <1for all primes p and f.p/ 2 < . Thus 2 j j p j j 1 P 1 f.p/ f.pk / j j A; j jD1 f.p/ k 1 XD j j independent of p. For example, .n ˛/ M2 for ˛>1 . 2 0 2 M2 The “quasi-norm” Mf .T /. Let f 0. The discussion above shows that ' .M2/ M2 but, typically, ' is not2 ‘bounded’ on M2 (if f l 1) in the sense f f 62 that5 ' a = a is not bounded by a constant for all a M2. A natural question k f k k k 2 is: how large can 'f a become as a function of a ? It therefore makes sense to define, for T 1, k k k k 'f a Mf .T / sup k k: D a 2 M2 a kakDT k k ˛ We shall consider only the case f.n/ n , although the result below can be ex- tended without any significant changesD to f completely multiplicative and such that 5 2 Here, and throughout this section 2 is the l -norm. kkDkk 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 111 f P is regularly varying of index ˛ with ˛>1=2in the sense that there exists j a regularly varying function f (of index ˛) with f.p/ f.p/for every prime p. Q Q D We shall write M .T / M˛.T / in this case. f D Theorem 3.7. As T !1 M1.T / e .log log T log log log T 2 log 2 1 o.1//; (3.23) D C C C 1 while for 2 <˛<1, 1 1 ˛ 1 ˛ B.˛ ;1 2˛ / .log T/ log M˛.T / o.1/ : (3.24) D .1 ˛/2˛ C .log log T/˛ 1 Sketch of proof for the case 2 <˛<1(for the full proof see [16]). We consider first upper bounds. The supremum occurs for a 0, which we now assume. Write a .an/, '˛a b .bn/.Define˛p and ˇp for prime p by D D D 1 2 1 2 ˛p a k and ˇp b k : D p D p k 1 k 1 XD XD 2 2 2 By the multiplicativity of a and b, T a .1 ˛p/ and b .1 Dk k D p C k k D p C ˇp/. Thus Q Q '˛a 1 ˇp k k C : a 1 ˛ D p s p k k Y C Now for k 1 k r˛ ˛ b k p a kr a k p b k1 ; p D p D p C p r 0 XD whence 2 2 ˛ 2˛ 2 b k a k 2p a k b k1 p b k1 : p D p C p p C p Summing from k 1 to and adding 1 to both sides gives D 1 ˛ 1 2˛ 1 ˇp 1 ˛p 2p a k b k1 p .1 ˇp/: (3.25) C D C C p p C C k 1 XD By Cauchy–Schwarz, 1=2 1 1 2 1 2 a k b k1 a k b k1 ˛p.1 ˇp/; p p p p D C k 1 k 1 k 1 XD XD XD q so, on rearranging, ˛ 2p ˛p.1 ˇp/ 1 ˛p .1 ˇp/ C C : C 1 p 2˛ 1 p 2˛ p 112 Titus Hilberdink Completing the square we obtain ˛ 2 p p˛p 1 ˛p 1 ˇp C : C 1 p 2˛ .1 p 2˛/2 q The term on the left inside the square is non-negative for every p since 1 ˇp 2˛ C 1 ˛p p ˛p 1 1 Cp2˛ , which is greater than .1 p2˛ /2 for ˛>2 . Rearranging gives 1 ˇp 1 1 ˛p C 2˛ 1 ˛ : 1 ˛p 1 p C p 1 ˛p s C r C Let ˛p . Taking the product over all primes p gives p 1 ˛p D C q ' a k f k .2˛/ 1 p .2˛/exp p : (3.26) a p˛ p˛ p C p k k Y nX o 1 2 Note that 0 p <1and p 2 T . The idea is to show now that the main 1 p D contribution to the above sumQ comes from the range p log T log log T . Let ">0and put P log T log log T . We split up the sum on the right-hand side of (3.26)intop aPD, aP < p AP ,andp>AP(for a small and A large). First, 1 ˛ 1 ˛ 1 ˛ ˛ ˛ a P .log T/ p p p <" ; (3.27) .1 ˛/ log P .log log T/˛ p aP p aP X X 2 1 2 for a sufficiently small. Next, using the fact that log T log p 2 p p, 1 p D we have Q P 1=2 1 2˛ 1 2˛ 1=2 ˛ 2˛ 2 2A P log T p p p . p .2˛ 1/ log P p>APX p>APX p>APX .log T/1 ˛.log log T/ ˛ .log T/1 ˛ <" (3.28) A˛ 1=2 ˛ 1=2 .log log T/˛ for A sufficiently large. This leavesp the range aP < p AP and the problem therefore reduces to maximising p p˛ aP p forQ primes p p By interpolation we may write p g.P /,whereg .0; / .0; 1/ is con- tinuously differentiable and decreasing.D Of course, g willW depend1 ! on P .Leth 1 D log 1 g2 , which is also decreasing. Note that p p 2 log T h h h.a/.aP / cah.a/log T; D P P p p aP X X for P sufficiently large, for some constant c>0. Thus h.a/ Ca (independently of T ). Now, for F .0; / Œ0; / decreasing, it follows from the Prime Number Theorem in theW form1.x/! li.x/1 O. x / that D C .log x/2 p x b xF.a/ F F O 2 ; (3.29) x D log x a C .log x/ ax p P A A 2 log T h h log T h: P log P aP Since a and A are arbitrary, 01 h must exist and is at most 2. Also, by (3.29), 1 R p p ˛ P 1 ˛ A g.u/ p g du: p˛ D P ˛ P P log P u˛ aP 1 where l h ,sincepug.u/ 0 as u . The final integral is, by H¨older’s inequality,D at most ! !1 h.0C/ ˛ h.0C/ 1 ˛ 1=˛ s0 l : (3.30) Z0 Z0 h.0C/ But l 1 uh .u/du 1 h 2,so 0 D 0 0 D 0 1 ˛ ˛ R R 1 g.u/ R 2 1 1=˛ du s0 : u˛ 1 ˛ Z0 Z0 6 1=˛ 1=˛ 1 1 A direct calculation shows that 01.s0/ 2 B.˛ ;1 2˛ /.Thisgivesthe upper bound. D The proof of the upper boundR leads to the optimal choice for g and the lower 1=˛ bound. We note that we have equality in (3.30)ifl=.s0/ is constant, i.e., l.x/ 1=˛ D cs0.x/ for some constant c>0— chosen so that 01 l 2. This means that we take D ˛ R 2˛ 1 x 1 1 c h.x/ .s0/ log 1 ; D c D 2 C 2 C x r from which we can calculateg. In fact, the required lower bound can be found by taking an completely multiplicative, with ap for p prime defined by p ap g0 ; D P where P log T log log T and g0 is the function D 2 g0.x/ 1 ; D v 1 1 . c /2˛ u x u C C t q with c 21 1=˛=B. 1 ;1 1 /. As such, by the same methods as before, we have D C ˛ 2˛ a T 1 o.1/ and k kD C 1 ˛ '˛a p P g0.u/ log k k p ˛g O.1/ 1 du: a 0 P log P u˛ D p C 0 k k X Z B. 1 ;1 1 /˛ By the choice of g , the integral on the right is ˛ 2˛ , as required. 0 .1 ˛/2˛ Remark. These asymptotic formulae bear a strong resemblance to the (conjectured) maximal order of .˛ iT/ . It is interesting to note that the bounds found here are j C j just larger than what is known about the lower bounds for Z˛.T / (see the interlude on upper and lower bounds on .s/, especially items (b) and (c)). We note that the 1 constant appearing in (3.24) is not Lamzouri’s C.˛/ since, for ˛ near 2 , the former is roughly 1 , while C.˛/ 1 . ˛ 1 p2˛ 1 2 q 1 1 6The integral is 21=˛ ex=˛.1 ex /1=2˛dx 21=˛ t 1=˛1.1 t/1=2˛dt. 0 D 0 R R 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 115 Lower bounds for '˛ and some further speculations. We can study lower bounds of '˛ via the function '˛a m˛.T / inf k k: D a 2 M2 a kakDT k k Using very similar techniques, one obtains results analogous to Theorem 3.7: 1 6e 2 .log log T log log log T 2 log 2 1 o.1// m1.T / D C C C and 1 B. 1 ;1 1 /˛.log T/1 ˛ ˛ 2˛ 1 log ˛ ˛ for 2 <˛<1: m˛.T / .1 ˛/2 .log log T/ We see that m˛.T / corresponds closely to the conjectured minimal order of .˛ iT/ (see [11]and[25]). j C Thej above formulae suggest that the supremum (respectively infimum) of 2 '˛a = a with a M and a T are close to the supremum (resp. infi- k k k k 2 k kD mum) of ˛ on Œ1; T . One could therefore speculate further that there is a close j j connection between '˛a = a (for such a)and .˛ iT/ . Heuristically, wek couldk arguek k as follows. Considerj C j T 2 1 2 1 it .˛ it/ ann dt: (3.31) T j j Z0 ˇn 1 ˇ ˇXD ˇ ˇ ˇ This is less than ˇ ˇ T 2 2 1 1 it 2 2 Z˛.T / ann dt Z˛.T / a ; T k k Z0 ˇn 1 ˇ ˇXD ˇ ˇ ˇ by the Montgomery–Vaughan meanˇ value theˇ orem (under appropriate conditions). On the other hand, (3.31) is expected to be approximately T 2 1 1 it 1 2 2 bnn dt bn '˛a : T j j Dk k Z0 ˇn 1 ˇ n 1 ˇXD ˇ XD ˇ ˇ Putting these together givesˇ ˇ '˛a k k . Z .T /: a ˛ k k The left-hand side, as a function of a , can be made as large as F. a /,where .log x/1˛ k k k k F.x/ exp c˛ ˛ . If the above continues to hold for a as large as T ,then D f .log log x/ g k k M˛.T / Z˛.T / would follow. Even if it holds for a as large as a smaller power of T , one would recover Montgomery’s -result. k k 116 Titus Hilberdink Alternatively, considering (3.31) over ŒT; 2T , 2T 2 2T 2 1 2 1 it 2 1 1 it .˛ it/ ann dt .˛ it0/ ann dt (3.32) T j j Dj C j T ZT ˇn 1 ˇ ZT ˇn 1 ˇ ˇXD ˇ ˇXD ˇ ˇ ˇ ˇ ˇ for some t0 ŒT; 2Tˇ and, usingˇ the Montgomery–Vaughanˇ mean valueˇ theorem (assuming it applies),2 this is approximately 2 1 2 2 2 .˛ it0/ an .˛ it0/ a : j C j j j Dj C j k k n 1 XD On the other hand, formally multiplying out the integrand, by writhing .˛ it/ nit D n1 1 n˛ and formally multiplying out the integrand, the left-hand side of (3.32) becomesD (heuristically) P 2T 2 1 1 it 1 2 2 bnn dt bn '˛a : T j j Dk k ZT ˇn 1 ˇ n 1 ˇXD ˇ XD ˇ ˇ Equating these gives ˇ ˇ '˛a k k .˛ it0/ : a j C j k k Clearly there are a number of problems with this. For a start, we need '˛a l 2. More importantly, the error term in the Montgomery–Vaughan theorem contains2 2 n1 1 n an , which may diverge. Also, an and hence a may depend on T ,and finally,D thej seriesj for .˛ it/ doesn’t converge for ˛ k1.k P If an 0 for n>N, the above argument can be made to work, even for N a (small) powerD of T (see for example [15]), but difficulties arise for larger powers of T . There seem to be some reasons to believe that the error from the Montgomery– Vaughan theorem should be much smaller when considering products. These occur when an is multiplicative. For example (with Q p P p so that log Q .P/ P by the Prime Number Theorem), D D Q 1 T 2 1 T 2 1 .1 pit/ dt d it dt 1 O d : 2T T C D 2T T D C T Z ˇp P ˇ Z ˇd Q ˇ d Q d Q ˇ Y ˇ ˇXj ˇ Xj Xj ˇ ˇ ˇ ˇ The ‘mainˇ term’ is d.Q/ˇ 2.P/, whileˇ the errorˇ is at least Q . However the left- D T hand side is trivially at most 4.P/, so the error dominates the other terms if P> .1 ı/ log T . If, say, P is of order log T log log T (which is the range of interest), thenC.P/ log T ,so2.P/ is like a power of T ,butQ is roughly like T log log T – far too large. Thus it may be that for an completely multiplicative, it holds that 2T 2 1 2 1 1 .˛ it/ it dt .˛ it0/ 2 T j j 1 app j C j 1 ap ZT ˇp P ˇ p P j j ˇ Y ˇ Y ˇ ˇ ˇ ˇ 3 Multiplicative Toeplitz Matrices and the Riemann zeta function 117 for P up to c log T log log T . This suggests the following might be true: (a) given a M2 with a T , there exists t ŒT; 2T such that 2 k kD 2 '˛a k k .˛ it/ : a j C j k k (b) given T 1, there exists a M2 with a T such that 2 k kD '˛a k k .˛ iT/ : a j C j k k Here, means something like log-asymptotic, , or possibly even . Thus (a) D implies M˛.T / . Z˛.T /, while (b) implies the opposite. Together they would imply 2 2 we can encode real numbers into M -functions with equal l -norm, such that '˛ has a similar action as ˛. M 2 Closure of BN? We finish these speculations with a final plausible conjecture 2 regarding the closure under multiplication of functions in BN with multiplicative co- efficients. M 2 M 2 M Let 0BN denote the subset BN of functions f for which fO 0. Recall M2 M2 M2 M2 2 that 0 is the subset of for which g f g . This suggests the following conjecture: 2 ) 2 2 2 2 Conjecture. Let f MB and g M0B .Thenfg MB . 2 N 2 N 2 N 2 2 2 2 1 In particular, MB McB MB .Since˛ McB for ˛> , this would N N D N 2 N 2 imply k MB2 for every k N and ˛>1 , which implies the Lindel¨of hypothesis. ˛ 2 N 2 2 4 Connections to matrices of the form 2 .f .ij=.i; j / //i;jN The asymptotic formulae for ˆf .N / in Theorem 3.1 can be used to obtain infor- mation on the largest eigenvalue of certain arithmetical matrices. Various authors have discussed asymptotic estimates of eigenvalues and determinants of arithmetical matrices (see for example [4], [5], [24]tonamejustafew). th Let AN .f / denote the N N matrix with ij -entry f.i=j/if j i and zero oth- j erwise. (i.e. AN .f / MN .f/O ). As noted in the introduction, these matrices behave much like Dirichlet seriesD with coefficients f.n/; namely, AN .f /AN .g/ AN .f g/: D 118 Titus Hilberdink In particular, AN .f / is invertible if f has a Dirichlet inverse, i.e., f.1/ 0,inwhich 1 1 ¤ case AN .f / AN .f /. Suppose forD simplicity that f is a real arithmetical function. (For complex values 2 we can easily adjust.) Observe that ˆf .N / is the largest eigenvalue of the matrix T AN .f / AN .f /: Indeed, we have (with bn d n f.d/an=d ) D j P2 n n b f f ai aj ; n D i j i;j n Xj so that, on noting i;j n if and only if Œi; j n (where Œi; j denotes the lcm of i and j ) j j N N 2 n n .N / b f f ai aj b ai aj ; n D i j D ij n 1 n 1 Œi;j n i;j N XD XD Xj X where (using .i; j /Œi; j ij ) D ki kj b.N / f f : ij D .i; j / .i; j / k N XŒi;j .N / th T But bij is also the ij -entry of AN .f / AN .f /, as an easy calculation shows. Thus 2 .N / ˆf .N / sup bij ai aj (3.33) D 2 2 a1 aN 1 i;j N CC D X T 7 is the largest eigenvalue of AN .f / AN .f /, i.e., ˆf .N / the largest singular value of AN .f /. Thus an equivalent formulation of Corollary 2.2 for f supported on N is: 1 For f l non-negative, the largest singular value of AN .f / tends to f 1. Now2 if f is completely multiplicative, then k k ij b.N / f f.k/2; ij D .i; j /2 k N XŒi;j which for large N is roughly f 2f. ij / for f l 2. This suggests that the matrix k k2 .i;j /2 2 f. ij / has its largest eigenvalue close to ˆ .N /2= f 2. This is indeed .i;j /2 i;j N f k k2 the case. 7The singular values of a matrix A are the square roots of the eigenvalues of AT A (or AA if A has complex entries). Bibliography 119 2 Corollary 4.1. Let f l be non-negative and completely multiplicative. Let ƒN 2 ij denote the largest eigenvalue of f..i;j /2 / i;j N .Then 2 3 2 ˆf .N / ˆf .N / ƒN : f 2 N 2 2 k 1 f.k/ k k D In particular, for f l 1, P 2 2 f 1 lim ƒN k k : N D f 2 !1 k k2 Proof. We have ij ƒN sup f 2 ai aj (3.34) D 2 2 .i; j / a1 aN 1 i;j N CC D X When f 0, the supremums in (3.33)and(3.34) are reached for an 0. Thus, 2 2 ˆ .N / f ƒN f k k2 follows immediately. On the other hand, for i;j N , Œi; j N 2 so 3 2 ij 2 ˆ .N / f ai aj f.k/ : f .i; j /2 i;j N k N X X 3 2 2 Taking the supremum over all such an gives, ˆf .N / ƒN k N f.k/ ,as required. 2 1 f 1 P Finally, if f l ,thenˆ .N / f and so ƒ k k follows. f 1 N f 2 2 !k k ! k k2 The approximate formulae for ˆ˛.N / in Theorem 3.1 lead to: ˛ Corollary 4.2. Let f.n/ n and let ƒN .˛/ denote the largest eigenvalue of ij D f..i;j /2 / i;j N .Then 6 2 ƒN .1/ .e log log N O.1// ; D 2 C 1 ˛ .log N/ 1 log ƒN .˛/ for <˛<1; log log N 2 1 log N log ƒN : 2 slog log N 120 Titus Hilberdink Bibliography [1] R. Balasubramanian and K. 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