Quasi -norm of an arithmetical convolution operator and the order of the Riemann zeta function 1

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1 Qusi -norm of n rithmeticl convolution oertor nd the order of the Riemnn zet function Titus Hilberdink Dertment of Mthemtics, University of Reding, Whiteknights, PO Box, Reding RG6 6AX, UK; t.w.hilberdink@reding.c.uk Abstrct In this er we study Dirichlet convolution with given rithmeticl function f s liner ming ϕ f tht sends sequence n to b n where b n = d n fd n/d. We investigte when this is bounded oertor on l nd find the oertor norm. Of rticulr interest is the cse fn = n α for its connection to the Riemnn zet function on the line Rs = α. For α >, ϕ f is bounded with ϕ f = ζα. For the unbounded cse, we show tht ϕ f : M M where M is the subset of l of multilictive sequences, for mny f M. Consequently, we study the qusi -norm su = T M ϕ f for lrge T, which mesures the size of ϕ f on M. For the fn = n α cse, we show this qusi-norm hs striking resemblnce to the conjectured mximl order of ζα + it for α >. AMS Mthemtics Subject Clssifiction: N37, M6. Keywords: Dirichlet convolution, mximl order of the Riemnn zet function. Introduction Given n rithmeticl function fn, the ming ϕ f sends n n N to b n n N, where b n = d n fd n/d.. Writing = n, ϕ f ms to f where is Dirichlet convolution. This is mtrix ming, where the mtrix, sy Mf, is of multilictive Toelitz tye; tht is, Mf = ij i,j where ij = fi/j nd f is suorted on the nturl numbers see, for exmle, [6], [7]. Toelitz mtrices whose ij th -entry is function of i j re most usefully studied in terms of symbol the function whose Fourier coefficients mke u the mtrix. Anlogously, the Multilictive Toelitz mtrix Mf hs s symbol the Dirichlet series fnn it. n= Our rticulr interest is nturlly the cse fn = n α when the symbol is ζα it. We re esecilly interested how nd to wht extent roerties of the ming relte to roerties of the symbol for α. These tye of mings were considered by vrious uthors for exmle Wintner [5] nd most notbly Toelitz [3], [4] lthough somewht indirectly, through his investigtions of so-clled To er in Functiones et Aroximtio Commentrii Mthemtici

2 D-forms. In essence, Toelitz roved tht ϕ f : l l is bounded if nd only if n= fnn s is defined nd bounded for ll Rs >. In rticulr, if fn then ϕ f is bounded on l if nd only if f l ; furthermore, the oertor norm is ϕ f = f. We rove this in Theorem. following Toelitz s originl ide. For exmle, for fn = n α, ϕ f is bounded on l for α > with oertor norm ζα. In this secil cse, the ming ws studied in [7] for α when it is unbounded on l by estimting the behviour of the quntity Φ f N = N / su b n = n= for lrge N. Aroximte formuls for Φ f N were obtined nd it ws shown tht, for < α, Φ f N is lower bound for mx t T ζα + it with N = T λ some λ > deending on α only. In this wy, it ws roven tht the mesure of the set { } t [, T ] : ζ + it e γ log log T A is t lest T ex { log T } log log T some > for A sufficiently lrge, while for < α < one hs { } clog T α mx ζα + it ex t T log log T for some c > deending on α only, s well roviding n estimte for how often ζα + it is s lrge s the right-hnd side bove. The method is kin to Soundrrjn s resonnce method nd incidentlly shows the limittion of this roch for α > since ζα + it is known to be of lrger order. In this er we study the unbounded cse in different wy, by restricting the domin. Thus in section, we show tht for mny multilictive f, in rticulr for f comletely multilictive, ϕ f M M even though ϕ f l l. Here M is the set of multilictive functions in l. As result we consider, for such f, the qusi -norm M f T = su = T M ϕ f nd obtin roximte formule for lrge T here is the usul l -norm. We find tht for the rticulr cse fn = n α α >, this qusi-norm hs striking similrity to the conjectured mximl order of ζα + it. For exmle, with α = i.e. fn = /n we rove while for < α < M f T = e γ log log T + log log log T + log + o,. log M f T B α, α α α α log T α log log T α, where Bx, y is the Bet function. Writing Z α T = mx t T ζα+it, Grnville nd Soundrrjn [3] roved tht Z T is t lest s lrge s. minus log log log log T term for some rbitrrily lrge T nd they conjectured tht it equls. ossibly with different constnt term. For < α <, Montgomery [9] found log Z α T α / log T α log log T α nd, using heuristic rgument, conjectured tht this is rt from the constnt the correct order of log Z α T. Further, in recent er see [8], Lmzouri suggests log Z α T Cαlog T α log log T α with some secific constnt Cα see lso the remrk fter Theorem 3..

3 Similrly one cn study the quntity m f T = inf = T M ϕ f. With fn = n α this is shown to behve like the known nd conjectured miniml order of ζα + it for α >. It should be stressed here tht, unlike the cse of Φ f N which ws shown to be lower bound for Z α T in [7], we hve not roved ny connection between ζα + it nd M f T. Even to show M f T is lower bound would be very interesting. Our results, though motivted by the secil cse fn = n α, extend nturlly to comletely multilictive f for which f P is regulrly vrying see section for the definition. Addendum. I would like to thnk the nonymous referee for some useful comments nd for ointing out recent er by Aistleitner nd Sei []. They del with n otimiztion roblem which is different yet curiously similr. The function ex{c α log T α log log T α } ers in the sme wy, lthough their c α is exected to remin bounded s α. It would be interesting to investigte ny links further.. Bounded oertors Nottion: Let l nd l denote the usul sces of sequences n n N, with norms = n nd = n / resectively. After section we shll, for ese of nottion, just write for since it is the norm we will use. A liner ming ϕ : l l is bounded if there exists C > such tht ϕx C x for ll x l. As such, we define the oertor norm by ϕx ϕ = su = su ϕx. x x x = We shll ssume from now on tht fn for ll n N. We re rticulrly interested in the cse where ϕ f cts on l. Define the function Φ f N = su b n, = n N where b n is given in terms of n by.. Note tht the suremum will occur when n for ll n nd when n N n =. Suose now tht f l ; i.e. f = n= fn <. Then b n = fd fdn/d fd fd n/d f fd n/d. d n d n d n d n Hence b n f fd n/d = f fd n f. n N n N d n d N n N/d Thus Following Toelitz [4], we show tht this inequlity is shr. Φ f N f.. Theorem. Let f be non-negtive rithmeticl function nd f l. Then Φ f N f s N. Thus 3

4 ϕ f : l l is bounded if nd only if f l, in which cse ϕ f = f. Proof. After., nd since Φ f N increses with N, we need only rovide lower bound for n infinite sequence of Ns. Let n = dn for n N nd zero otherwise N to be chosen lter, where d is the divisor function. Thus N = nd Φ f N n N n b n = dn n N d n fd = dn d N N fdd,. d sy. We choose N such tht it hs ll divisors d u to some lrge number, nd tht dn/d dn close to for ech such divisor d of N. Tke N of the form is N = P α where α = [ log P log ]. Thus every nturl number u to P is divisor of N. For divisor d = P β dn/d dn = P β. α + of N, we hve If we tke d log P, then β log P for every rime divisor of d. Hence, for such, log log P β log nd β = if > log P. Thus for d log P, dn/d dn = log P β log log P log log P π log P, = α + log P log P log P where πx is the number of rimes u to x. Since πx = O x log x, it follows tht for ll P sufficiently lrge, the exression in. is t lest A fd log P d log P for some constnt A. The sum cn be mde s close to f s we lese by incresing P.. Unbounded oertors on l Now we investigte when ϕ f is unbounded on l i.e. f l. In similr generlistion of Theorem. of [7], one cn redily show tht both ϕ f : l l nd ϕ f : l l re bounded if nd only if f l, with ϕ f = f in either cse. So here we shll ssume tht f l \ l. In the endix we see tht, for ll cses of interest t lest, if f l, then ϕ f l for ll excet =. For unbounded oertors, there re different wys of mesuring the unboundedness. One wy, which ws done in [7] for the cse fn = n α, is to restrict the rnge by looking t restricted norm; i.e. by considering Φ f N for given N. Another wy is to restrict the domin to set S sy, such tht ϕ f S l nd to consider the size of ϕ f su S,=N for lrge N. For f comletely multilictive one is nturlly led to consider S = M the set of squre summble multilictive functions. It is lso nturl to consider regulrly vrying functions. Regulr Vrition. A function l : [A, R is regulrly vrying of index ρ if it is mesurble nd lλx λ ρ lx s x for every λ > 4

5 see [] for detiled tretise on the subject. For exmle, x ρ log x τ is regulrly-vrying of index ρ for ny τ. The Uniform Convergence Theorem sys tht the bove symtotic formul is utomticlly uniform for λ in comct subsets of,. Note tht every regulrly vrying function of non-zero index is symtotic to one which is strictly monotonic nd continuous. We shll mke use of Krmt s Theorem: for l regulrly vrying of index ρ, x l xlx ρ + if ρ >, x l xlx ρ + if ρ <, while if ρ =, x l is slowly vrying regulrly vrying with index nd x l xlx. Nottion. Let M nd M c denote the subsets of l of multilictive nd comletely multilictive functions resectively. Further, write M + for the non-negtive members of M nd similrly for M c+.. The size of ϕ f on M Now we consider ϕ f on the subset M of multilictive functions in l. We suose, s in section, tht f l \ l so tht ϕ f is unbounded. This imlies there exist l such tht ϕ f l by the closed grh theorem. However, if f is multilictive then, s we shll see, ϕ f M l in mny cses nd hence ϕ f M M. Lemm. Let f, g M be non-negtive. Then f g M if nd only if f m g n f m+k g n+k converges.. m,n k= Proof. Let h = f g. Since h is multilictive, Let k nd rime. Then h k = k r= n= hn < h k <. k k f r g k r = f k + g k + f r g k r. Using the inequlity + b + c + b + c 3 + b + c we hve k k. f r g k r h k 3f k + 3g k + 3 f r g k r r= Since,k fk nd,k gk converge we find tht,k hk converges if nd only if k f r g k r converges. But k= k r= k= r= f r g k r = = k= r,s k k k= s= r= r= k= s= r= r= f r f s g k r+ g k s+. s f r f s g k r+ g k s+ f r f s+r g k+s g k. 5

6 On the other hnd, the RHS of. is greter thn k s f r f s g k r+ g k s+. Hence h M if nd only if k= s= r= f m g n f m+k g n+k m,n k= converges. Let M denote the set of M functions f for which f g M whenever g M ; tht is, M = {f M : g M = f g M }. Thus for f M, ϕ f M M. We shll see tht it my hen tht f, g M but f g M. So M M. The following gives criterion for multilictive functions to be in M. Proosition. Let f M be such tht k= fk converges for every rime nd tht k= fk A for some constnt A indeendent of. Then f M. On the other hnd, if f M with f nd for some rime, f k decreses with k nd k= fk diverges, then f M. Proof. Without loss of generlity we cn tke f. Let g M gin w.l.o.g. g with α = k= gk. Thus α converges. By the Cuchy-Schwrz inequlity, g n g n+k g n n= Thus by Lemm., f g M if n= n= g n+k α α = α. α m= f m f m+k k= converges. By ssumtion, the inner sum over k is bounded by constnt indeendent of, nd hence so is the sum over m. This imlies the convergence of the bove. Hence f g M. Now suose k= fk diverges for some rime. Then with g M nd g k decresing to zero we hve f g k = k r= f r g k r g k k f r = g k c k, where c k. Thus k f gk k gk c k. But we cn lwys choose gk decresing so tht k gk converges while, for the given sequence c k, k gk c k diverges. Choose g k = c k c k. Thus f g M ; i.e. f M. r= 6

7 Thus, in rticulr, M c M. For f M c if nd only if f < for ll rimes nd f <. Thus f k = f f A, indeendent of since f. k= The qusi-norm M f T Let f M. From bove we see tht ϕ f M M but, tyiclly, ϕ f is not bounded on M if f l in the sense tht ϕ f / is not bounded by constnt for ll M. It therefore mkes sense to define, for T, M f T = su M = T ϕ f. We im to find the behviour of M f T for lrge T. We shll consider f comletely multilictive nd such tht f P is regulrly vrying of index α with α > / in the sense tht there exists regulrly vrying function f of index α with f = f for every rime. Our min result here is the following: Theorem.3 Let f M c, such tht f nd f P is regulrly vrying of index α where α,. Then log M f T cα flog T log log T log T where f is ny regulrly vrying extension of f P nd cα = B α, α α α α nd Bx, y = tx t y dt is the Bet function. For the roof, we obtin uer nd lower bounds for log M f T which re symtotic to ech other. For the lower bounds, we require formul for ϕ f when M c. This follows from the following rther elegnt formul: Lemm.4 For f, g M c, f g f, g = f g fg, where, denotes the usul inner roduct for l. Proof. We hve f g = f gn = n= = = c,d m= fcfd n= c,d n m[c, d] g c m= c,d n n fcfdg g c d g m[c, d] gm d c fcfdg g. c, d c, d d 7

8 Collecting those terms for which c, d = k, writing c = km, d = kn, nd using comlete multilictivity of f f g = fk fmfngmgn. g But f, g = m,n k= fmfngmgn = m, n m, n = fdgd d= m, n m, n = fmfngmgn, so the result follows. Thus for f, M c, ϕ f = f n= fn n n= fn n. / Since n, s corollry we hve: Corollry.5 For f, M c, fn n ϕ f n= Note tht by comlete multilictivity, n= fn n = f = nd f f = O, so tht f fn n. n= { } ex f + O f, log ϕ f = R f + O..3 Proof of Theorem.3. We consider first uer bounds. The suremum occurs for which we now ssume. Write = n, ϕ f = b = b n. Define α nd β for rime by α = nd β k = b k. k= By multilictivity of nd b we hve T = = + α nd b = + β. Thus Now for k Thus b k = b k ϕ f = k= + β + α. k f r k r = k + fb k. r= = k + f kb k + f b k. Summing from k = to nd dding to both sides gives + β = + α + f kb k + f + β..4 k= 8

9 By Cuchy-Schwrz, so, on rerrnging kb k k= k= k k= b k / = α + β, + β f α + β f + α f. Comleting the squre we find + β f α + α f f. The term on the left inside the squre is non-negtive for sufficiently lrge since f ; in fct from.4, + β +α f which is greter thn f α f if f /. Rerrnging gives + β α + α f + f. + α Let γ = α +α. Tking the roduct over ll rimes gives ϕ f A f { } + fγ A ex fγ.5 for some constnts A, A deending only on f. We cn tke A = if f /. Note tht γ < nd γ = T. Let ε > nd ut P = log T log log T. We slit u the sum on the RHS of.5 into P, P < AP nd > AP for smll nd A lrge. First P fγ P f α P fp α log P < ε flog T log log T log T,.6 for sufficiently smll. Next, using the fct tht log T = log f is regulrly-vrying of index α >AP >AP >AP γ fγ f / A γ α P fp log T α log P flog T log log T log T A α / α / γ, we hve since / < ε flog T log log T log T.7 for A sufficiently lrge. This leves the rnge P < AP. Note tht the result follows from the cse fn = n α. For, by the uniform convergence theorem for regulrly vrying functions P α f fp < εf.8 for P < AP nd P sufficiently lrge, deending only on ε. The roblem therefore reduces to mximising γ α Using x f x ft log t dt P < AP x fx α log x, since f is regulrly-vrying of index α. 9

10 γ subject to γ < nd = T. The mximum clerly occurs for γ decresing if γ > γ for rimes <, then the sum increses in vlue if we sw γ nd γ. Thus we my ssume tht γ is decresing. By interoltion we my write γ = g P where g :,, is continuously differentible nd decresing. Of course g will deend on P. Let h = log g, which is lso decresing. Note tht log T = h h hπp ch log T, P P P for P sufficiently lrge, for some constnt c >. Thus h C indeendent of T. Now, for F :, [, decresing, x< bx F = x x log x b xf F + O log x,.9 where the imlied constnt is indeendent of F nd x. For, on writing πx = lix + ex, the LHS is bx t b F t b F dπt = x x x log xt dt + F t dext = x b [ ] b b F + F text ext df t some θ [, b] x log θx = x log x b xf F + O log x, on using ex = O log x nd the fct tht F is decresing. Thus by.9 log T P < AP h P P log P A h log T Since nd A re rbitrry, h must exist nd is t most. Also, by.9 Hence by.8, P < AP P < AP γ α = P α P < AP fγ fp P α A h. α P α A gu g P P log P u α du. P < AP γ α P fp A gu log P u α du. As, A re rbitrry, it follows from bove nd.5,.6,.7 tht log ϕ f gu u α du + o flog T log log T log T. Thus we need to mximize guu α du subject to h over ll decresing g :,,. Since h is decresing, x xhx h. x/ The RHS cn be mde s smll s we lese for x sufficiently smll or lrge s h converges. In rticulr, xhx s x nd s x +. In fct, for the suremum, we cn consider just those g nd h which re continuously differentible nd strictly decresing, since we cn

11 roximte rbitrrily closely with such functions. On writing g = s h where sx = e x, we hve gu u α [ guu α du = α = α ] α g uu α du s huh uu α du = α h + s xlx α dx, where l = h, since ugu s u. The finl integrl is, by Hölder s inequlity t most h + But h + l = uh udu = h, so gu u α α h + s /α α du α l α.. α /α s. A direct clcultion shows tht 3 s /α = /α B α, α. This gives the uer bound. The roof of the uer bound leds to the otimum choice for g nd the lower bound. We note tht we hve equlity in. if l/s /α is constnt; i.e. lx = cs x /α for some constnt c > chosen so tht l =. This mens we tke hx = s x α = log c + + c x α. from which we cn clculte g. In fct, we show tht we get the required lower bound by just considering n comletely multilictive. To this end we use.3, nd define by: = g, P where P = log T log log T nd g is the function g x = + + c, x α with c = +/α /B α, +o α. As such, by the sme methods s before, we hve = T nd log ϕ α = fg + O P fp g u P log P u α du. By the choice of g, the integrl on the right is B α, α α α α, s required. Remrk. From the bove roof, we see tht the suremum of ϕ f / over M c is roughly the sme size s the suremum over M ; i.e. they re log-symtotic to ech other. Is it true tht these resective surem re closer still; eg. re they symtotic to ech other for < α <? 3. The secil cse fn = n α. In this cse we cn tke fx = x α which is regulrly vrying of index α. Here we shll write ϕ α for ϕ f nd M α for M f. 3 The integrl is /α e x/α e x /α dx = /α t/α t /α dt.

12 Theorem 3. We hve while for < α <, M T = e γ log log T + log log log T + log + o, 3. log M α T = B α, α α log T α α α + o log log T α. 3. Remrk. As noted in the introduction, these symtotic formule ber strong resemblnce to the conjectured mximl order of ζα + it. It is interesting to note tht the bounds found here re just lrger thn wht is known bout the lower bounds for Z α T = mx t T ζα + it. In recent er see [8], Lmzouri suggests log Z α T Cαlog T α log log T α with some secific function 4 Cα for < α <. We note tht the constnt ering in 3. is not Cα since, for α ner, the former is roughly, while Cα α. For α =, see the comment α in the introduction. It would be very interesting to be ble to extend these ides nd results to the α = cse. As we show in the endix, we cnnot do this by restricting ϕ to smller domins in l. Somehow the nlogy if such exists between M α nd Z α breks down just here. Proof of Theorem 3.. For < α < the result follows from Theorem.3, so we only concern ourselves with α =. For n uer bound we use.5 with f = / nd A =. Thus ϕ ζ + γ. γ Agin, the mximum of the RHS subject to γ < nd = T occurs for γ decresing. Let P = log T log log T nd, A be rbitrry constnts such tht A > > >. Slit the roduct into the rnges P, P < AP nd > AP. We hve ζ + γ ζ + = e γ log P + o >AP P P by Merten s Theorem, while the roduct over > AP is t most { } { γ } { ex ex γ ex log T >AP >AP But >AP / /AP log P /A log T log log T, so + γ + A log log T >AP >AP for ll lrge enough T. Combining the bove two estimtes gives ζ + γ e γ log T + log 3 T + log + + o. A P > AP }. For the remining rnge P < AP we write, s before, γ = g P where g :,, is decresing. Then log P < AP + g/p P < AP g/p A gu log P u du, 4 Lmzouri hs Cα = G α α α α α α, where G x = u /x log n= u/ n n! du.

13 by.9. Thus ϕ A eγ gu log T + log 3 T + u du u du + + o A for ll A > > >. We need to minimise the constnt term. Since gu <, the minimum occurs for rbitrrily smll. On the other hnd gu A u du A A g / = o/ A, so the constnt is minimized for rbitrrily lrge A; i.e. it is t most gu u du gu u du. Thus M T e γ log log T + log log log T + κ + o where κ = su{lg : g G}. Here Lg = gu u du gu u du nd G is the set of ll decresing g :,, for so tht g = s h where which log g. As in the roof of Theorem.3, let h = log g sx = e x. Now we show κ = log. Trivilly, by Cuchy-Schwrz, we hve Lg u du gu du h, so κ. Note tht the suremum is chieved for h =. For if h <, then we cn lwys increse g by smll mount while keeing it less thn nd decresing, while h is incresed by rescribed mount just tke g = k g where k :,, is incresing nd kx > x. With kx x sufficiently smll, h while Lg > Lg. Further, we my tke the suremum over g for which g is continuously differentible nd strictly decresing, since they cn roximte functions in G rbitrrily closely. Now, for Lg to be finite i.e. > we need gu u du to converge. For x,, x x gu u The LHS tends to s x +, so we must hve x du gx x u du = gx log x. gx log x s x +. In rticulr, gx s x + so hx s x +. Also, s in Theorem.3, xhx s x. Now, with g = s h, gu u du = [gu log u] where l = h is the inverse function of h. Also, gu u du = [ gu log u] + s huh u log u du = h s y log ly dy, s huh u log u du = s y log ly dy. h Hence Lg = s log l nd l =. Now, using Jensen s inequlity log fdµ log fdµ for µ robbility mesure [],.6, we hve s logl/s = logl/s ds log l/s ds = log l = log. 3.3 Hence s log l log + s log s = log + 3 u log du = log, u

14 fter some clcultion. The roof of the uer bound leds to the otimum choice for g nd the lower bound. We note tht we hve equlity in 3.3 if l/s is constnt; i.e. lx = cs x for some constnt c > chosen so tht l = i.e. we tke c =. Thus, ctully κ = log nd the suremum is chieved for the function g, where g x =. + + x In fct, we show tht we get the required lower bound by just considering n comletely multilictive. To this end we use Corollry.5, nd define by: = g, P where P = log T log log T. As such, by the sme methods s before, we hve = T +o. Let > nd P = log T log log T. By Corollry.5 ϕ = P P + >P. 3.4 Using Merten s Theorem, the first roduct on the right is e γ log P + o, while the second roduct is greter thn { ex log P P } P The sum is symtotic to The third roduct in 3.4 is greter thn { } { + o ex = ex log P by.9. Thus ϕ eγ >P g /P. g u u du < ε log P, for ny given ε >, for sufficiently smll. g u u } du g u log P + du + log ε e γ log P + Lg ε u for sufficiently smll. As Lg = log nd ε rbitrry, this gives the required lower bound. Lower bounds for ϕ α nd some further secultions We cn study lower bounds of ϕ α vi the function m α T = inf M = T ϕ α. Using very similr techniques, one obtins nlogous results to Theorem 3.: m T = 6eγ log log T + log log log T + log + o π nd log m α T B α, α α log T α α α log log T α for < α <. 4

15 We see tht m α T corresonds closely to the conjectured miniml order of ζα + it see [3] nd [9]. We omit the roofs, but just oint out tht for n uer bound for /m α T we use ϕ α + γ α, which cn be obtined in much the sme wy s.5. For the lower bound, we choose s times the choice in Theorem 3. nd use Corollry.5. The bove formule suggest tht the suremum resectively infimum of ϕ α / with M nd = T re close to the suremum res. infimum of ζ α on [, T ]. One could therefore seculte further tht there is close connection between ϕ α / for such nd ζα + it, nd hence between Z α T nd M α T. Recent ers by Gonek [4] nd Gonek nd Keting [5] suggest this my be ossible, or t lest tht M α is lower bound for Z α. On the Riemnn Hyothesis, it ws shown in [4] Theorem 3.5 tht ζs my be roximted for σ > u to height T by the truncted Euler roduct P s for P T. Thus one might exect tht, with M c+ mximizing ϕ α subject to = T, nd As = P with P T, s T T ζα it Ait dt T it α it dt = T P T B it dt T P where B s = k b k ks. The heuristics of Gonek nd Keting now suggests this is symtotic to T b T ϕ α k P k if P log T log log T for the lst ste. Thus it would follow tht Z α T T T ζα it Ait dt T T Ait dt T ϕ α T M α T nd hence Z α T M α T. As mentioned before, this would contrdict Lmzouri s suggestion tht log Z α T Cαlog T α log log T α since Cα < cα nottion from Theorem.3 for α sufficiently close to t lest. It is uncler to the uthor which ossibility is more likely. References [] C. Aistleitner nd K. Sei, GCD sums from Poisson integrls nd systems of dilted functions, rerint, see rxiv:.74v [mth.nt] 9 Oct. [] N. H. Binghm, C. M. Goldie, nd J. L. Teugels, Regulr vrition, Cmbridge University Press, 987. [3] A. Grnville nd K. Soundrrjn, Extreme vlues of ζ + it, Rmnujn Mth. Soc. Lect. Notes Ser, Rmnujn Mth. Soc., Mysore [4] S. Gonek, Finite Euler roducts nd the Riemnn Hyothesis, Trns. Amer. Mth. Soc [5] S. M. Gonek nd J. P. Keting, Men vlues of finite Euler roducts, J. London Mth. Soc [6] T. W. Hilberdink, Determinnts of Multilictive Toelitz mtrices, Act Arith

16 [7] T. W. Hilberdink, An rithmeticl ming nd lictions to Ω-results for the Riemnn zet function, Act Arith [8] Y. Lmzouri, On the distribution of extreme vlues of zet nd L-functions in the stri < σ <, Int. Mth. Res. Not., [9] H. L. Montgomery, Extreme vlues of the Riemnn zet-function, Comment. Mth. Helv [] G. Póly, Uber eine neue Weise, bestimmte Integrle in der nlytischen Zhlentheorie zu gebruchen, Göttinger Nchr [] W. Rudin, Rel nd comlex nlysis 3 rd -edition, McGrw-Hill, 986. [] K. Soundrrjn, Extreme vlues of zet nd L-functions, Mth. Ann [3] O. Toelitz, Zur Theorie der qudrtischen und bilineren Formen von unendlichvielen Veränderlichen, Mth. Ann [4] O. Toelitz, Zur Theorie der Dirichletschen Reihen, Amer. J. Mth [5] A. Wintner, Diohntine roximtions nd Hilbert s sce, Amer. J. Mth Aendix Here we show tht if f l, we cnnot hoe to cture ϕ f by considering the ming on some non-trivil subset of l. Proosition A Suose f diverges, where rnges over the rimes. Then ϕ f l for l if nd only if =. Proof. Suose there exists l with such tht ϕ f l. Let m be the first non-zero coordinte for. Let b = b n = ϕ f l. Consider b m for rime such tht m. We hve b m = d m fd m/d = mf + k, where k = d m fd m/d. Since k fd m/d A m/d <, d m d m d m nd b m converges, we must hve m f <. This is contrdiction. 6