Bounding sums of the Möbius function over arithmetic progressions

Transcription

1 Bounding sums of the Möbius function over arithmetic progressions Lynnelle Ye Abstract Let Mx = n x µn where µ is the Möbius function It is well-known that the Riemann Hypothesis is equivalent to the assertion that Mx = Ox /+ɛ for all ɛ > 0 There has been much interest and progress in further bounding Mx under the assumption of the Riemann Hypothesis In 009, Soundararajan established the current best bound of Mx x exp log x / log log x c setting c to 4, though this can be reduced Halupczok and Suger recently applied Soundararajan s method to bound more general sums of the Möbius function over arithmetic progressions, of the form Mx; q, a = µn n x n a mod q They were able to show that assuming the Generalized Riemann Hypothesis, Mx; q, a satisfies Mx; q, a ɛ x exp log x 3/5 log log x 6/5+ɛ for all q exp log log x3/5 log log x /5, with a such that a, q =, and ɛ > 0 In this paper, we improve Halupczok and Suger s work to obtain the same bound for Mx; q, a as Soundararajan s bound for Mx with a / in the exponent of log x, with no size or divisibility restriction on the modulus q and residue a Introduction The Riemann Hypothesis RH is a conjecture about the zeros of the Riemann zeta function, a meromorphic function ζ : C C defined as follows: for s C with Rs >, we let ζs = For Rs, ζs is the value of the unique analytic continuation of this function to the rest of the complex plane The function ζ is holomorphic everywhere except at s =, where n= n s

2 it has a simple pole For Rs >, we can also write ζs in its Euler product representation ζs = p s p prime The Riemann zeta function satisfies an important functional equation relating it to the Γ function, which is defined by Γs = x s e x dx for s C such that Rs > 0 and 0 analytically continued to the rest of the complex plane The functional equation is s s π s/ Γ ζs = π s/ Γ ζ s The Γ function is nonzero everywhere, and it has simple poles exactly at the nonpositive integers From this and the above equation, we can see that ζs = 0 when s is a negative even integer From the Euler product of ζ, we may compute for Rs > that ζs = n= µn n s where µn is the multiplicative function, called the Möbius function, defined by 0 if p n for some prime p, µn = if n = p p k for distinct primes p i, if n = p p k for distinct primes p i Since µn n= is absolutely convergent for Rs >, we know ζs 0 for Rs >, and n s hence from the functional equation that ζs 0 for Rs < 0, except for the previously mentioned negative even integers So all other zeros of ζ lie in the strip 0 Rs referred to as the critical strip and are known as the nontrivial zeros The behavior of the zeros of the Riemann zeta function is deeply connected to the behavior of the primes in N Let πx be the number of primes no greater than x; then the Prime πx Number Theorem, which states that lim x =, follows from the fact that ζ has x/ log x no zeros on the line Rs = More specific information about the distribution of primes follows from more specific information about the zeros of ζ, such as would be given by the Riemann Hypothesis Conjecture Riemann Hypothesis All nontrivial zeros of ζs lie on the line Rs =, called the critical line From the fact that ζ has no zeros on the line Rs =, the more specific version of the Prime Number Theorem as proven by Hadamard and de la allée Poussin is that πx = x log t dt + Ox exp a log x

3 for some a > 0 Assuming the Riemann Hypothesis, this would be improved to πx = x log t dt + O x log x The Mertens function Mx is defined by Mx = n x µn where µ is the Möbius function It is well-known that the Riemann Hypothesis is equivalent to the assertion that Mx = Ox /+ɛ for all ɛ > 0 see [5], for example, for details There has been much interest and progress in further bounding Mx under the assumption of the Riemann Hypothesis Landau [6] showed in 94 that RH implies Mx = Ox /+ɛ with ɛ log log log x/ log log x, and Titchmarsh [] in 97 improved this to ɛ / log log x In 009, Maier and Montgomery [7] gave a much improved bound of Mx x exp Clog x 39/6 Finally, in [], Soundararajan established the current best bound of Mx x exp log x / log log x c where Soundararajan set c to 4, though Balazard and de Roton decrease it to 5/ + ɛ in [] Gonek has conjectured see Ng in [9] for details that the true range of Mx goes up precisely to the much smaller O xlog log log x 5/4 Halupczok and Suger recently applied Soundararajan s method to bound more general sums of the Möbius function over arithmetic progressions, of the form Mx; q, a = µn n x n a mod q This requires a generalized version of the Riemann Hypothesis, appropriately called the Generalized Riemann Hypothesis GRH The GRH deals with Dirichlet L-functions, defined by n Ls, = n s and analytically continued to the complex plane, where is a certain multiplicative function called a Dirichlet character There are ϕq distinct characters for each modulus q, each with range contained in the qth roots of unity, satisfying ϕq na = n= { if n a mod q 0 otherwise Like ζs, Ls, has a functional equation relating it to Γ, which looks like q s/ κ + s q s/ κ + s Γ Ls, = ɛ Γ L s, π π where κ is equal to 0 if = and if = Consequently its zeros also have a similar structure 3

4 Conjecture Generalized Riemann Hypothesis For every, all nontrivial zeros of Ls, lie on the line Rs = As with the consequences of RH for the distribution of primes, GRH would imply, among other things, similar consequences for the distribution of primes in arithmetic progressions Halupczok and Suger showed that assuming GRH, given ɛ > 0, the bound Mx; q, a ɛ x exp log x 3/5 log log x 6/5+ɛ holds for all progressions a mod q with q exp log log x3/5 log log x /5 and a, q = In this paper, we improve Halupczok and Suger s work to obtain a bound of strength equal to that of Soundararajan, with no restriction on the modulus q or the residue a Our main theorem is as follows Theorem Assuming GRH, given ɛ > 0, the bound Mx; q, a ɛ x/d exp logx/d / log logx/d 3+ɛ holds uniformly for all progressions a mod q, where d = gcda, q Since µn, the trivial bound on Mx; q, a is x/q nontrivial whenever q x / ɛ Our bound therefore remains In Section, we provide preliminaries on the explicit formula for sums over zeros of L- functions, consequent bounds on the deviation from average of the number of zeros in an interval of the critical line, and the large sieve In Section 3, we bound the number of wellseparated ordinates on the critical line with an unusual accumulation of zeros of Ls, nearby In Section 4, we give bounds for Ls, depending on the number of zeros near s In Section 5, we prove Theorem We assume GRH for the rest of this paper Acknowledgements I would like to thank Kannan Soundararajan for advising me on this project, and the Stanford Undergraduate Research Institute in Mathematics for its support 4

5 Preliminaries First consider the case a, q = Using the Dirichlet characters mod q and Perron s formula, we may write Mx; q, a = ϕq = ϕq a mod q n x a πi mod q nµn +/ log x+i x +/ log x i x More generally, let a = bd, q = rd with b, r = We have Mx; q, a = µn = µdm = µd n x n bd mod rd so the sum we are interested in is really just m x/d m b mod r d,m= µm = ϕr m x/d m b mod r mod r b x s ds + Olog x sls, m x/d d,m= m x/d m b mod r d,m= mµm µm where = ϕr mod r b πi +/ logx/d+i x/d +/ logx/d i x/d l d s, = p d x/d s ds + Ologx/d sls, l d s, p p s x/d s sls,l d s, We are now concerned with bounding the average value of the integrand over all, as this suffices to bound Mx; q, a We first require the following lemmas The first constructs smooth approximations to the characteristic function of an interval Lemma Let [ h,h] be the characteristic function of the interval [ h, h] There are even entire functions F + and F depending on h and which satisfy the following properties: F u [ h,h] u F + u for u R F ±u [ h,h] u du 3 ˆF± x = 0 for x, and ˆF ± x = sinπhx πx 4 If z = x + iy with z h then F ± z eπ y z + O/ 5

6 Proof Such functions were originally constructed by Selberg [0] using Beurling s approximation to the sign function They are discussed in detail in [3] We will just state the relevant formulas Let Kz = sin πz πz, Hz = sin πz π n= sgnn z n +, z where sgnx is the sign function, with values for x > 0, for x < 0, and 0 for x = 0 Then the functions F ± are F ± z = H z + h ± K z + h + H h z ± K h z Next we need the following form of the explicit formula for sums over zeros of L-functions Lemma Let be a primitive character mod q Let fs be analytic in the strip Is / + ɛ for some ɛ > 0, real-valued on the real line, and such that fs + s δ for some δ > 0 Then if = / + iγ runs over the nontrivial zeroes of Ls,, we have fγ = π ˆf0 log q + κf + fur Γ / + κ + iu du π i π Γ Λnn log n ˆf + π n π ˆf log n π n= where κ is equal to 0 if = and if = Proof This is a specialization of Theorem 5 in [5] We reproduce the proof here We begin by proving a version of this identity for Mellin transforms This states that for a C function ϕ : 0, + C with compact support and Mellin transform ϕs = ϕxx s dx, and setting ψx = x ϕx, we have 0 ϕ =ϕ log q + κ ϕ + ϕur Γ / + κ + iu du π π Γ n Λnnϕn + Λnnψn To obtain this, we recall that the functional equation for Ls, takes the form q s/ κ + s q s/ κ + s Γ Ls, = ɛ Γ L s, π π 6

7 from which log q π + Γ Γ κ + s + Γ Γ κ + s + L L L s, + s, = 0 L We multiply this equation by ϕs, integrate along the line Rs = δ, and divide by πi The first term becomes ϕ log q by Mellin inversion The line of integration for the second and third terms with Γ /Γ may be shifted to Rs = / while picking up a single pole at s = if κ = 0 and nothing otherwise giving the term κ ϕ, after which the two terms /+κ+iu in the integrand are conjugates and cancel to give ϕur Γ for s = / + iu We Γ shift the line of integration for the fourth and fifth terms with L /L to Rs = δ, picking up a pole with residue ϕ for each zero of L on the /-line Finally, the remaining integrals may be written as n Λnnϕn + Λnnψn by Mellin inversion The Fourier transform version follows from setting ϕx = π x / ˆf log x π, x = e πy Let be a primitive character mod q and let Nt, be the number of zeros / + iγ of Ls, lying in the interval 0 γ t on the critical line, so that the average value of Nt + h, Nt h, is h qt log We now apply the explicit formula to our characteristic π π function approximations F ± to estimate the deviation of the actual value of Nt + h, Nt h, from the average as a Dirichlet polynomial Lemma 3 Let t 5,, and 0 < h t Let be a primitive character mod q Then Nt+h, Nt h, h qt log π π logqt π π R p log p log p p /+it ˆF+ +Olog π p e π and Nt+h, Nt h, h qt log π π logqt π π R p log p p /+it ˆF p e π where F ± are the functions from Lemma log p +Olog π Proof Let fs = F ± s t, so that ˆfx = ˆF ± xe πixt We have F γ t Nt + h, Nt h, = [ h,h] γ t F + γ t Since F ± is real and even on the real line, ˆF± is also real and even Hence log n ˆf + π ˆf log n log n = n it ˆF± + n it ˆF± log n = R π π π n ˆF log n it ± π 7

8 so that using the explicit formula from Lemma, we find F ± γ t = π ˆF ± 0 log q π + κf ± i t π R n= / + κ + iu du + π F ± u tr Γ Γ Λnn n /+it ˆF± log n We have F ± t from Property 4 of Lemma Also, as shown in [3], we have i t that F ± u tr Γ / + κ + iu du = π Γ π log t ˆF ± 0 + O We reproduce the proof here, using Stirling s approximation for Γ /Γ From Property 4 of Lemma, we have F ± u tr Γ / + κ + iu logu + log t du du Γ t+4 t u t t t+4 t and similarly for the integral from to t 4 t, while t+4 t t 4 t F ± u tr Γ Γ / + κ + iu du = = = = t+4 t t 4 t t+4 t t 4 t log t π F ± u t log u + O/u du F ± u t log t + O/t du F ± u t log t logt/ + h du + O t ˆF 0 + O Finally, since ˆF log n ± π = 0 for n e π, we have Λnn log n R n /+it ˆF± = R Λnn π n /+it n= n e π log n ˆF± π = R Λpp log p p /+it ˆF± + R Λpp log p π p +it ˆF± + O π p e π p e π = R Λpp log p p /+it ˆF± + Olog π p e π Finally, we need the following version of the large sieve inequality 8

9 Definition A set of points {s r = σr + it r }, where r =,, R and ranges over the characters mod q, is well-separated if, for any given, we have t i t j for all i j Proposition 4 Let As, = p N appp s be a twisted Dirichlet polynomial Let T be large and let {s r } be a set of well-separated points with T < t < t < < t R T for all and σr α Then for any k with N k qt, we have R k As r, k ϕqt log T k! ap p α r= p N Proof The proof follows that of Proposition 4 in [7]; it is identical except for the inclusion of a summation over characters Let Bs, = As, k = n N k c n nn s and let D r s denote the disk of radius r centered at s By the mean-value property of harmonic functions and the Cauchy-Schwarz inequality, we have Bs, log T Bx + iy, dxdy π D / log T s for all s Since the disks D / log T s r are disjoint and lie in the half-strip σ α / log T, between the lines t = T and t = T +, we can write R Bs r, log T r= α / log T T + By the proof of Theorem 64 of [8], specifically Equation 64, we have T Bσ + it, dtdσ T + T Bσ + it, dt = ϕqt + ON k /q n c n n σ which, using the condition that N k qt, plugging back into the previous inequality, and integrating with respect to σ, gives R Bs r, ϕqt log T n r= c n n α log n ϕqt log T n c n n α since c = 0 If n has prime factorization p e p em m with e = e + e m, then we can explicitly write e m c n = ap i e i e,, e m i= 9

10 so n c n n α = e m ap i e i e n,, e m p k iα i= i k! e m ap i e i e n,, e m p k iα i i= k = k! ap p α p N and we are done 3 Point count bounds In this section, we use Lemma 3 and Lemma 4 to upper-bound the frequency with which an ordinate t may have an abnormal number of zeros of Ls, near it We introduce some quick notation Definition 3 Let at, q = log qlog logqt, bt, q = logqt log logqt We will also need the following elementary inequality Lemma 3 Let at, q bt, q, η = / log, k = +η Then we have klogk log log qt logη log + log log + log log qt Proof This calculation is Proposition 4 in [] We reproduce the proof here Since k, we have logk log logqt logη log + log log 0 log logqt but also k η, so +η klogk log logqt logη η log + log log log logqt log + log log +η log = log log logqt log logqt + log log + 0

11 All our statements about the distribution of zeros of Ls, assume that is primitive, but of course Perron s formula for Mx; q, a includes the imprimitive characters mod q as well To deal with this, we just note that there is not much difference between the behavior of and where is the primitive character inducing Lemma 3 For any character mod q, let be the primitive character inducing and q be the conductor of For any twisted Dirichlet polynomial As, = p N appp s and σ /, we have log q Aσ + it, Aσ + it, max ap p log log q If l d s, = p d, p then we have p s and, as a consequence, log d log l d s, = O log log d log Ls, log Ls, = O log q log log q Proof We can write Aσ + it, Aσ + it, = p q ap p p σ+it max ap p p q p / We then have p q p / p log q p / log q log log q We can write log l d s, = p d log p p s p d log d p = O / log log d as in part, and Ls, = Ls, p q p = Ls, p s l q s,

12 We are now ready to formulate the analog for Dirichlet L-functions of the key estimate in Soundararajan s proof Definition 3 Let q N and let be a character mod q induced by primitive mod q Let T > e, 0 < δ, and at, q bt, q An ordinate t [T, T ] is, δ, -typical of order T if it satisfies the following conditions: i Let y = qt / For all σ, we have n y nλn n σ+it log n logy/n log y ii Every sub-interval of t, t + of length δπ logqt zeroes of Ls, iii Every sub-interval of t, t + of length zeroes of Ls, contains at most + δ ordinates of π log logqt contains at most ordinates of Note that this definition differs from the one used in [4], in that it sets y = qt / rather than T / When there is no risk of confusion, we will shorten, δ, -typical to - typical Our next two propositions give an upper bound for the size of any set of well-separated -atypical ordinates First we bound the size of any set of well-separated ordinates with a given accumulation of zeros in intervals of given size centered on the ordinates Proposition 33 Let be a character mod q with conductor q, T be large, h be such that 0 < h T, at, q bt, q, and {t r } be well-separated ordinates with T < t < t < < t R T for all, such that for all and r we have Let R = R Then Nt r + h Nt r h, h π log q t r π + O R ϕqt exp Proof By Lemma 3, we have for all that log + log log + O log logqt + O Nt r + h Nt r h, h π log q t r π logqt π + p log p log p ˆF + + Olog π p /+it r π p e π

13 Let ap = log p π ˆF log p + π, so that ap 4 and ap p p /+it r p e logqt + Olog π π which implies by Lemma 3 that app p /+it r p e logqt π π + O log + log q log log q Let η = / log and = log logqt Also +η logqt π We have expπ = qt +η/ and log η = log log qlog logqt log log q + 4 log log logqt log q so for q sufficiently large, we have log q log log q log q η On the other hand, for small q, we also have log q log log q 00 η for T sufficiently large Therefore, as long as qt is sufficently large, we have logqt log q π + O log + logqt log log q + η logqt + η 00 + O Let k = +η R η η + η log + η log log T + η 00 + O η, so that in fact expπ k T and we can apply Proposition 4 This gives k Using Lemma 3, this implies R ϕqt exp r p qt +η/ ap p /+it r k ϕqt log T Ck log logqt k log + log log + O log logqt 3

14 We now translate this bound into one for -atypical ordinates Proposition 34 Let be a character mod q, let T be large, let at, q bt, q, and let {t r } be well-separated -atypical ordinates with T < t < t < < t R T for all Let R = R Then R T exp ϕq log log logqt + log log + O Proof Let R be the number of t r in the list failing criterion i For such t r there is a σr such that nλn logy/n n σ r +it log n log y > n y The sum over n = p α with α is p α p σ r +it p α y,α logy/n log y p y p + log log y log logqt p α y,α 3 p α/ The error log q from subtracting off the contribution from imprimitive characters is a small log log q fraction of for qt sufficiently large, as argued in the proof of Proposition 33, so it suffices to count t r such that p logy/p p σr+it log y Since y k qt for all k, applying Proposition 4 gives and we have so R k p y p logy/p p σr+it log y r R p y p y log y/p p log y p y Next let R be the number of t r some t r with t r t r and N t r + πδ logqt, p k ϕqt log T k! p y log log y log logqt, R ϕqt exp log log logqt + O k log y/p p log y in the list failing criterion ii For each such t r there is N t r πδ logqt, > + δ 4

15 from which N t r + πδ logqt, N t r πδ logqt, δ logqt log q t r + O π Applying Proposition 33 on the sets {t 3s+j } for j = 0,, gives the desired bound on R The computation for R 3, the number of t r failing criterion iii, is effectively the same We will additionally find it convenient to obtain the following absolute bound on the amount by which Nt + h, Nt h, can deviate from average This is an analog of Theorem of [3], and is also proven as Propositon 7 of [4] Proposition 35 Let qt be sufficiently large, 0 < h t, and be a primitive character mod q Then Nt + h, Nt h, h qt log π π logqt logqt log log logqt log logqt + + o log logqt logqt Proof We apply Proposition 3 with δ = log Since π log logqt π R p log p log p p /+it ˆF+ π p e p eπ, π p e π we can compute Nt + h, Nt h, h qt log π π logqt log logqt log log logqt + O logqt/ log logqt log logqt log log logqt = logqt k log log logqt logqt + O log logqt log logqt log logqt k=0 = logqt log logqt logqt log log logqt + + o log logqt The above bound allows us to state, for a given t, a choice of for which t is guaranteed to be -typical Proposition 36 For between /+o logqt / log logqt and logqt / log logqt, all ordinates t [T, T ] are -typical of order T 5

16 Proof We have Λn n n log n u log u fu = n u and consequently Λn n y log = n y n log n n Λn n j + log n y n log n j n j y = Λn n j + log = j+ Λn n du j y n j n log n j j y j n j n log n u y = fu du y u log y We conclude that n y nλn n σr+it log n logy/n log n The other two follow directly from Proposition 35 y log y = qt / logqt logqt = o log logqt 4 Ls, bounds Our next goal is to find lower bounds for Ls, given that s has -typical imaginary part, which we will obtain from bounds for the logarithmic derivative L s, Let F s, = L R where runs over the nontrivial zeroes of Ls, We can write L /L in terms s of F Proposition 4 Let be a primitive character mod q, T be sufficiently large, σ, T t T, and σ be such that Lσ + it, 0 Then R L L σ + it, = F σ + it, logqt + O Proof We start with Equation 7 of [], which states that for primitive, L L s, = log q π Γ s + κ + B + Γ s + where RB = R/ From Stirling s formula for Γ /Γ, we have R L L s, = log q π σ + it + κ RΓ + RB + R Γ σ + it + = log q π log σ + it + κ + O σ + it + κ + F σ + it, = F σ + it, logqt + O 6

17 We also need L /L in terms of a sum over primes together with another sum over zeros of L Proposition 4 Let be a primitive character mod q and y Let z C be such that Rz 0 and T Iz T, and assume that z is not a pole of L s, Then L nλn y log = L L z, log y s, y z n z n L L z + OT n y Remark This is Proposition 3 of [4], with the condition y T removed Proof Proposition in [4] states that for Rz > /, nλn y log = L L z, log y z, n z n L L n y The proof is as follows We begin with the identity πi c+i c i y z z n 0 L z + w, yw L w dw = Λnn log y n z n, n y y n κ z z + n + κ which is valid for Rz > / and c > /, and can be obtained by expanding L z + L w, as a Dirichlet series and integrating term by term Then we shift the line of integration to Rw = N κ Rz where N is a positive integer, picking up z, from the pole at 0 and the sums from the nontrivial and trivial zeroes of L respectively Since y z z the residue L z, log y L and L L y n κ z 0 n N+κ z+n+κ L s, logq s for Rs excluding circles of radius / around the trivial zeroes L see page 6 of [], the integral c +i L yw z + w, dw approaches 0 as N approaches c i L w infinity, giving the identity above Finally we note that since R n κ z 0 for n 0, κ {0, }, and Rz 0, it is clear that y n κ z z + n + κ T + n + κ T n 0 n 0 From the two previous propositions, we can write an explicit expression for a lower bound for log Lσ + it, Proposition 43 Let be a character mod q induced by mod q Let T be sufficiently large and t [T, T ] Then we have for all σ and y that Λn n logy/n y / σ F σ + it, log Lσ + it, R + n σ+it log n log y σ / log y log y n y 7

18 log q +O log log q Proof First assume is primitive We integrate the equation from Proposition 4 from z = σ + it to z = + it This gives Λnn log y n it n log n n σ it log n n y = log L + it, + log Lσ + it, log y L L + it, + σ + it, L L σ y u it u it du + OT or, rearranging and absorbing small terms into the error term, log y log Lσ+it, = n y Λnn n σ+it log n log y n L L σ+it, + σ y u it du+olog y u it We divide by log y, take real parts, and use Proposition 4 to turn this into log Lσ + it, = R n y Λnn logy/n F σ + it + logqt n σ+it log n log y log y log y + log y R σ y u it du + O u it where σ y/ σ log y y u it u it du σ it σ y / u du σ it = y / σ F σ + it, σ / log y This gives the inequality for primitive Otherwise we have log q log Ls, = log Ls, + O log log q by Lemma 3 Now we assume that t is -typical and get two explicit bounds for log Lσ+it,, depending on the size of σ The first is for σ not too close to /, and is like Proposition 5 of [4], with the condition T q removed 8

19 Proposition 44 Let be a character mod q Let T be sufficiently large, at, q bt, q, and t [T, T ] be, δ, -typical of order T Then for σ such that + σ logqt and some C > 0, we have log Lσ + it, C δ + log q log log q Proof For y = qt /, we have y / σ exp σ / log y logqt log qt logqt log qt = exp < so by Proposition 43, given that t is, δ, -typical, we have log q log Lσ + it, F σ + it, + O logqt log log q The proof of Proposition 5 in [4] states that F σ + it, = O log qt δ ; we reproduce this computation here For 0 n N = logqt, let I 4πδ n be the set of zeros = / + iγ such that πnδ πn + δ t γ logqt logqt Note that N a n=0 + a dt = + π Then since t is -typical, we use a +c n a 0 a +c t a c Property of Definition 3 to get γ I n R σ + it / iγ N σ / + δ n=0 σ / + logqt + δ πnδ logqt = γ I n σ / σ / + t γ + δ + logqt 4δ σ / + logqt 4δ = OlogqT /δ For the remaining zeros, we note that if t γ / and / σ then σ / + t γ + t γ 5 σ / 8 so citing Equation 63 of [], we have R = σ + it / iγ t γ / 9 t γ / σ / σ / + t γ

20 t γ / 8 + t γ Combining these gives F σ+it, = O log qt δ it, is O/δ, and we are done 8 + t I logqt as desired, so the middle term logqt F σ+ If σ is very close to /, we instead compute the following Proposition 45 Let be a character mod q, T be sufficiently large, at, q bt, q, and t [T, T ] be -typical of order T Then we have for all < σ σ 0 = + logqt that log Lσ+it, log Lσ 0 +it, log σ 0 / log q +δ log log +O σ / δ + log log q Proof This is Proposition 6 of [4]; we reproduce the proof here First assume is primitive Using Proposition 4, we have σ0 log Lσ 0 + it, log Lσ + it, = R L σ0 u + it, du F u + it, du σ L σ = γ σ0 σ u / u / + t γ du = γ log σ 0 / + t γ σ / + t γ Again we separate the sum according to the distance of γ from t By Property 3 of Definition 3, we have t γ <π/log logqt Next, for 0 n N = πδn + log σ 0 / + t γ σ / + t γ t γ <π/log logq T log σ 0 / σ / logqt, let J 4πδ n be the set of γ such that π log logqt Then, by Property of Definition 3, t γ log σ 0 / + t γ + δ σ / + t γ γ J n = + δ log + log + + δ π 0 πδn + + N n= N log n=0 π log log σ 0 / σ / logqt + πδn + π/ log πδn + π/ log log + πδn + π/ log

21 Finally, since t γ 5 +t γ t γ / + δ log log + O/δ for t γ /, we see, again by Equation 63 of [], that log σ 0 / + t γ σ / + t γ t γ / t γ / σ 0 / logqt t γ logqt log + σ 0 / t γ log log qt This gives the bound for primitive For imprimitive, we get the same inequality by Lemma 3 Now we put Propositions 45 and 44 together to get a bound of the right size for x z Lz, for all z in the range we need Proposition 46 Let be a character mod q Let t be sufficiently large, x t, at/, q bt/, q so that t is -typical of order T, and Then for z such that Rz / log x and Iz = t, we have x z Lz, x exp log log x + + δ log log + O logqt δ + log x log log x Remark This is Proposition 9 of [4], with the condition t q removed Proof If Rz +, we can apply Proposition 45 to get logqt log Lz, log / logqt Rz / + + δ log log + O δ + log / logqt / log x = log log q log log q log q + + δ log log + O δ + log log q log q δ + log log q log x logqt + + δ log log + O If Rz > +, we can apply Proposition 44, from which it is clear that log Lz, logqt still satisfies the above bound We conclude that log x z Lz, = Rz log x log Lz, log x + + log x log q log logqt + + δ log log + O δ + log log q log x log q log x + log + + δ log log + O logqt δ + log log q

22 When t is small, the -typicality of t becomes less useful, so we supplement with the following simple bound It is similar to Proposition 8 of [4], but with the q-dependence appearing explicitly in the bound Proposition 47 Let x and T be large and σ = + Then there exists a C > 0 such log x that for all t T and mod q we have Lσ + it, exp C logqt log log x Proof Assume first that is primitive From Equation 64 of [], we can write +it L +it s + it, ds = + Ologq Is + ds L s σ+it σ+it Is I which implies log L + it, log Lσ + it, = t I log + it t I log σ + it + Ologq t + We can see that t I log + it Nt +, Nt, logq t and since σ + it = + it I log x t I log x, we get log σ + it logq t log log x This gives the desired inequality for log Lσ + it, For imprimitive with conductor q, following the same approximation procedure as before, we can write log q log Lσ + it, logq t + O log log q We will find it convenient to write out the results of Propositions 45 and 44 for such that t is guaranteed to be -typical, as in Proposition 7 of [4] with the condition t q removed Proposition 48 Let be a character mod q, t sufficiently large, and < σ Then log Lσ + it, logq t log logq t log σ / log log logq t 3logq t log logq t

23 logq t Proof Let = and δ = /, so that by Proposition 36 t is -typical of order t log logq t Then by Proposition 45 and Proposition 44, we have / logq t log q log Lσ + it, log + δ log log + O σ / δ + log log q logq t log logq t log log log logq t logq t logq t + O σ / log logq t log logq t logq t log logq t log log log logq t 3logq t σ / log logq t 5 Proof of theorem We proceed to our estimation of x/d s sls,l d ds For the sake of brevity, we write x in place s, of x/d We first introduce some notation Definition 5 Let Ax, = +/ log x+i x πi +/ log x i x Definition 5 We set K = log x log x s sls,l d s, ds, l = log x / log log x c where c will be determined later if q exp log x and otherwise l = C for some large constant C, and T k = k for l k K For any, k with l k < K, and n with T k n < T k, let n be the smallest integer in the interval at k, q bt k, q such that all points in [n, n + ] are n, δ, -typical ordinates of order T k We are going to split the line segment of integration into dyadic intervals, then, within each interval, deform the line of integration according to the -typicality of the ordinates inside Lemma 5 Let x and c > Let be a character mod q and 0 < δ Then Ax, x δ exp log x / log log x c++δ + Bx, where T K log x Bx, = n exp n log + + δn log log log x n + D logqn log log x n=t l Proof We deform the path of integration for Ls, as follows It is symmetric across the real axis In the upper half-plane, it consists of the following five groups of segments: 3

24 A vertical segment from + log x to + log x + it l if q exp log x, and otherwise from + ɛ to + ɛ + it l where ɛ is some positive constant, ertical segments from + n + in to + n + in +, log x log x 3 A horizontal segment from + + it log x l to + l + it log x l if q exp log x, and otherwise from + ɛ + it l to + l + it log x l 4 Horizontal segments from + n + in + to + n+ + in +, log x log x 5 A horizontal segment from + T K log x + it K to + log x + it K We now estimate the contribution from each segment By Lemma 3, the factor l d s, contributes a factor of at most expc log dlog log d to all integrands By Proposition 47, in the case q exp log x, segment gives a contribution of /+/ log x+itl x s π sls, l d s, ds /+/ log x e Tl x expc log dlog log d max L/ + / log x + it, dt π t T l 0 /4 + t x expc log dlog log d log T l max t T l L + log x + it, x log T l expc logq T l log log x + O log qlog log q x exp log xlog log x c+ In the case q > exp log x, our choice of segment contributes a constant only By Proposition 46, a segment from group gives a contribution of /+ n / log x+in+ x s π sls, l d s, ds x exp n log n /+ n / log x+in πn expc log dlog log d log x logqn max z [/+ n / log x+in,/+ n / log x+in+] + + δ n log log n + O n δ + x z Lz, log x log log x From segment 3 we get, by Proposition 48, + l log x +it l x s π sls, l d s, ds xtl 3 expc log dlog log d + log x +it l 4

25 A segment from group 4, by Proposition 46, gives /+ n / log x+in+ x s π sls, l d s, ds log x x exp n log n logqn + x exp n+ log n + /+ n / log x+in log x logqn δn log log n + O n δ δ n+ log log n+ + O log x log log x n+ + δ By Proposition 44, for segment 5, we have + log x +it K x s π T K sls, l d s, ds δ x expc log dlog log d + log x +it K log x log log x Before we proceed to bound Bx, and complete the proof, we need one final elementary inequality Lemma 5 Let A and C be positive numbers such that A 4C 4 + Then for all > e C we have A log + C log log e A A C Proof This is Proposition 3 in [] We reproduce the proof here Let f = A log + C log log We compute that and f = A log + C log log + f = + C log C log C log We can see from this that f < 0 for > e C, that f e C = A C + C log C + 4C 4 + C + C log C > 0, and that f = Hence there is a unique 0 > e C such that f 0 = 0, and we have max f = f 0 = 0 A log 0 + C log log 0 = 0 C/ log 0 0 e C Let = e A A C Then f = A A + C log A + C loga + C log A + 5 C A + C log A

26 C log + C log A + CA A C log + A C + C A C A + C A 0 Therefore A log + C log log 0 = e A A C, as desired Now we apply our count of -atypical ordinates from Section 3 to bound Bx, Lemma 53 We have Bx, ɛ ϕq exp log x / log log x max{3+ɛ,5 c+ɛ} Proof Let BT, x, = T n<t log x n exp n log + + δn log log n logqn We first rearrange the sum to put together the terms corresponding to the same value of Let M, T, = {n N T n < T, n = } Then BT, x, = log x n exp log + + δ log log logqn T at,q bt,q at,q bt,q exp log T n<t n = log x logqt + + δ log log M, T, For at, q + we note that trivially M, T, T, and therefore log x exp log + + δ log log M, T, T logqt at,q at,q+ ϕq at,q at,q+ ɛ ϕq exp exp C log qlog log qt log log x log xlog log x 3+ɛ For larger, note that since n is the smallest integer such that all t [n, n + ] are n, δ, -typical of order T, there is some t n [n, n + ] which is n, δ, -atypical of order T Then for i {0, }, the set N i, T = {t n n M, T,, n i mod } is a set of well-separated ordinates satisfying the hypotheses of Proposition 34, so we conclude that N i, T δ ϕqt exp log log logqt δ log log n

27 But M, T, = N 0, T + N, T Thus we also have M, T, δ ϕqt exp log log logqt + + δ log log Plugging this in gives BT, x, δ,ɛ Oϕq exp log xlog log x 3+ɛ +ϕq at + bt exp log log x log logqt log δ log log logqt We apply Lemma 5 with A = log log x log logqt and C = 4 + 5δ, noting that A logqt log log logqt 4C 4 + and > e C for qt sufficiently large Then we get log x log logqt log log δ log log logqt 4+5δ log logqt log logqt log x log log x logqt logqt By our choice of l we have qt explog x / log log x c, so this is 4+5δ log log x c log log x c log x log log x log x / log log x 5 c+5δ log x / log x / We plug back in to BT, x, to get BT, x, δ,ɛ Oϕq explog x / log log x 3+ɛ logqt + ϕq log logqt explog x/ log log x 5 c+5δ and hence δ,ɛ ϕq exp Bx, = T =T k l k K Setting δ = ɛ/5 gives the bound we want log x BT, x, exp D log log x log x / log log x 3+ɛ + log log x 5 c+5δ + D log log x Our main theorem now follows immediately 7

28 Proof of Theorem From the preliminaries we have after setting c = Mx; q, a ϕq Ax/d, + Ologx/d ɛ x/d exp logx/d / log logx/d c++ɛ + x/d exp logx/d / log logx/d 3+ɛ + log logx/d 5 c+ɛ ɛ x/d exp logx/d / log logx/d 3+ɛ References [] M Balazard and A de Roton, Notes de lecture de l article Partial sums of the Möbius function de Kannan Soundararajan, arxiv: [mathnt] [] H Davenport, Multiplicative Number Theory, Springer erlag, GTM, 974 [3] D Goldston and S Gonek, A note on St and the zeros of the Riemann zeta-function, Bull London Math Soc olume , [4] K Halupczok and B Suger, Partial sums of the Möbius function in arithmetic progressions assuming GRH, Funct Approx Comment Math 48 03, 6-90 [5] H Iwaniec and E Kowalski, Analytic number theory, Amer Math Soc Coll Publ [6] E Landau, 80 Über die Möbiussche Funktion, Rend Circ Mat Palermo 48 94, 77- [7] H Maier and H L Montgomery, The sum of the Möbius function, Bull London Math Soc 4 009, 3-6 [8] H L Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics 7, Springer, Berlin, 97 [9] N Ng, The distribution of the summatory function of the Möbius function, Proc London Math Soc , [0] A Selberg, Lectures on sieves, Collected Papers, ol, Springer-erlag, Berlin 989, [] K Soundararajan, Partial sums of the Möbius function, J reine angew Math , 4-5 8

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