Surpassing the Ratios Conjecture in the 1-level density of Dirichlet L -functions
aa r X i v : . [ m a t h . N T ] A p r SURPASSING THE RATIOS CONJECTURE IN THE 1-LEVEL DENSITYOF DIRICHLET L -FUNCTIONS DANIEL FIORILLI AND STEVEN J. MILLER
Abstract.
We study the -level density of low-lying zeros of Dirichlet L -functions in thefamily of all characters modulo q , with Q/ < q ≤ Q . For test functions whose Fouriertransform is supported in ( − / , / , we calculate this quantity beyond the square-rootcancellation expansion arising from the L -function Ratios Conjecture of Conrey, Farmerand Zirnbauer. We discover the existence of a new lower-order term which is not predictedby this powerful conjecture. This is the first family where the 1-level density is determinedwell enough to see a term which is not predicted by the Ratios Conjecture, and proves thatthe exponent of the error term Q − + ǫ in the Ratios Conjecture is best possible. We alsogive more precise results when the support of the Fourier Transform of the test function isrestricted to the interval [ − , . Finally we show how natural conjectures on the distributionof primes in arithmetic progressions allow one to extend the support. The most powerfulconjecture is Montgomery’s, which implies that the Ratios Conjecture’s prediction holds forany finite support up to an error Q − + ǫ . Contents
1. Introduction 22. Background and New Results 62.1. Background and Previous Results 62.2. Unconditional Results 72.3. Results under GRH 92.4. Results beyond GRH 103. The Explicit Formula and Needed Sums 123.1. The Explicit Formula for fixed q q ∈ ( Q/ , Q ] Date : October 10, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Dirichlet L -functions, low-lying zeros, primes in progressions, random matrixtheory, ratios conjecture.The authors thank Andrew Granville, Chris Hughes, Jeffrey Lagarias, Zeév Rudnick and Peter Sarnak formany enlightening conversations, and the referee for many helpful suggestions. The first named author wassupported by an NSERC doctoral, and later on postdoctoral fellowship, as well as NSF grant DMS-0635607,and pursued this work at the Université de Montréal, the Institute for Advanced Study and the University ofMichigan. The second named author was partially supported by NSF grants DMS-0600848, DMS-0970067and DMS-1265673. . Results under Montgomery’s Hypothesis (Theorem 2.13) 29References 301. Introduction
In this paper we study the 1-level density of Dirichlet L -functions with modulus q . Thegoal is to compute this statistic for large support and small error terms, providing a testof the predictions of the lower order and error terms in the L -function Ratios Conjecture.In this introduction we assume the reader is familiar with low-lying zeros of families of L -functions and the Ratios Conjecture, and briefly describe our results. For completeness weprovide a brief review of the subject in §2.1, and state our results in full in §2.2 to §2.4.We let η ∈ L ( R ) be an even real function such that b η is C and has compact support.Denoting by ρ χ = + iγ χ the non-trivial zeros of L ( s, χ ) (i.e., < ℜ ( ρ χ ) < ) and choosing Q a scaling parameter close to q , the 1-level density is D q ( η ) := 1 φ ( q ) X χ mod q X γ χ η (cid:18) γ χ log Q π (cid:19) ; (1.1) throughout this paper a sum over χ mod q always means a sum over all characters, includingthe principal character. If we assume GRH then the γ χ are real. As η ( y ) = c (ˆ η )( y ) is definedfor complex values of y , it makes sense to consider (1.1) for complex γ χ , in case GRH is false(in other words, GRH is only needed to interpret the 1-level density as a spacing statisticarising from an ordered sequence of real numbers, allowing for a spectral interpretation). Wealso study the average of (1.1) over the moduli Q/ < q ≤ Q , which is easier to understandin general: D Q/ ,Q ( η ) := 1 Q/ X Q/ Assume GRH. If the Fourier Transform of the test function η is supportedin ( − , ) , then D Q/ ,Q ( η ) equals b η (0) − log(4 πe γ ) + 1log Q − P p log pp ( p − log Q ! + Z ∞ b η (0) − b η ( t ) Q t/ − Q − t/ dt + Q − / log Q S η ( Q ) , (1.5) where S η ( Q ) = C b η (1) + C b η ′ (1)log Q + O (cid:18)(cid:16) log log Q log Q (cid:17) (cid:19) , (1.6) To rescale we multiply (1.3) by log q/ log Q , replace q t/ − q − t/ with Q t/ − Q − t/ and average over Q/ < q ≤ Q . The term log q averages to log Q + log 2 − O (log Q/Q ) , explaining the “additional” term (log 2 − / log Q . Moreover the average of P p | q log pp − over this range is easily shown to be P p p ( p − + O (log Q/Q ) . ith C := (2 − √ ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) C := C √ − ζ ′ ζ (cid:18) (cid:19) − X p log p ( p − p / + 1 !! . (1.7)We can give a more precise formula for the term S η ( Q ) : see Remark 2.5. While Theorem1.2 is conditional on GRH, in Theorem 2.1 we prove a more precise and unconditional resultfor test functions η whose Fourier transform has support contained in [ − , .The first two terms in (1.5) agree with the Ratios Conjecture’s Prediction. As for theterm Q − S η ( Q ) / log Q , its presence confirms that the error term Q − + o (1) in the RatiosConjecture is best possible, and suggests more generally that the -level density of a familyought to contain a (possibly oscillating) arithmetical term of order Q − + o (1) , a statementwhich should be tested in other families. Interestingly this new term contains the factors ˆ η (1) and b η ′ (1) , and is zero when b η is supported in ( − , . In this case we give a moreprecise estimate for the -level density in Theorem 2.1, in which a lower-order term of order Q σ/ − o (1) appears, where σ = sup( supp b η ) . This term is a genuine lower-order term, andshows that for such test functions the Ratios Conjecture’s prediction is not best possible.We thus show that a transition happens when σ is near . Indeed looking at the differencebetween the -level density and the Ratios Conjecture’s prediction, that is defining E Q ( η ) := D Q/ ,Q ( η ) − b η (0) − log(4 πe γ ) + 1log Q − P p log pp ( p − log Q ! − Z ∞ b η (0) − b η ( t ) Q t/ − Q − t/ dt, (1.8)our results imply that E Q ( η ) = Q − µ ( σ )+ o (1) , where µ ( σ ) = ( σ − if σ ≤ − if ≤ σ < . (1.9)We conjecture that µ ( σ ) should equal − / for all σ ≥ , and that our new lower-order term Q − S η ( Q ) / log Q should persist in this extended range. Conjecture 1.3. Theorem 1.2 holds for test functions η whose Fourier transform has arbi-trarily large finite support σ . In Figure 1, the solid curve represents our results (Theorems 1.2 and 2.1), and the dashedline represents Conjecture 1.3; note the resemblance between this graph and the one appear-ing in Montgomery’s pair correlation conjecture [Mon2]. We prove in Theorem 2.13 thatMontgomery’s Conjecture on primes in arithmetic progressions implies that µ ( σ ) ≤ − / for all σ ≥ .We believe that this phenomenon is universal and should also happen in different families,in the sense that we believe that the Ratios Conjecture’s prediction should be best possiblefor σ ≥ , and should not be for σ < . For example, in [Mil4] it is shown that if the Fourier For σ > , this holds for test functions η for which either b η (1) = 0 or b η ′ (1) = 0 (see Theorem 1.2); seeTheorem 2.1 if σ ≤ . If b η ( u ) vanishes in a small interval around u = 1 , then Theorem 2.6 gives the correctanswer. igure 1. The graph of µ ( σ ) .transform of the involved test function is supported in ( − , , then the Ratios Conjecture’sprediction is not best possible and one can improve the remainder term; however, in thisregion of limited support there are no new, non-zero lower order terms unpredicted by theRatios Conjecture. These results confirm the exceptional nature of the transition point σ = 1 , as is the case in Montgomery’s Pair Correlation Conjecture [Mon2]. Indeed if thislast conjecture were known to hold beyond the point α = 1 , then this would imply thenon-existence of Landau-Siegel zeros [CI].Our plan of attack is to use the explicit formula to turn the -level density into an averageof the various terms appearing in this formula. The bulk of the work is devoted to carefullyestimating the contribution of the prime sum, which when summing over χ mod q becomes asum over primes in the residue class q , averaged over q ∼ Q . Accordingly, the proof ofTheorem 1.2 is based on ideas used in the recent results of the first named author [Fi1], whichimprove on results of Fouvry [Fo], Bombieri, Friedlander and Iwaniec [BFI], Friedlander andGranville [FG2] and Friedlander, Granville, Hildebrandt and Maier [FGHM]. Theorem 1.1of [Fi1] cannot be applied directly here, since this estimate is only valid when looking atprimes up to x modulo q with q ∼ Q , where Q is not too close to x . Additional estimates areneeded, including a careful analysis of the range x − ǫ < Q ≤ x , which required a combinationof divisor switching techniques and precise estimates on the mean value of smoothed sums ofthe reciprocal of Euler’s totient function. Additionally, in our analysis of the -level densityafter using the explicit formula and executing the sum over the family we obtain a sum overprimes in the arithmetic progressions q ; this is one of the cases in which one obtainsan asymptotic in Theorem 1.1 of [Fi1], which explains the occurrence of the lower-order term Q − S η ( Q ) / log Q in Theorem 1.2.The paper is organized as follows. In §2.1 we review previous results on low-lying zerosin families of L -functions and describe the motivation for the Ratios Conjecture. See forexample [GJMMNPP, Mil4] for a detailed description of how to apply the Ratios Conjectureto predict the 1-level density. We describe our unconditional results in §2.2, and then improveour results in §2.3 by assuming GRH. In previous families there often is a natural barrier, andextending the support is related to standard conjectures (for example, in [ILS] the authorsshow how cancelation in exponential sums involving square-roots of primes leads to largersupport for families of cuspidal newforms). A similar phenomenon surfaces here, where in ( − , is related to conjectures on thedistribution of primes in residue classes. We analyze the increase in support provided byvarious conjectures. These range from a conjecture on the variance of primes in the residueclasses, which allow us to reach ( − , , to Montgomery’s conjecture for a fixed residue,which gives us any finite support. The next sections contain the details of the proof; westate the explicit formula and prove some needed sums in §3, and then prove our theoremsin the remaining sections. 2. Background and New Results Background and Previous Results. Assuming GRH, the non-trivial zeros of anynice L -function lie on the critical line, and therefore it is possible to investigate statistics ofits normalized zeros. These zeros are fundamental in many problems, ranging from the distri-bution of primes in congruence classes to the class number [CI, Go, GZ, RubSa]. Numericaland theoretical evidence [Hej, Mon2, Od1, Od2, RS] support a universality in behavior ofzeros of an individual automorphic L -function high above the central point, specifically thatthey are well-modeled by ensembles of random matrices (see [FM, Ha] for histories of theemergence of random matrix theory in number theory). The story is different for the low-lying zeros, the zeros near the central point. A convenient way to study these zeros is viathe 1-level density, which we now describe. Let η ∈ L ( R ) be an even real function whoseFourier transform ˆ η ( y ) = Z ∞−∞ η ( x ) e − πixy dx (2.1)is C and has compact support. Let F N be a (finite) family of L -functions satisfying GRH. The -level density associated to F N is defined by D F N ( η ) = 1 |F N | X g ∈F N X j η (cid:18) log c g π γ ( j ) g (cid:19) , (2.2)where + iγ ( j ) g runs through the non-trivial zeros of L ( s, g ) . Here c g is the “analytic conduc-tor” of g , and gives the natural scale for the low zeros. As η decays, only low-lying zeros (i.e.,zeros within a distance / log c g of the central point s = 1 / ) contribute significantly. Thusthe -level density can help identify the symmetry type of the family. To evaluate (2.2), oneapplies the explicit formula, converting sums over zeros to sums over primes.Based in part on the function-field analysis where G ( F ) is the monodromy group associatedto the family F , Katz and Sarnak conjectured that for each reasonable irreducible family of L -functions there is an associated symmetry group G ( F ) (one of the following five: unitary U ,symplectic USp, orthogonal O, SO(even), SO(odd)), and that the distribution of critical zerosnear / mirrors the distribution of eigenvalues near . The five groups have distinguishable -level densities. To date, for suitably restricted test functions the statistics of zeros of manynatural families of L -functions have been shown to agree with statistics of eigenvalues ofmatrices from the classical compact groups, including Dirichlet L -functions, elliptic curves,cuspidal newforms, Maass forms, number field L -functions, and symmetric powers of GL automorphic representations [AM, AAILMZ, DM1, FI, Gao, Gü, HM, HR, ILS, KaSa1, We often do not need GRH for the analysis, but only to interpret the results. If the GRH is true, thezeros lie on the critical line and can be ordered, which suggests the possibility of a spectral interpretation. aSa2, Mil1, MilPe, RR, Ro, Rub, ShTe, Ya, Yo2], to name a few, as well as non-simplefamilies formed by Rankin-Selberg convolution [DM2].In addition to predicting the main term (see for example [Con, KaSa1, KaSa2, KeSn1,KeSn2, KeSn3]), techniques from random matrix theory have led to models that capturethe lower order terms in their full arithmetic glory for many families of L -functions (see forexample the moment conjectures of [CFKRS] or the hybrid model in [GHK]). Since themain terms agree with either unitary, symplectic or orthogonal symmetry, it is only in thelower order terms that we can break this universality and see the arithmetic of the familyenter. These are therefore natural and important objects to study, and can be isolated inmany families [HKS, Mil2, Yo1]. We thus require a theory that is capable of making detailedpredictions. Recently the L -function Ratios Conjecture [CFZ1, CFZ2] has had great successin determining lower order terms. Though a proof of the Ratios Conjecture for arbitrarysupport is well beyond the reach of current methods, it is an indispensable tool in currentinvestigations as it allows us to easily write down the predicted answer to a remarkable levelof precision, which we try to prove in as great a generality as possible.To study the 1-level density, it suffices to obtain good estimates for R F N ( α, γ ) := 1 |F N | X g ∈F N L (1 / α, g ) L (1 / γ, g ) . (2.3)(In the current paper, the parameter Q plays the role of |F N | .) Asymptotic formulasfor R F N ( α, γ ) have been conjectured for a variety of families F N (see [CFZ1, CS1, CS2,GJMMNPP, HMM, Mil3, Mil4, MilMo]) and are believed to hold up to errors of size |F N | − / ǫ for any ǫ > . The evidence for the correctness of this error term is limitedto test functions with small support (frequently significantly less than ( − , ), though insuch regimes many of the above papers verify this prediction. Many of the steps in theRatios Conjecture’s recipe lead to the addition or omission of terms as large as those beingconsidered, and thus there was uncertainty as to whether or not the resulting predictionsshould be accurate to square-root cancelation. The results of the current paper can be seenas a confirmation that this is the right error term for the final predicted answer, at least inthis family. Further, the novelty in our results resides in the fact that we are able to go be-yond square-root cancelation and we find a smaller term which is unpredicted by the RatiosConjecture (see Theorem 1.2). For a precise explanation on how to derive the Ratios Con-jecture’s prediction in our family, we refer the reader to [GJMMNPP], and also recommend[CS1] for an accessible overview of the Ratios Conjecture.2.2. Unconditional Results. We now describe our unconditional results. We remind thereader that η is a real even function such that b η is C and has compact support. Theorem 2.1. Suppose that the Fourier transform of the test function η is supported onthe interval [ − , , so σ = sup( supp b η ) ≤ . There exists an absolute positive constant c (coming from the Prime Number Theorem) such that the 1-level density D q ( η ) (from (1.1) ith scaling parameter Q = q ) equals b η (0) − log(8 πe γ )log q − P p | q log pp − log q ! + Z ∞ b η (0) − b η ( t ) q t/ − q − t/ dt − φ ( q ) Z q u/ (cid:18) b η ( u )2 − b η ′ ( u )log q (cid:19) du − q X p ν k qp e ≡ q/p ν e,ν ≥ log pφ ( p ν ) p e/ b η (cid:18) log p e log q (cid:19) + O (cid:18) q σ − e c √ σ log q (cid:19) . (2.4) Remark 2.2. The average over Q/ < q ≤ Q of the fourth term in (2.4) can be shown tobe O ( Q − ) , and is therefore negligible when considering D Q/ ,Q ( η ) (see (3.16) ). However,the term involving the second integral in (2.4) is of size q σ/ − − o (1) , and thus constitutes agenuine lower-order term, smaller than the error term in (1.3) predicted using the RatiosConjecture. Theorems 1.2 and 2.1 should be compared to the main result of Goes, Jackson, Miller,Montague, Ninsuwan, Peckner and Pham [GJMMNPP], where they show one can extendthe support of b η to [ − , and still get the main term, as well as the lower order terms downto a power savings. They only consider q prime, and thus the sum over primes p dividing q below in Theorem 2.3 is absorbed by their error term. We briefly discuss how one can easilyextend their results to the case of general q . First note that L ( s, χ ) and L ( s, χ ∗ ) have thesame zeros in the critical strip if χ ∗ is the primitive character of conductor q ∗ inducing thenon-principal character χ of conductor q . We now have log q ∗ , which can be converted to asum over primes p dividing q by the same arguments as in the proof of Proposition 3.1. Therest of the expansion follows from expanding the digamma function Γ ′ / Γ in the integral inTheorem 1.3 of [GJMMNPP] and then standard algebra (along the lines of the computationsin §3). We use Lemma 12.14 of [MonVa2], which in our notation says that for a, b > wehave Z ∞−∞ Γ ′ ( a ± ibτ )Γ( a ± ibτ ) η ( t ) dt = Γ ′ ( a )Γ( a ) b η (0) + 2 πb Z ∞ exp( − πax/b )1 − exp( − πx/b ) ( b η (0) − b η ( ∓ x )) dx, (2.5)and the identity Γ ′ (1 / / 4) + Γ ′ (3 / / 4) = − γ − , (2.6)with γ the Euler-Mascheroni constant. We then extend to q ∈ ( Q/ , Q ] by rescaling the zerosby log Q and not log q and summing over the family; recall the technical issues involved inthe rescaling are discussed in Footnote 2. Theorem 2.3 (Goes, Jackson, Miller, Montague, Ninsuwan, Peckner, Pham [GJMMNPP]) . If < σ ≤ , then the 1-level density D q ( η ) (from (1.1) with scaling parameter Q = q )equals b η (0) − log(8 πe γ )log q − P p | q log pp − log q ! + Z ∞ b η (0) − b η ( t ) q t/ − q − t/ dt + O (cid:18) log log q log q q σ − (cid:19) , (2.7) and this agrees with the Ratios Conjecture. emark 2.4. Goes et al. [GJMMNPP] actually proved (2.7) for any σ ≤ , with theadditional error term O ( q − / ǫ ) . We prefered not to include the case σ ≤ , as Theorem2.1 is more precise in this range. Results under GRH. We first mention a more precise version of Theorem 1.2. Remark 2.5. If in addition to the hypotheses of Theorem 1.2 we assume that the Fouriertransform of the test function η is K + 1 times continuously differentiable, then we can givea more precise expression for the term S η ( Q ) appearing in (1.5) : S η ( Q ) = K X i =0 a i ( η )(log Q ) i + O ǫ,K (cid:18) Q ) K +1 − ǫ (cid:19) , (2.8) where the a i ( η ) are constants depending (linearly) on the Taylor coefficients of b η ( t ) at t = 1 .In fact, S η ( Q ) is a truncated linear functional, which composed with the Fourier Transformoperator is supported on { } (in the sense of distributions). Our next result is an extension of Theorem 1.2, in the case where ˆ η ( u ) vanishes in a smallinterval to the right of u = 1 . Theorem 2.6. Assume GRH. (1) If ˆ η is supported in ( − , − − κ ] ∪ [ − , ∪ [1 + κ, ) for some κ > , then for any ǫ > the average 1-level density D Q/ ,Q ( η ) equals ˆ η (0) − πe γ )log Q − P p log pp ( p − log Q ! + Z ∞ b η (0) − b η ( t ) Q t/ − Q − t/ dt − Q ζ (2) ζ (3) ζ (6) Z Q u/ (cid:18) ˆ η ( u )2 − ˆ η ′ ( u )log Q (cid:19) du − Z / κ (( u − 1) log Q + C ) Q − u/ (cid:18) ˆ η ( u )2 − ˆ η ′ ( u )log Q (cid:19) du + O ǫ ( Q − − κ + ǫ + Q − log Q + Q σ − log Q ) , (2.9) with C := log( π/ 2) + 1 + γ + P p log pp ( p − .Note that for σ ≥ , unless ˆ η ( x ) has some mass near x = λ for some < λ < − σ ,the fourth term in (2.9) goes in the error term (and hence (2.9) reduces to (2.10) ).However, if < σ < , it is always a genuine lower-order term of size Q − σ/ o (1) . (2) If f is supported in ( − , − a ] ∪ [ − , ∪ [ a, for some ≤ a < (if a = 1 , then wehave the full interval ( − , ), then we have that D Q/ ,Q ( η ) equals ˆ η (0) − πe γ )log Q − P p log pp ( p − log Q ! + Z ∞ b η (0) − b η ( t ) Q t/ − Q − t/ dt − Q ζ (2) ζ (3) ζ (6) Z Q u/ (cid:18) b η ( u )2 − b η ′ ( u )log Q (cid:19) du + O ( Q − a + Q σ − log Q ) . (2.10) Unless a > and σ < , the third term of (2.10) goes in the error term. .4. Results beyond GRH. As the GRH is insufficient to compute the 1-level densityfor test functions supported beyond [ − , , we explore the consequences of other standardconjectures in number theory involving the distribution of primes among residue classes.Before stating these conjectures, we first set the notation. Let ψ ( x ) := X n ≤ x Λ( n ) , ψ ( x, q, a ) := X n ≤ xn ≡ a mod q Λ( n ) , (2.11) E ( x, q, a ) := ψ ( x, q, a ) − ψ ( x ) φ ( q ) . (2.12)If we assume GRH, we have that ψ ( x ) = x + O ( x (log x ) ) , E ( x, q, a ) = O ( x (log x ) ) . (2.13)Our first result uses GRH and the following de-averaging hypothesis, which depends on aparameter δ ∈ [0 , . Hypothesis 2.7. We have X Q/ Theorem 2.8. Assume GRH and Hypothesis 2.7 for some δ ∈ (0 , . The average 1-leveldensity D Q/ ,Q ( b η ) equals ˆ η (0) − πe γ )log Q − P p log pp ( p − log Q ! + Z ∞ b η (0) − b η ( t ) Q t/ − Q − t/ dt + O ( Q δ − (log Q ) + Q σ +2 δ − (log Q ) ) , (2.15) which is asymptotic to b η (0) provided the support of b η is contained in ( − δ, − δ ) . The proof of Theorem 2.8 is given in §6. It uses a result of Goldston and Vaughan [GV],which is an improvement of results of Barban, Davenport, Halberstam, Hooley, Montgomeryand others. Note that we only need this de-averaging hypothesis for the special residue class a = 1 . emark 2.9. In Theorem 2.8 we study the weighted 1-level density D Q/ ,Q ( η ) := X Q/ Hypothesis 2.10 (Montgomery) . For any a, q such that ( a, q ) = 1 and q ≤ x , we have ψ ( x ; q, a ) − ψ ( x ) φ ( q ) ≪ ǫ x ǫ (cid:18) xq (cid:19) / . (2.18)It is by gaining some savings in q in the error E ( x, q, a ) that we can increase the support forfamilies of Dirichlet L -functions. The following weaker version of Montgomery’s Conjecture,which depends on a parameter θ ∈ (0 , / , also suffices to increase the support beyond [ − , . Hypothesis 2.11. For any a, q such that ( a, q ) = 1 and q ≤ x , we have ψ ( x ; q, − ψ ( x ) φ ( q ) ≪ ǫ x + ǫ q θ . (2.19) Hypothesis 2.12. Fix ǫ > . We have for x ǫ ≤ q ≤ √ x that X n ≤ xn ≡ q Λ( n ) (cid:16) − nx (cid:17) − φ ( q ) X n ≤ x Λ( n ) (cid:16) − nx (cid:17) = o ( x / ) . (2.20)Note that this is a weighted version of ψ ( x ; q, − ψ ( x ) φ ( q ) ; that is, we added the weight (cid:0) − nx (cid:1) . The reason for this is that it makes the count smoother, and this makes it easier toanalyze in general since the Mellin transform of g ( y ) := 1 − y in the interval [0 , is decayingfaster in vertical strips than that of g ( y ) ≡ .Amongst the last three hypotheses, Hypothesis 2.12 is the weakest, but it is still sufficientto derive the asymptotic in the -level density for test functions with arbitrary large support. Theorem 2.13. For η whose Fourier Transform has arbitrarily large (but finite) support,we have the following: (1) If we assume Hypothesis 2.12, then the 1-level density D q ( η ) equals b η (0) + o (1) ,agreeing with the scaling limit of unitary matrices. If we assume Hypothesis 2.11 for some < θ ≤ , then D q ( η ) equals b η (0) − log(8 πe γ )log q − P p | q log pp − log q ! + Z ∞ b η (0) − b η ( t ) q t/ − q − t/ dt + O ǫ ( q − θ + ǫ ) . (2.21) Remark 2.14. Under GRH, the left hand side of (2.20) is O ( x / log q ) . Therefore, if wewin by more than a logarithm over GRH, then we have the expected asymptotic for the 1-leveldensity for b η of arbitrarily large finite support.Interestingly, if we assume Montgomery’s Conjecture (Hypothesis 2.10), then we can take θ = 1 / in (2.21) , and doing so we end up precisely with the Ratios Conjecture’s prediction(see (1.3) ). We derive the explicit formula for the families of Dirichlet characters in §3, as well assome useful estimates for some of the resulting sums. We give the unconditional results in§4, Theorems 2.1 and 2.3. The proofs of Theorems 1.2 and 2.6 are conditional on GRH,and use results of [FG2] and [Fi1]; we give them in §5. We conclude with an analysis of theconsequences of the hypotheses on the distribution of primes in residue classes, using thede-averaging hypothesis to prove Theorem 2.8 in §6 and Montgomery’s hypothesis to proveTheorem 2.13 in §7. 3. The Explicit Formula and Needed Sums Our starting point for investigating the behavior of low-lying zeros is the explicit formula,which relates sums over zeros to sums over primes. We follow the derivation in [MonVa2](see also [ILS, RS], and [Da, IK] for all needed results about Dirichlet L -functions). Wefirst derive the expansion for Dirichlet characters with fixed conductor q , and then extendto q ∈ ( Q/ , Q ] . We conclude with some technical estimates that will be of use in provingTheorem 1.2. Here and throughout, we will set f := b η . Note that η is real and even, andthus so is the case for f , and moreover we have b f = η .3.1. The Explicit Formula for fixed q .Proposition 3.1 (Explicit Formula for the Family of Dirichlet Characters Modulo q ) . Let f be an even, twice differentiable test function with compact support. Denote the non-trivialzeros of L ( s, χ ) by ρ χ = 1 / iγ χ . Then the 1-level density D ,q ( b f ) equals φ ( q ) X χ mod q X γ χ b f (cid:18) γ χ log Q π (cid:19) = f (0)log Q log q − log(8 πe γ ) − X p | q log pp − + Z ∞ f (0) − f ( t ) Q t/ − Q − t/ dt − Q X p ν k qp e ≡ q/p ν e,ν ≥ log pφ ( p ν ) p e/ f (cid:18) log p e log Q (cid:19) − Q X n ≡ q − φ ( q ) X n ! Λ( n ) n / f (cid:18) log n log Q (cid:19) + O (cid:18) φ ( q ) (cid:19) . (3.1) Proof. We start with Weil’s explicit formula for L ( s, χ ) , with χ mod q a non-principal char-acter (we add the contribution from the principal character later). We can replace L ( s, χ ) by ( s, χ ∗ ) (where χ ∗ is the primitive character of conductor q ∗ inducing χ ), since these havethe same non-trivial zeros. Taking F ( x ) := π log Q f (cid:16) πx log Q (cid:17) in Theorem 12.13 of [MonVa2](whose conditions are satisfied by our restrictions on f ), we find Φ( s ) = b f (cid:16) log Q π ( s − ) i (cid:17) , and X ρ χ b f (cid:18) log Q π γ χ (cid:19) = f (0)log Q (cid:18) log( q ∗ /π ) + Γ ′ Γ (cid:18) 14 + a ( χ )2 (cid:19)(cid:19) − Q ∞ X n =1 Λ( n ) ℜ ( χ ∗ ( n )) n / f (cid:18) log n log Q (cid:19) + 4 π log Q Z ∞ e − (1+2 a ( χ )) πx − e − πx (cid:18) f (0) − f (cid:18) πx log Q (cid:19)(cid:19) dx, (3.2)where a ( χ ) = 0 for the half of the characters with χ ( − 1) = 1 and for the half with χ ( − 1) = − . Making the substitution t = πx log Q in the integral and summing over χ = χ ,we find X χ = χ X γ χ b f (cid:18) γ χ log Q π (cid:19) = f (0)log Q X χ = χ log( q ∗ /π ) + φ ( q )2 Γ ′ Γ (cid:18) (cid:19) + φ ( q )2 Γ ′ Γ (cid:18) (cid:19)! + φ ( q ) Z ∞ Q − t/ + Q − t/ − Q − t ( f (0) − f ( t )) dt − Q φ ( q ) X n ≡ q − X n ! Λ( n ) n / f (cid:18) log n log Q (cid:19) − Q X χ = χ X n Λ( n ) ℜ ( χ ∗ ( n ) − χ ( n )) n / f (cid:18) log n log Q (cid:19) + O (1) . (3.3)To get (3.3) from (3.2) we added zero by writing χ ∗ ( n ) as ( χ ∗ ( n ) − χ ( n )) + χ ( n ) . Summing χ ( n ) over all χ mod q gives φ ( q ) if n ≡ q and otherwise; as our sum omits theprincipal character, the sum of χ ( n ) over the non-principal characters yields the sum on thethird line above. We also replaced ( φ ( q ) − / by φ ( q ) / in the first term, hence the O (1) .We use Proposition 3.3 of [FiMa] for the first term (which involves the sum over theconductor of the inducing character). We then use the duplication formula of the digammafunction ψ ( z ) = Γ ′ ( z ) / Γ( z ) to simplify the next two terms, namely ψ (1 / 4) + ψ (3 / . As ψ (1 / 2) = − γ − (equation 6.3.3 of [AS]) and ψ (2 z ) = ψ ( z ) + ψ ( z + ) + ln 2 (equation6.3.8 of [AS]), setting z = 1 / yields ψ (1 / 4) + ψ (3 / 4) = − γ − . We keep the next twoterms as they are, and then apply Proposition 3.4 of [FiMa] (with r = 1 ) for the last term,obtaining that it equals − Q X n Λ( n ) n / f (cid:18) log n log Q (cid:19) Re X χ = χ ( χ ∗ ( n ) − χ ( n )) ! . (3.4)Writing n = p e , this term is zero unless p | q . If p | q , then it is zero unless p e ≡ q/p ν ,where ν ≥ is the largest ν such that p ν | q . Therefore this term equals − Q X p X e,ν ≥ p ν k q,p e ≡ q/p ν Λ( p e ) φ ( p ν ) p e/ f (cid:18) log p e log Q (cid:19) . (3.5) ombining the above and some elementary algebra yields φ ( q ) X χ = χ X γ χ b f (cid:18) γ χ log Q π (cid:19) = f (0)log Q log q − log(8 πe γ ) − X p | q log pp − + Z ∞ f (0) − f ( t ) Q t/ − Q − t/ dt − Q X n ≡ q − φ ( q ) X n ! Λ( n ) n / f (cid:18) log n log Q (cid:19) − Q X p ν k qp e ≡ q/p ν e,ν ≥ log pφ ( p ν ) p e/ f (cid:18) log p e log Q (cid:19) + O (cid:18) φ ( q ) (cid:19) . (3.6)Finally, since the non-trivial zeros of L ( s, χ ) coincide with those of ζ ( s ) , the differencebetween the left hand side of (3.1) and that of (3.6) is φ ( q ) X γ ζ b f (cid:18) γ ζ log Q π (cid:19) ≪ φ ( q ) (3.7)(since f is twice continuously differentiable, b f ( y ) ≪ /y ), completing the proof. (cid:3) The Averaged Explicit Formula for q ∈ ( Q/ , Q ] . We now average the explicitformula for D q ( b f ) (Proposition 3.1) over q ∈ ( Q/ , Q ] . We concentrate on deriving usefulexpansions, which we then analyze in later sections when we determine the allowable support. Proposition 3.2 (Explicit Formula for the Averaged Family of Dirichlet Characters Modulo q ) . The averaged 1-level density, D Q/ ,Q ( b f ) , equals D Q/ ,Q ( b f ) = 1 Q/ X Q/ 1) ; (3.14)however, it can be as large as C √ q log Q for other values of q (such as q = 2(2 e − ). This ismore or less as large as it can get, since for general q we have T ( q ) ≪ Q X p ν k qe,ν ≥ p e ≤ Q σ log pφ ( p ν )( q/p ν ) / ≪ (log q ) q log log q . (3.15)On average, however, T ( q ) is very small: Q/ X Q/ It will be convenient later that in the averaged case ψ and ψ are both evaluatedat ( Q u ; q, and not ( q u ; q, ; this is because we are rescaling all L -function zeros by the samequantity (a global rescaling instead of a local rescaling). Technical Estimates. In the proof of Theorem 2.6, we need the following estimationof a weighted sum of the reciprocal of the totient function. Lemma 3.4. Let φ be Euler’s totient function. We have X r ≤ R φ ( r ) (cid:16) R / + rR / − r / (cid:17) = D R / log R + D R / + D + O (cid:18) log RR / (cid:19) , (3.27) where D := ζ (2) ζ (3) ζ (6) , D := D γ − − X p log pp − p + 1 ! ,D := − ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) . (3.28) More generally, if P ( u ) := P di =0 a i u i is a polynomial of degree d and of norm k P k := max i | a i | , (3.29) then X r ≤ R φ ( r ) Z log r log R P ( u ) (cid:16) R u − rR u (cid:17) du = E log R Z −∞ R u uP ( u ) du + E Z −∞ R u P ( u ) du + d +1 X j =1 F j ( P )(log R ) j + O d ( R − k P k ) (3.30) here E := ζ (2) ζ (3) ζ (6) , E := E γ − − X p log pp − p + 1 ! , (3.31) and the F j ( P ) are constants depending on P which can be computed explicitly. For example, F ( P ) = − ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) d X i =0 ( − i P ( i ) (1) F ( P ) = − ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) ζ ′ ζ (cid:18) (cid:19) − X p log p ( p − p / + 1 ! d X i =1 ( − i P ( i ) (1) . (3.32) Finally, X r ≤ R φ ( r ) Z log( r/ R/ P ( u ) (cid:18) ( R/ u − r R/ u (cid:19) du = E log( R/ Z −∞ ( R/ u uP ( u ) du + ( E + E log 2) Z −∞ ( R/ u P ( u ) du + d +1 X j =1 F (2) j ( P )(log( R/ j + O d ( R − k P k ) , (3.33) where the first two constants are given by F (2)1 ( P ) := F ( P ) √ F (2)2 ( P ) := − √ ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) × ζ ′ ζ (cid:18) (cid:19) − X p log p ( p − p / + 1 + log 2 ! d X i =1 ( − i P ( i ) (1) . (3.34) Remark 3.5. It is possible to improve the estimates in (3.27) , (3.30) and (3.33) to oneswith an error term of O ǫ,d ( R − / ǫ k P k ) ; however, this is not needed for our purposes.Proof. By Mellin inversion, for c ≥ the left hand side of (3.27) equals πi Z ℜ ( s )= c Z ( s ) R s + s + R s + s + 1 − R s + s + ! ds = 12 πi Z ℜ ( s )= c Z ( s ) R s + s ( s + )( s + 1) ds, (3.35)where Z ( s ) := X n ≥ n s φ ( n ) . (3.36)Taking Euler products, Z ( s ) = ζ ( s + 1) ζ ( s + 2) Z ( s ) , (3.37) here Z ( s ) := Y p (cid:18) p ( p − (cid:18) p s +1 − p s +2 (cid:19)(cid:19) , (3.38)which converges for ℜ ( s ) > − . We shift the contour of integration to the left to the line ℜ ( s ) = − + ǫ . By a standard residue calculation, we get that (3.35) equals D R / log R + D R / + D + D log RR / + D R / + 12 πi Z ℜ ( s )= − + ǫ Z ( s ) R s + s ( s + )( s + 1) ds (3.39)for some constants D and D . The proof now follows from standard bounds on the zetafunction, which show that this integral is ≪ ǫ R − ǫ . See the proof of Lemma 6.9 of [Fi1] formore details.We now move to (3.30). The Mellin transform in this case is (for ℜ ( s ) > ) α ( s ) := Z R r s − Z log r log R P ( u ) (cid:16) R u − rR u (cid:17) dudr = Z −∞ P ( u ) Z R u r s − (cid:16) R u − rR u (cid:17) drdu = Z −∞ P ( u ) R u ( s + ) s ( s + 1) du, (3.40)which is now defined for ℜ ( s ) > − / . To meromorphically extend α ( s ) to the whole complexplane, we integrate by parts n times: α ( s ) = R s + s ( s + 1) n X i =0 ( − i P ( i ) (1)( s + ) i +1 (log R ) i +1 , (3.41)which is a meromorphic function with poles at the points s = 0 , − / , − . The integral weneed to compute is πi Z ℜ ( s )=1 Z ( s ) α ( s ) ds. (3.42)We remark that α ( − / ǫ + it ) ≪ ǫ,d R − ǫ t k P k , (3.43)hence the proof is similar as in the previous case, since by shifting the contour of integrationto the left, we have πi Z ℜ ( s )=1 Z ( s ) α ( s ) ds = A + O ǫ,d ( R − ǫ k P k ) , (3.44)where A is the sum of the residues of Z ( s ) α ( s ) for − / ǫ ≤ ℜ ( s ) ≤ . Note that if β ( s ) := s ( s + 1) α ( s ) , then β (0) = Z −∞ R u P ( u ) du, β ′ (0) = log R Z −∞ R u uP ( u ) du, (3.45) o the residue at s = 0 equals ζ (2) ζ (3) ζ (6) β (0) β ′ β (0) + γ − − X p log pp − p + 1 ! . (3.46)For the pole at s = − / , we need to use the analytic continuation of α ( s ) provided in (3.41),which shows that this residue equals n +1 X j =1 F j ( P )(log R ) j , (3.47)where the F j ( P ) are constants depending on P which can be computed explicitly. Forexample, F ( P ) = − ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) d X i =0 ( − i P ( i ) (1) F ( P ) = − ζ (cid:18) (cid:19) Y p (cid:18) p − p / (cid:19) ζ ′ ζ (cid:18) (cid:19) − X p log p ( p − p / + 1 ! d X i =1 ( − i P ( i ) (1) . (3.48)Moreover, F i ( P ) ≪ d k P k for all i .At s = − , we have a double pole with residue R − n +1 X j =0 G j ( P )(log R ) j , (3.49)for some constants G j ( P ) ≪ d k P k , hence the the proof of (3.30) is complete.For the proof of (3.33), we proceed in the same way, noting that the Mellin transform is α ( s ) = 2 s s ( s + 1) Z −∞ P ( u )( R/ u ( s + ) du. (3.50) (cid:3) Unconditional Results (Theorems 2.1 and 2.3) Using the expansion for the 1-level density D ,q ( b f ) and the averaged 1-level density D Q/ ,Q ( b f ) from Propositions 3.1 and 3.2, we prove our unconditional results. Proof of Theorem 2.1. We start from Proposition 3.1. The only term of (3.1) we need tounderstand is the last one (the “prime sum”), which is given by T ( q ) := 2 Z (cid:18) f ( u )2 − f ′ ( u )log q (cid:19) ψ ( q u ; q, − ψ ( q u ) φ ( q ) q u/ du. (4.1)(We used that the support of f is contained in [ − , and we made the substitution t = q u .)However, since there are no integers congruent to q in the interval [2 , q u ] when u ≤ (this is also true when q u is replaced by Q u , with Q/ < q ≤ Q ), the ψ ( q u ; q, term equals ero. By the Prime Number Theorem there is an absolute, computable constant c > suchthat T ( q ) = − Z (cid:18) f ( u )2 − f ′ ( u )log q (cid:19) ψ ( q u ) q u/ φ ( q ) du = − φ ( q ) Z q u/ (cid:18) f ( u )2 − f ′ ( u )log q (cid:19) du + O (cid:18) φ ( q ) Z σ q u/ e c √ u log q du (cid:19) , (4.2)and the error term is ≪ q σ/ φ ( q ) Z σ/ e − c √ u log q du + e − c √ σ log q φ ( q ) Z σσ/ q u/ du ≪ q σ/ − e c ′ √ σ log q (4.3)for q large enough (in terms of σ ), completing the proof. (cid:3) Proof of Theorem 2.3. Starting again from Proposition 3.1, we have that − Q X p ν k qp e ≡ q/p ν e,ν ≥ log pφ ( p ν ) p e/ f (cid:18) log p e log Q (cid:19) ≪ (log q ) q log log q (4.4)(see (3.15)), hence this goes in the error term and the only term we need to worry about isthe last one.As our support exceeds [ − , , the ψ ( q u ; q, no longer trivially vanishes, and the lastterm is T ( q ) = 2 Z (cid:18) f ( u )2 − f ′ ( u )log q (cid:19) ψ ( q u ; q, − ψ ( q u ) φ ( q ) q u/ du. (4.5)In the proof of Theorem 2.1 above we showed that the contribution from the integral where ≤ u ≤ is O ( q − / ) .For any fixed ǫ > , trivial bounds for the region ≤ u ≤ ǫ yield a contribution thatis ≪ Z ǫ ( u log q ) q u − du ≪ q − + ǫ . (4.6)We use the Brun-Titchmarsh Theorem (see [MonVa1]) for the region where ǫ ≤ u ≤ ,which asserts that for q < x , π ( x ; q, a ) ≤ xφ ( q ) log( x/q ) . (4.7)We first bound the contribution from prime powers as follows. First there are at most e ω ( q ) residue classes b mod q such that b e ≡ q , and so using that ω ( q ) ≪ log q/ log log q we ompute X e ≥ X p ≤ x /e p e ≡ q log p ≪ X ≤ e ≤ ǫ e ω ( q ) max b mod q X p ≤ x /e p ≡ b mod q log p ! + X ǫ ≤ e ≤ x X p ≤ x /e log p ≪ X ≤ e ≤ ǫ e ω ( q ) (cid:18) x /e q (cid:19) log x + X ǫ ≤ e ≤ x x /e ≪ (cid:18) ǫ (cid:19) ω ( q )+1 (cid:18) x / q (cid:19) log x + x ǫ/ log x ≪ ǫ x ǫ (cid:18) x / q (cid:19) , (4.8)provided q is large enough in terms of ǫ .Thus, for ǫ ≤ u ≤ , we have ψ ( q u ; q, ≪ ǫ q u − log( q u ) log log q ( u − 1) log q + q ǫ + q u − ǫ ≪ ǫ q u − log log q, (4.9)which bounds the integral from ǫ to σ by ≪ Z σ ǫ q u − log log qdu ≪ log log q log q q σ − , (4.10)completing the proof. (cid:3) Results Under GRH (Theorems 1.2 and 2.6) In this section we assume GRH (but none of the stronger results about the distributionof primes among residue classes) and prove Theorems 1.2 and 2.6. The main ingredient inthe proofs are the results of [Fo, BFI, FG2, Fi1]. The following is the needed conditionalversion. Theorem 5.1. Assume GRH. Fix an integer a = 0 and ǫ > . We have for M = M ( x ) ≤ x that X x M To prove (2.10), we need to understand the part of the integral in(5.4) with a ≤ u ≤ . Arguing as in [FG2] (see also the proof of Proposition 6.1 of [Fi1]),we have that for x / ≤ Q ≤ x , X Q/ 1) = X n ≤ xn − qrQ/ We need to study the part of the integral in (5.4) with κ ≤ u ≤ . We first see that by (5.7), the part of the integral with ≤ u ≤ is ≪ Q − + Q σ − log Q. (5.8) e turn to the part of the integral with κ ≤ u ≤ . We have by Theorem 5.1 (setting x := Q u and M := Q u − ) that it is = 2 Z κ (cid:18) f ( u )2 − f ′ ( u )log Q (cid:19) Q − u/ (cid:16) − 12 log( Q u − ) − C O ǫ (cid:16) Q − u (1 − ǫ ) + Q u − (log Q u ) (cid:17) (cid:17) du = − Z κ (( u − 1) log Q + C ) Q − u/ (cid:18) f ( u )2 − f ′ ( u )log Q (cid:19) du + O ǫ Q − − κ (1 − ǫ ) log Q + Q − log Q ! , (5.9)hence (2.9) holds. (cid:3) Proof of Theorem 1.2. We now turn to (1.5), with f supported in ( − , ) . Set κ := A log log Q log Q with A ≥ a constant. As the big-Oh constant in (5.9) is independent of κ , we may use(5.9) to estimate the contribution to (5.4) from u ∈ [1 + κ, ] . This part of the integralcontributes − Z / κ (( u − 1) log Q + C ) Q − u/ (cid:18) f ( u )2 − f ′ ( u )log Q (cid:19) du + O ǫ (cid:18) Q − / (log Q ) A (1 − ǫ )+1 (cid:19) ≪ Q − / (log Q ) A/ . (5.10)The part of the integral with ≤ u ≤ was already shown to be ≪ Q − + Q σ − log Q , andhence is absorbed into the error term since σ < / .We now come to the heart of the argument, the part of the integral where ≤ u ≤ κ .Since f ∈ C ( R ) , we have that in our range of u , g ( u ) := f ( u )2 − f ′ ( u )log Q satisfies g ( u ) = f (1)2 + f ′ (1)2 ( u − 1) + O (( u − ) − f ′ (1)log Q + O (cid:18) u − Q (cid:19) = P ( u − 1) + O (cid:18) (log log Q ) (log Q ) (cid:19) , (5.11)where P ( u ) := f (1)2 − f ′ (1)log Q + f ′ (1)2 u . At this point, if f were C K ( R ) , we could take its Taylorexpansion and get an error of O ǫ,A (cid:16) (log log Q ) K (log Q ) K (cid:17) .We cannot apply Theorem 5.1 directly since the error term is not got enough for moderatevalues of M . Instead, we argue as in the proof of Proposition 6.1 of [Fi1]. Slightly modifyingthe proof and using GRH, we get that X Q/ In this section we assume the de-averaging hypothesis (Hypothesis 2.7), which relates thevariance in the distribution of primes congruent to 1 to the average variance over all residueclasses. Explicitly, we assume (2.14) holds for some δ ∈ (0 , , and show how this allows usto compute the main term in the averaged 1-level density, D Q/ ,Q ( b f ) , for test functions f supported in [ − δ, − δ ] . (Remember that this hypothesis is trivially true for δ = 1 ,and expected to hold for any δ > .) roof of Theorem 2.8. Starting from (3.23), we have that T ( q ) = 2 Z ∞ (cid:18) f ( u )2 − f ′ ( u )log Q (cid:19) ψ ( Q u ; q, − ψ ( Q u ) φ ( q ) Q u/ du. (6.1)Feeding this into Proposition 3.2, we are left with determining Q/ X Q/ We continue our investigations beyond the GRH, and assume a smoothed version of Mont-gomery’s hypothesis, Hypothesis 2.12. Interestingly, this assumption allows us to computethe main term of the 1-level density, D q ( b f ) , for test functions of arbitrarily large (but finite)support. While similar results have been previously observed [MilSar], we include a proofboth for completeness and because these observations are not in the literature. Proof of Theorem 2.13. As we are fixing the modulus, we take Q := q . By the explicitformula from Proposition 3.1, we have D q ( b f ) = f (0)log q log q − log(8 πe γ ) − X p | q log pp − + Z ∞ f (0) − f ( t ) q t/ − q − t/ dt − q X n ≡ q − φ ( q ) X n ! Λ( n ) n / f (cid:18) log n log q (cid:19) + O (cid:18) φ ( q ) (cid:19) . (7.1)Let σ := sup( supp f ) < ∞ . We proved in §4 that the only terms that are not O (1 / log q ) are the leading term f (0) and possibly the prime sum, which we now study. We have T ( q ) = 2 Z ∞ (cid:18) f ( u )2 − f ′ ( u )log q (cid:19) ψ ( q u ; q, − ψ ( q u ) φ ( q ) q u/ du. (7.2)In the proof of Theorem 2.1 we determined that the part of the integral with ≤ u ≤ is O ( q − / ) . From the proof of Theorem 2.3, the part with ≤ u ≤ is O ( log log q log q ) .(1) Proof of Theorem 2.13(1). For the rest of the integral, we use Hypothesis 2.12. Notethat u ≥ , so x = q u ≥ q with u ≤ σ , hence we can replace o x →∞ by o q →∞ . An ntegration by parts gives that the rest of the integral is = 0 − (cid:18) f (2)2 − f ′ (2)log q (cid:19) ψ ( q ; q, − ψ ( q ) φ ( q ) q − Z ∞ (cid:18) f ( u )4 − f ′ ( u )log q + f ′′ ( u )(log q ) (cid:19) ψ ( q u ; q, − ψ ( q u ) φ ( q ) q u/ du = o ( q ) q + Z σ ( | f ( u ) | + | f ′ ( u ) | + | f ′′ ( u ) | ) o ( q u/ ) q u/ du = o (1) , (7.3)proving the claim. Note that we are using the smoothed version of the prime sum.(2) Proof of Theorem 2.13(2). We already know that the part of the integral with ≤ u ≤ is ≪ q − / . 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. What we need is some control over biasesof primes congruent to q . For the residue class a mod q , (cid:12)(cid:12)(cid:12) ψ ( x ; q, a ) − ψ ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) is thevariance; the above conjecture can be interpreted as bounding (cid:12)(cid:12)(cid:12) ψ ( x ; q, − ψ ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) in termsof the average variance. Under these hypotheses, we show how to extend the support to a wider but still limitedrange.
small enough, u +1 ≥ max(7 / u/ ǫ ( u − , u/ ,so (6.5) implies that X q ≤ Q X ≤ a ≤ q ( a,q )=1 (cid:18) ψ ( x ; q, a ) − ψ ( x ) φ ( q ) (cid:19) ≪ Qx (log x ) (6.6)(which, up to x ǫ , follows from Hooley’s original result [Ho]), so we get that the part of (6.4)with ≤ u ≤ is ≪ Q δ − Z Q − u/ Q u +12 (log Q ) / du ≪ Q δ − (log Q ) / , (6.7)which is o (1) if δ < . e now examine the second interval, that is ≤ u ≤ σ . In this range, (6.5) becomes X q ≤ Q X ≤ a ≤ q ( a,q )=1 (cid:18) ψ ( x ; q, a ) − ψ ( x ) φ ( q ) (cid:19) ≪ x / (log x ) / (log log x ) (6.8)(which, up to a factor of x ǫ , follows from Hooley’s original result [Ho]). We thus get thatthe part of (6.4) with ≤ u ≤ σ is ≪ Q δ Q/ Z σ Q − u/ Q u/ ( u log Q ) / log log( Q u ) du ≪ Q σ +2 δ − (log Q ) / log log Q. (6.9)If σ < − δ then the above is o (1) , completing the proof. (cid:3) Results under Montgomery’s Hypothesis (Theorem 2.13)