SUMS OF MULTIPLICATIVE FUNCTIONS. Andrew Granville

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1 SUMS OF MULTIPLICATIVE FUNCTIONS Andrew Granville Abstract In these six hours of lectures we will discuss three themes that have been central to multiplicative number theory in the last few years: The uncertainty principle, bounds for character sums, and mean values of multiplicative functions I will discuss many recent developments, in historical context, involving some proofs The notes here are pieced together from several old articles of mine mostly with Soundararajan, some alone, and some with Friedlander, so apologies to the reader for any disjointedness The Uncertainty Principle, in the twentieth century The distribution of Primes 2 Maier matrices 3 And beyond 2 The Uncertainty Principle, in the twenty-first century 2 A general phenomenon 22 The new framework 23 Oscillations in mean-values of multiplicative functions 24 More Examples 3 Halász s theorem 3 The Halász-Montgomery theorem 32 Integral delay equations a model for mean values 33 An integral delay equation version of Proposition 3 34 An uncertainty principle for integral equations 35 Spectra 36 Sieving extrema 4 Character Sums 4 The Pólya-Vinogradov Theorem 42 Burgess s Theorem 43 Improving Burgess s Theorem? Integral delay equations 5 Pretentiousness 5 Proof of the prime number theorem 52 Interlude: Distance and beyond 53 A new and easy application of pretentiousness 54 Pólya-Vinogradov revisited 6 Recent work 6 Pretentiousness is indeed repulsive 62 Multiplicative functions in arithmetic progressions 63 Exponential sums 64 The prime number theorem in terms of multiplicative functions 65 Periodic functions pretending to be characters Typeset by AMS-TEX

2 2 ANDREW GRANVILLE The Uncertainty Principle, in the twentieth century The distribution of Primes As a boy of 5 or 6, Gauss determined, by studying tables of primes, that the primes occur with density log x at around x This translates into the guess that πx := #{primes x} Lix where Lix := The existing data lend support to Gauss s guesstimate: x 2 dt log t x πx = #{primes x} Overcount: [Lix πx] x log x Notice how the entries in the final column are always positive and always about half the width of the entries in the middle column: So it seems that Gauss s guess is always an overcount by about x We believe that this observation is both right and wrong: Although the last column looks to be positive and growing we believe that it eventually turns negative, and changes sign infinitely often see, eg [GM] On the other hand we believe that the error in Gauss s guess is never much more than x correctly formulated this statement is equivalent to the Riemann Hypothesis: In 859, Riemann, in a now famous memoir, illustrated how the question of estimating πx could be turned into a question in analysis: Define ζs := n n s for Res >, and then analytically continue ζs to the rest of the complex plane We have, for sufficiently large x, x b ɛ max πy Liy y x xb+ɛ where b := sup β ζβ+iγ=0 The as yet unproven Riemann Hypothesis RH asserts that b = /2 in fact that β = /2 whenever ζβ + iγ = 0 with 0 β leading to the sharp estimate 2a πx = Lix + Ox /2 log x This section is an edited version of [Gr]

3 OTTAWA NOTES 3 It was not until 896 that Hadamard and De La Vallée Poussin independently proved that β < whenever ζβ + iγ = 0, which implies The Prime Number Theorem: that is, Gauss s prediction that πx Lix x log x The best version known has an error term in which we do not even save a power of x: / clog x 3/5 πx = Lix + O x exp log log x /5 In 94, Littlewood showed, unconditionally, that 2b πx Lix = Ω ± x /2 log log log x log x the first proven irregularities in the distribution of primes 2 Since Gauss s vague density assertion was so prescient, Cramér [Cra] decided, in 936, to interpret Gauss s statement more formally in terms of probability theory, to try to make further predictions about the distribution of prime numbers: Let Z 2, Z 3, be a sequence of independent random variables with, ProbZ n = = log n and ProbZ n = 0 = log n Let S be the space of sequences T = z 2, z 3, and, for each x 2 define π T x = 2 The sequence P = π 2, π 3,, where π n = if and only if n is prime, belongs to S Cramér wrote: In many cases it is possible to prove that, with probability, a certain relation R holds for sequences in S Of course we cannot in general conclude that R holds for the particular sequence P, but results suggested in this way may sometimes afterwards be rigorously proved by other methods For example Cramér was able to show, with probability, that max y x π T y Liy z n 2x log log x log x, which corresponds well with the estimates in 2; and if true for T = P implies RH, by Gauss s assertion was really about primes in short intervals, and so is best applied to πx + y πx, where y is small compared to x The binomial random variables Z n are more-or-less the same for all integers n in such an interval If we take y = λ log x so that 2 fx = Ω ± gx means that there exists a constant c > 0 such that fx + > cgx + and fx < cgx for certain arbitrarily large values of x ±

4 4 ANDREW GRANVILLE the expected number of primes, λ, in the interval is fixed then we would expect that the number of primes in such intervals should follow a Poisson distribution Indeed, we can prove that for any fixed λ > 0 and integer k 0, we have λ λk 3 #{integers x X : π T x + λ log x π T x = k} e k! X as X, with probability for T S In 976, Gallagher [Ga2] showed that this holds for the sequence of primes that is, for P under the assumption of a reasonable uniform version of Hardy and Littlewood s Prime k-tuplets conjecture This conjecture is the case where we take each f j x to be a linear polynomial in Schinzel and Sierpiński s Hypothesis H Let F = {f x, f 2 x,, f k x} be a set of irreducible polynomials with integer coefficients Then the number of integers n x for which each f j n is prime is x π F x = {C F + o} log f x log f 2 x log f k x where C F = ω / F p k, p p p prime and ω F p counts the number of integers n, in the range n p, for which f nf 2 n f k n 0 mod p 3,4 Estimates analogous to 2a should hold for the number of primes in intervals of various lengths, if we believe that what almost always occurs in S, should also hold for P Specifically, if 0 log 2 x y x then 4 π T x + y π T x = Lix + y Lix + Oy /2 with probability for T S In 943 Selberg [Sl] showed that primes do, on the whole, behave like this by proving, under the assumption of RH, that 5 πx + y πx y log x for almost all integers x, provided y/ log 2 x as x Cramér s model does seem to accurately predict what we already believe to be true about primes for more substantial reasons 5 To be sure, one can find small discrepancies 6 3 Elementary results on prime ideals guarantee that the product defining C F converges if the primes are taken in ascending order 4 The asymptotic formula proposed here for π F x has a local part C F, which has a factor corresponding to each rational prime p, and an analytic part x/ Q i log f ix This reminds one of formulae which arise when counting points on varieties 5 Though Hardy and Littlewood remarked thus on probabilistic models: Probability is not a notion of pure mathematics, but of philosophy or physics 6 As has been independently pointed out to me by Selberg, Montgomery and Pintz: for example, Pintz noted that the mean square of ψy y for y x, is x 2b ε with b as in, in fact x assuming RH, whereas the probabilistic model predicts x log x

5 OTTAWA NOTES 5 but the probabilistic model usually gives one a strong indication of the truth Cramér made one conjecture, based on his model, which does not seem to be attackable by other methods: If p = 2 < p 2 = 3 < p 3 = 5 < is the sequence of prime numbers then max p n+ p n log 2 x p n x This statement or the weaker Olog 2 x is known as Cramér s Conjecture ; there is some computational evidence to support it: p n p n+ p n p n+ p n / log 2 p n Record-breaking gaps between primes, up to In 985 Maier [Mai] surprisingly proved that, despite Selberg showing 5 holds almost all the time when y = log B x assuming RH for fixed B > 2, it cannot hold all of the time for such y This not only radically contradicts what is predicted by the probabilistic model, but also what most researchers in the field had believed to be true, whether or not they had faith in the probabilistic model Specifically, Maier showed the existence of a constant δ B > 0 such that for occasional, but arbitrarily large, values of x + and x, 6 πx + + log B x + πx + > + δ B log B x +, and πx + log B x πx < δ B log B x Outline of the Proof There are e γ x/ log z integers x, all of whose prime factors are > z, provided z is not too large Among these we have all but πz of the primes x, and so the probability that a randomly chosen such integer is prime is e γ log z/ log x Thus in a specific interval x, x + y] we should expect e γ Φ log z/ log x primes, where Φ is the number of integers in the interval that are free of prime factors z Now if we can select our interval so that Φ e γ y/ log z then our new prediction is not the same as that in 5 If x is divisible by P = p z p then y Φ = Φy, z := #{ n y : p n p > z} ωu log z for y = z u with u fixed, where ωu, the Buchstab function, equals 0 if 0 < u < and satisfies the differential-delay equation u ωu = + u ωtdt if u Obviously

6 6 ANDREW GRANVILLE lim u ωu = e γ Iwaniec showed that ωu e γ oscillates, crossing zero either once or twice in every interval of length Thus if we fix u > B, chosen so that ωu > e γ or < e γ as befits the case of 6, select y = log B x and z = y /u, and adjust x so that it is divisible by P, then we expect 5 to be false 2 Maier matrices To convert this heuristic into a proof, Maier considered a progression of intervals of the form rp, rp + y], with R r < 2R for a suitable value of R Visualizing this as a matrix, with each such interval represented by a different row, we see that the primes in the matrix are all contained in those columns j for which j, P = RP + RP + 2 RP + 3 RP + y R + P + R + P + 2 R + P + 3 R + P + y R + 2P + R + 2P + 2 R + 3P + i, jth entry : R + ip + j 2R P + 2R P + 2 2R P + y The Maier Matrix for πx + y πx Now, for any integer q >, the primes are roughly equi-distributed amongst those arithmetic progressions a mod q with a, q = : in fact up to x we expect that the number of such primes 7 πx; q, a πx φq If so, then the number of primes in the jth column, when j, P =, is π2x; P, j πx; P, j X φp log X where X = RP To get the total number of primes in the matrix we sum over all such j, and then we can deduce that, on average, a row contains Φy, z R φp RP log RP ωu y log z P φp log RP y eγ ωu log RP primes Maier s result follows provided we can prove a suitable estimate in 7 In general, it is desirable to have an estimate like 7 when x is not too large compared to q It has been proved that 7 holds uniformly for 7 i All q log B x and all a, q =, for any fixed B > 0 Siegel-Walfisz 8 7 We believe, but cannot prove, that 7 holds uniformly for all q x ɛ based on what computations appear to reveal 8 It is impossible to give all the constants explicitly in the resulting version of 7 because the proof works in two parts: first, if GRH is true, and second, by using a putative counterexample to GRH, if one exists Hence if GRH is true but remains unproved one cannot, by the very nature of this proof, determine the constants involved An alternate argument does provide all the constants involved in the more limited range B 2

7 OTTAWA NOTES 7 ii All q x/ log 2+ɛ x and all a, q =, assuming GRH 9 In fact 7 then holds with error term O x log 2 qx iii Almost all q x/ log 2+ɛ x and all a, q = Bombieri-Vinogradov 0 iv Almost all q x /2+o with q, a =, for fixed a 0 Bombieri-Friedlander- Iwaniec, Fouvry v Almost all q x/ log 2+ɛ x and almost all a, q = Barban-Davenport- Halberstam, Montgomery, Hooley Thus, when GRH is true, we get a good enough estimate in 7 with R = P 2 to complete Maier s proof However Maier, in the spirit of the Bombieri-Vinogradov Theorem, showed how to pick a good value for P, so that 7 is off by, at worst, an insignificant factor when R is a large, but fixed, power of P thus proving his result unconditionally In [GS7], we extended the range for y in the proof above, establishing that there are intervals x, x + y] in every interval [X, 2X] for which 4 fails to hold for some y > exp clog x/ log log x /2 It is plausible that 5 holds uniformly if log y/ log log x as x ; and that 4 holds uniformly for T = P if y > exp log x /2+ɛ at least, we can t disprove these statements as yet We conjecture, presumably safely, that 4 and 5 hold uniformly when y > x ɛ One can show that there are more than x/ exp log x c B integers x ± x satisfying the unexpected inequalities in 6 Maier s work suggests that Cramér s model should be adjusted to take into account divisibility of n by small primes It is plausible to define small to mean those primes up to a fixed power of log n Then we are led to conjecture that there are infinitely many primes p n with p n+ p n > 2e γ log 2 p n, contradicting Cramér s conjecture, as 2e γ > 2 If we analyze the distribution of primes in arithmetic progressions using a suitable analogue of Cramér s model, then we would expect 7, and even x 7 πx; q, a = πx /2 logqx φq + O, q to hold uniformly when a, q = in the range 8 q Q = x/ log B x, for any fixed B > 2 However the method of Maier is easily adapted to show that neither 7 nor 7 cannot hold in at least part of the range 8: For any fixed 9 The Generalized Riemann Hypothesis GRH states that if β+iγ is a zero of any Dirichlet L-function then β /2 0 This result is often referred to as GRH on average See section 64 for another result of this type One has to be careful about the meaning of small here, since if we were to take into account the divisibility of n by all primes up to n, then we would conclude that there are e γ x/ log x primes up to x 2 It is unclear what the correct conjecture here should be since, to get at it with this approach, we would need more precise information on sifting limits than is currently available

8 8 ANDREW GRANVILLE B > 0 there exists a constant δ B > 0 such that for any modulus q, with not too many small prime factors, there exist arithmetic progressions a ± mod q and values x ± [φq log B q, 2φq log B q] such that 9 πx + ; q, a + > + δ B πx + φq and πx ; q, a < δ B πx φq The proof is much as before, though now using a modified Maier matrix : RP RP + q RP + 2q RP + yq R + P R + P + q R + P + 2q R + P + yq R + 2P R + 2P + q i, jth entry : R + 3P R + ip + jq 2R P 2R P + q 2R P + yq The Maier Matrix for πyq; q, a The Bombieri-Vinogradov Theorem is usually stated in a stronger form than above: For any given A > 0, there exists a value B = BA > 0 such that 0 max max a,q= y x q Q πy πy; q, a φq x log A x where Q = x/ log B x It is possible [FG] to take the same values of R and P in the Maier matrix above for many different values of q, and thus deduce that there exist arbitrarily large values of a and x for which Q q 2Q q,a= { πx; q, a πx φq } x; thus refuting the conjecture that for any given A > 0, 0 should hold in the range 8 for some B = BA > 0 In [FG2] we showed that 0 even fails with Q = x/ exp A ɛlog log x 2 /log log log x In [GS7] we showed that, for any q x/ exp log x /2 ɛ, the bound 7 cannot hold for every integer a prime to q It seems plausible that 7 holds uniformly if logx/q/ log log q as q ; and that 0 holds uniformly for Q < x/ exp log x /2+ɛ At least we can t disprove

9 OTTAWA NOTES 9 these statements as yet, though we might play it safe and conjecture only that they hold uniformly for q, Q < x ɛ Notice that in the proof described above, the values of a increase with x, leaving open the possibility that 7 might hold uniformly for all a, q = in the range 8 if we fix a 3 However, in [FG3] we observed that when a is fixed one can suitably modify the Maier matrix, by forcing the elements of the second column to all be divisible by P : + RP + 2RP + yrp + R + P + 2R + P + yr + P + R + 2P i, jth entry : + jr + ip + 2R P + y2r P The Maier Matrix for πyq; q, Notice that the jth column here is now part of an arithmetic progression with a varying modulus, namely j mod jp With this type of Maier matrix we can deduce that, for almost all 0 < a < x/ log B x including all fixed a 0, there exist q x/ log B x, 2x/ log B x], coprime to a, for which 7 does not hold However 7 cannot be false too often like in, since this would contradict the Barban-Davenport- Halberstam Theorem So for which a is 7 frequently false? It turns out that the answer depends on the number of prime factors of a: In [FG5], extending the results of [BFI], we show that for any given A > there exists a value B = BA > 0 such that, for any Q x/ log B x and any integer a which satisfies 0 < a < x and has log log x distinct prime factors 4, we have { 2 πx; q, a πx } φq x Q q 2Q log A x q,a= On the other hand, assuming GRH, for every given A, B > 0, there exists Q x/ log B x and an integer a for which 2 does not hold, where 0 < a < x and a has < log log x +ɛ distinct prime factors Finding primes in x, x + y] is equivalent to finding integers n y for which fn is prime, where ft is the polynomial t + x Similarly, finding primes x which belong to the arithmetic progression a mod q, is equivalent to finding integers n y := x/q for which fn is prime, where ft is the polynomial qt + a Define the height, hf, of a given 3 Which would be consistent with the Barban-Davenport-Halberstam Theorem 4 which includes almost all integers a once the inexplicit constant here is > by a famous result of Hardy and Ramanujan

10 0 ANDREW GRANVILLE polynomial ft = i c it i to be hf := i c2 i In the cases above, in which the degree is always, we proved that we do not always get the asymptotically expected number of prime values fn with n y = log B hf, for any fixed B > 0 In [FG4] we showed that this is true for polynomials of arbitrary degree d, which is somewhat ironic since it is not known that any polynomial of degree 2 takes on infinitely many prime values, nor that the prime values are ever well-distributed Nair and Perelli [NP] showed that some of the polynomials F R n = n d + RP attain more than, and others attain less than, the number of prime values expected in such a range, by considering the following Maier matrix: F R F R 2 F R 3 F R y F R+ F R+ 2 F R+ 3 F R+ y F R+2 F R+2 2 i, jth entry : F R+3 F R+i j F 2R F 2R 2 F 2R y The Maier Matrix for π F y Notice that the jth column here is part of the arithmetic progression j d mod P Using Maier matrices it is possible to prove bad equi-distribution results for primes in other interesting sequences, such as the values of binary quadratic forms, and of prime pairs For example, if we fix B > 0 then, once x is sufficiently large, there exists a positive integer k log x such that there are at least + δ B times as many prime pairs p, p + 2k, with x < p x+log B x, as we would expect from assuming that the estimate in Hypothesis H holds uniformly for n log B ht + xt + x + 2k We have now seen that the asymptotic formula in Hypothesis H fails when x is an arbitrary fixed power of log hf := i log hf i, for many different non-trivial examples F Presumably the asymptotic formula does hold uniformly as log x/ log log hf However, to be safe, we only make the following prediction: Conjecture Fix ε > 0 and positive integer k The asymptotic formula in Hypothesis H holds uniformly for x > hf ε as hf Our work here shows that the random-like behaviour exhibited by primes in many situations does not carry over to all situations It remains to discover a model that will always accurately predict how primes are distributed, since it seems that minor modifications of Cramér s model will not do We thus agree that: It is evident that the primes are randomly distributed but, unfortunately, we don t know what random means RC Vaughan February And beyond Armed with Maier s ideas it seems possible to construct incorrect conclusions from, more-or-less, any variant of Cramér s model This flawed model may still be used to make conjectures about the distribution of primes, but one should be very cautious of such predictions!

11 OTTAWA NOTES There are no more than Ox 2 / log 3B x arithmetic progressions a mod q, with a < q < x/ log B x and a, q =, for which 7 fails, by the Barban-Davenport- Halberstam Theorem However our methods here may be used to show that 7 does fail for more than x 2 / exp log x ɛ such arithmetic progressions Maier s matrix has been used in other problems too: Konyagin used it to find unusually large gaps between consecutive primes Maier used it to find long sequences of consecutive primes, in which there are longer than average gaps between each pair Shiu has used it to show that every arithmetic progression a mod q with a, q = contains arbitrarily long strings of consecutive primes Recently Thorne 5 generalized these ideas to function fields, and to Gaussian primes Despite the efforts of several mathematicians we have still not got a particularly good model to replace the now discredited model of Cramér Certainly not a well-motivated one, so there are still many questions about the distribution of primes in which we cannot even guess at the right answer with much confidence This is not a new complaint: Mathematicians have tried in vain to discover some order in the sequence of prime numbers but we have every reason to believe that there are some mysteries which the human mind will never penetrate L Euler The Uncertainty Principle, in the twenty-first century 2 A general phenomenon I ended my 994 ICM talk with Euler s quote it seemed fitting since these counterintuitive extensions of Maier s results seemed to be something to do with Euler s observation that primes refuse, in almost every way, to be easily understood There was a hiatus in developments in this subject, until a paper of Balog and Wooley [BW] appeared in 2000 There they applied the same circle of ideas to the integers that are the sum of two squares, and proved an analogous result to that of Maier My first reaction when I heard about their work was that it was nice they had generalized things beyond primes, but was this really so interesting? Especially as it was probably technically easier than Maier s work on primes? They sent me a reprint of their paper and it sat unread on my desk for an embarrassingly long time After two years I picked it up and almost immediately realized that what they had done was a big surprise and their work surely pointed the way to an important phenomenon The point is that I had thought that irregularities in the distribution of primes came about because primes are always strange, but here Balog and Wooley had taken a relatively civilized sequence, one with very predictable multiplicative structure even, and found similar irregularities in the distribution This meant that Maier s proof was surely nothing to do with primes and must be the harbinger of a much more general phenomenon Indeed this intuition was correct and Sound and I were able to prove a very general uncertainty principle which establishes that most arithmetic sequences of interest are either not-so-well distributed in longish arithmetic progressions, or are not-so-well distributed in both short intervals and short arithmetic progressions 5 In his 2008 PhD at Madison

12 2 ANDREW GRANVILLE With probability, there are no Maier-type irregularities in the distribution of randomly chosen subsets of the integers Indeed such irregularities seem to depend on our sequence coming from arithmetic Sieve theory already provides a framework for considering analytic properties of arithmetic sequences, so this is our starting point: A could be a set of integers but, more generally, let A denote a sequence an of non-negative real numbers, and Ax = an If the an are well-distributed in short intervals then we expect 2 Ax + y Ax y Ax x, for suitable y Suppose that the proportion of A which is divisible by d is approximately hd/d where h is a non-negative multiplicative function; in other words, 22 A d x := d n n a mod q an hd d Ax, for each d 6 The reason for taking hd to be a multiplicative function is that for most sequences that appear in arithmetic one expects that the criterion of being divisible by an integer d should be independent of the criterion of being divisible by any integer d 2 which is coprime with d If the asymptotic behavior of A in the arithmetic progression a mod q, depends only on a, q when q, S = then, by 22, we arrive at the prediction that, for q, S =, 23 Ax; q, a := an f qa Ax, qγ q where γ q = p q p /p hp and f qa is a certain non-negative multiplicative function of a for which f q a = f q a, q thus f q a is periodic mod q 7 Example We take an = for all n, so that hn = for all n Then f q a = for all q and all a, and γ q = Clearly both 23 and 2 are good approximations with an error of at most Example 2 We take an = if n is prime and an = 0 otherwise, so that hn = if n = and hn = 0 if n > Further f q a = if a, q = and f q a = 0 otherwise, and γ q = φq/q The approximation 23 is then the prime number theorem for arithmetic progressions for small q log x A Friedlander and Granville s result sets limitations to 23, and Maier s result sets limitations to 2 Example 3 Take an = if n is the sum of two squares and an = 0 otherwise Here we take S = {2}, and for odd prime powers p k we have hp k = if p k mod 4 and hp k = /p otherwise Balog and Wooley s result places restrictions on the validity of 2 A weak form of our main results are given in the next two Theorems: 6 Or perhaps when d, S =, where S is a finite set of bad primes 7 We prove this in section 22 below, giving an explicit description of f q in terms of h

13 OTTAWA NOTES 3 Theorem 2 Let A, S, h, f q and γ q be as above For given α > 0 there exist constants c > 0, c > such that, for sufficiently large x, if either 24 p log x max{0, hp} p log p α log log x, or there exists n x with hn log x c, and if u log x c then there exists y x/4, x and an arithmetic progression a mod l with l x/log x u and l, S = such that Ay; f l a l, a y Ax cu Ax u lγ l x φl The condition 24 ensures that hp is not always close to ; this is essential in order to eliminate the very well behaved Example One can show that Ax φl lγ l y Ax x It is good to not have f la in the lower bound since it may well be 0 Theorem 2 applies to the sequences of primes with α = + o and sums of two squares with α = /2 + o, two results already known Surprisingly it also applies to any subset of the primes: Example 4 Let A be any subset of the primes Fix u For any x there exists y x/4, x such that either 25a Ay/y Ax/x u Ax/x meaning that the subset is poorly distributed in short intervals, or there exists some arithmetic progression a mod l with a, l = and l x/log x u, for which 25b Ay; l, a Ay Ax u φl φl In other words, we find Maier type irregularities in the distribution of any subset of the primes 8 A similar result holds for any subset of the numbers that are sums of two squares Let S ɛ be the set of integers n having no prime factors in the interval [log n ɛ, log n], so that S ɛ N ɛn Notice that the primes are a subset of S ɛ Theorem 2 with α ɛ + o implies that any subset A of S ɛ is poorly distributed in that for any x there exists y x/4, x such that either 25a holds, or there exists some arithmetic progression a mod l and l x/log x u with a, l =, for which a suitably modified 25b holds that is with φl replaced by l p l, log x ɛ <p<log x /p Our next result gives an uncertainty principle implying that we either have poor distribution in long arithmetic progressions, or in short intervals, generalizing Maier s original result: 8 If we had chosen A to be the primes 5 mod 7 then this is of no interest when we take a =, l = 7 To avoid this minor technicality we can add For a given finite set of bad primes S, we can choose such an l for which l, S =, where l, S = means that l, p = for all p S

14 4 ANDREW GRANVILLE Theorem 22 Let A, S, h, f q and γ q be as above For given α > 0 there exist constants c > 0, c > such that, for sufficiently large x, if either 24 holds or there exists n x with hn log x c, and if u log x c then at least one of the following two assertions holds: i There exists an interval v, v + y x/4, x with y log x u such that Av + y Av y Ax u cu y Ax x x ii There exists y x/4, x and an arithmetic progression a mod q with q, S = and q exp2log x η such that Ay; q, a f qa y Ax cu Ax u qγ q x φq This is aptly named an uncertainty principle for we can construct sequences which are well distributed in short intervals and then must have fluctuations in arithmetic progressions, and primes are known to be well-distributed in these long arithmetic progressions and so exhibit fluctuations in short intervals Our proofs develop Maier s matrix method : In the earlier work on primes and sums of two squares, the problem then reduced to showing oscillations in certain sifting functions arising from the theory of the half dimensional for sums of two squares and linear for primes sieves In our case the problem boils down to proving oscillations in the mean-value of the more general class of multiplicative functions satisfying 0 fn for all n as we will discuss in section 23 In section 34 we will prove an analogy of such oscillation results for a wide class of related integral equations, which has the flavor of a classical uncertainty principle from Fourier analysis and hence the nomenclature 22 The new framework We may assume that hp k < p k for all prime powers p k without any significant loss of generality We hypothesize that, for q, S =, the asymptotics of Ax; q, a depends only on the greatest common divisor of a and q Writing q, a = m, since {b mod q : b, q = m} = ϕq/m, we then guess that Ax; q, a ϕq/m q,n=m an = an µd = ϕq/m d q m m n d n m ϕq/m µda dm x d q m Using now 22 we would guess that 26 Ax; q, a Ax ϕq/m d q m µd hdm dm =: f qa Ax, qγ q where 27 γ q = p q hp/p /p = p + f qp + f qp 2 p p p 2 +,

15 OTTAWA NOTES 5 and f q a, a multiplicative function with f q a = f q a, q so that it has period q, is defined as follows: f q p k = if p q If p divides q, indeed if p e is the highest power of p dividing q then hp k hpk+ f q p k p := hp e p hp p hp p if k < e if k e Note that if q is squarefree and hp then f q p k for all prime powers p k We may assume that 0 hn for all n, since such results are easy if hp > often: 9 Proposition 23 Suppose that q x is an integer for which hq > 6 Then either 20 Ax; q, 0 f q0 Ax qγ q 2 f q 0 Ax qγ q or, for every prime l in the range x l 3x + q/hq which does not divide q, there is an arithmetic progression b mod l such that Proof If the first option fails then /q Ax; l, nq Ax; l, b f lb Ax lγ l 2 /q anq = Ax; q, 0 2 f l b Ax lγ l f q 0 Ax = hq qγ q 2q Ax On the other hand, if prime l q then f l nq = if l n, and f l nq = hlγ l if l n Therefore for any N, n N f l nq lγ l = n N l n + lγ l n N l n hl l = [N] [N/l] hl l ll Combining this taking N = x/q with the display above yields /q Ax; l, nq hq 2q 3x + q Ax Ax 3 2ql 2 N + l{n/l} {N} l /q f l nq Ax, lγ l 9 Proposition 23 implies Theorems 2 and 22 when there exists n x with hn log x c 20 Note that this criterion is equivalent to A q x hq/qax 2 hq/qax, since f q0/qγ q = f q q/qγ q = hq/q

16 6 ANDREW GRANVILLE which implies the Proposition with b = nq for some n x/q From now on assume 0 hn for all n Suppose that q, S = and define q = q x by 28 q x := max max Ay; q, a f qa y / Ax x/4 y x a mod q qγ q x Ax φq We can now formulate our main principle Proposition 24 Let x be large and let A, S h, f q and q be as above Let q x l x/4 be positive coprime integers with q, S = l, S = Then q φq qx + l φl lx + x 8 [x/2l] s x/2l f q s γ q Proof Let R := [x/4q] x/5 and S := [x/2l] < x/2 We sum the values of an as n varies over the integers in the following R S Maier matrix R + q + l R + q + 2l R + q + Sl R + 2q + l R + 2q + 2l R + 2q + Sl R + 3q + l r, sth entry : R + 4q + l R + rq + sl 2Rq + l 2Rq + Sl We sum the values of an in two ways: first row by row, and second column by column Note that the n appearing in our matrix all lie between x/4 and x The r-th row contributes AR+rq +ls; l, R+rq AR+rq; l, R+rq Using 28, and noting that f l R + rq = f l R + r as l, q =, this is f l R + r lγ l ls x Ax + O l φl Ax Summing this over all the rows we see that the sum of a n with n ranging over the Maier matrix above equals 29a ls x Ax 2R r=r+ f l r lγ l l + O φl AxR The contribution of column s is A2Rq + ls; q, ls ARq + ls; q, ls By 28, and since f q ls = f q s as l, q =, we see that this is f q s qγ q Rq x Ax + O q φq Ax

17 OTTAWA NOTES 7 Summing this over all the columns we see that the Maier matrix sum is 29b Rq x Ax S s= f q s qγ q Comparing 29a and 29b we deduce that q + O φq AxS 20 Sγ q S s= q q f q s + O = φq Rγ l 2R r=r+ l l f l r + O φl Write f l r = d r g ld for a multiplicative function g l Note that g l p k = 0 if p l We also check easily that g l p k p + /p for primes p l, and note that γ l = d= g ld/d Thus Rγ l 2R r=r+ f l r = Rγ l d 2R R g l d d +O = +O γ l We see easily that the error terms above are bounded by R 3 γ l g l d d= d 2 3 R 3 since l x, and R x We conclude that p l d>2r g l d + d Rγ l + O, p 2 3 R 4 d 2R g l d Rγ l 2R r=r+ f l r = + OR 4 Combining this with 20 we obtain the Proposition In Proposition 24 we compared the distribution of A in two arithmetic progressions We may also compare the distribution of A in an arithmetic progression versus the distribution in short intervals Define y = y, x by 2 y, x := max v,v+y x/4,x Av + y Av y Ax x / y Ax x Proposition 25 Let x be large and let A, S, h, f q, q and be as above Let q x with q, S = and let y x/4 be positive integers Then q φq qx + x, y f q s γ q y Proof The argument is similar to the proof of Proposition 24, starting with an R y Maier matrix again R = [x/4q] whose r, s-th entry is R + rq + s We omit the details s y

18 8 ANDREW GRANVILLE 23 Oscillations in mean-values of multiplicative functions Assume that q is an integer all of whose prime factors are less than or equal to large z Let f q n be a multiplicative function with f q p k = for all p q, and 0 f q n for all n Note that f q n = f q n, q is periodic mod q Define F q s = n= f q n n s = ζsg q s, where G q s = p s + f qp p s + f qp 2 p 2s + p q To start with, F q is defined in Res >, but the above furnishes a meromorphic continuation to Res > 0 Now γ q = G q, and define Eu := z u n z u f q n G q, which is what we need to consider in Propositions 22 and 23 above For complex number ξ, let Hξ := H 0 ξ where H j ξ := p q f q p logz/p j for each j 0, and Jξ := p ξ/ log z log z p q p 2 ξ/ log z One can easily show that for π ξ 2 3 log z one has The new development is to show that if Eu exphξ ξu + 5Jξ τ := 5H 2 ξ + 9Jξ + 5/Hξ /2 then there exist points u ± in the interval [Hξ 2τ, Hξ + 2τ] such that ±Eu ± 20ξHξ exp{hξ ξu ± 5H 2 ξ 5Jξ} This gives us, in some generality, a best possible oscillation result up to a small factor We will not prove this result here, but we will later in section 34 indicate how we proved it, though first in section 3 we will need to discuss the theory of mean values of multiplicative functions in some detail By a judicious choice of ξ, one can deduce Theorems 2 and 22 from Propositions 24 and 25 respectively 24 More Examples Sieves Let B be a given set of x integers and P be a given set of primes Define SB, P, z to be the number of integers in B which do not have a prime factor p P with p z Sieve theory is concerned with estimating SB, P, z under certain natural

19 OTTAWA NOTES 9 hypothesis for B, P and u := log x/ log z The fundamental lemma of sieve theory see [HR] implies for example, when B is the set of integers in an interval that SB, P, z x p P,p z p u + o x u log u p P,p z p for u < z /2+o It is known that this result is essentially best-possible in that one can construct examples for which the bound is obtained both as an upper and lower bound However these bounds are obtained in quite special examples, and one might suspect that in many cases which one encounters, those bounds might be significantly sharpened It turns out that these bounds cannot be improved for intervals B, when P contains at least a positive proportion of the primes: We proved in [GS7] that if P is a given set of primes for which #{p P : p y} πy for all y z, z], then there exists a constant c > 0 such that for any u z there exist intervals I ± of length z u for which and SI +, P, z SI, P, z { c + { u log u c u log u u } I + u } I p P,p z p P,p z p p Moreover if u o log log z/ log log log z then our intervals I ± have length z u+2 The reduced residues mod q are expected to be distributed much like random numbers chosen with probability φq/q Indeed when φq/q 0 this follows from work of Hooley, and of Montgomery and Vaughan [MV2] who showed that #{n [m, m + h : n, q = } has Gaussian distribution with mean and variance equal to hφq/q, as m varies over the integers, provided h is suitably large This suggests that #{n [m, m + h : n, q = } should be { + o}hφq/q provided h log 2 q However, by a modification of our argument, we can show that this is not true for h = log A q for any given A > 0, provided that p q log p/p log log q a condition satisfied by many highly composite q In fact Montgomery and Vaughan showed that m k mh+ n m+h n,q= φq 2r q h 2r! 2 r r! q h φq q for integers r Our work places restrictions on the uniformity with which this estimate can hold: Given h and q with hφq/q large, define η by q/φq = hφq/q η If the above holds then r loghφq/q 2+4η+o Wirsing Sequences Let P be a set of primes of logarithmic density α for a fixed number α 0, ; that is log p = α + o log x, p p x p P r

20 20 ANDREW GRANVILLE as x Let A be the set of integers not divisible by any prime in P and let an = if n A and an = 0 otherwise Wirsing proved that see page 47 of [Ten] 22 Ax eγα Γ α x p p x p P We can show oscillatory results for any such A: Fix u maxe 2/α, e 00 and suppose x is sufficiently large Then there exists y x/4, x and an arithmetic progression a mod l with l x3/ log x u such that Ay; l, a f la Ay exp ulog u + Olog log u Ay lγ l φl Moreover suppose MN = u with M, N Then at least one of the following is true: Either there exists y x/4, x and an arithmetic progression a mod q with q explog x M such that Ay; q, a f qa Ay exp ulog u + Olog log u Ay qγ q φq Or there exists y > 3 log xn and an interval v, v + y x/4, x such that Av + y Av y Av exp ulog u + Olog log uy Av v v Examples of Wirsing sequences include sums of two squares, in fact norms of integral ideals belonging to a given ideal class in a given number field 3 Halász s theorem 3 The Halász-Montgomery theorem Given a multiplicative function f with fn for all n, define Θf, x := p x + fp + fp2 p p 2 + p We are concerned with understanding the mean value of f up to x, that is x For real-valued f it turns out that 3a fn Θf, as x x fn In 944 Wintner proved this when Θf, 0, which is equivalent to the hypothesis that p fp/p converges In 967, Wirsing [Wrs] settled the harder remaining case

21 OTTAWA NOTES 2 when Θf, = 0, thereby establishing an old conjecture of Erdős and Wintner that every multiplicative function f with fn has a mean-value On the other hand not all complex valued multiplicative functions have a mean value tending to a limit; for example, the function fn = n iα, with α R \ {0}, since x niα x iα /+iα In the early seventies, Gábor Halász [Hal, Hal2] brilliantly realized that the correct question to ask is whether p Refpp iα /p converges for all real numbers α His fundamental result states: I If p Refp/piα /p diverges for all α then x fn 0 as x II If there exists α for which p Refp/piα /p converges then 3b fn x xiα + iα Θf α, x where f α n := fn/n iα Now Θf α, x Θf α, as x so we can rewrite the above as fn xiα x + iα Θf α, e irx where rx = arg Θf α, x which varies very slowly, for example rx 2 = rx + o Also note that if p fp/piα /p converges then II holds and Θf α, x Θf α, as x Moreover one should note the Corollary that if II holds then for any real number β one has fnn iβ x xiα+β + iα + β Θf α, x In case I, Halász also quantified how rapidly the limit is attained His method was modified and refined by Montgomery [Mon], Tenenbaum [Ten, p343], and Sound and I [GS4]: Throughout define 32 Mx, T := min y 2T p x Refpp iy p Theorem Halász-Montgomery Let f be a multiplicative function with fn for all n Let x 3 and T be real numbers, and let M = Mx, T If f is completely multiplicative then fn x M + 2 e γ M + O 7 T log log x + log x If f is multiplicative then fn x p + 2 M + 4 e γ M + O pp 7 T log log x + log x

22 22 ANDREW GRANVILLE This is essentially best possible up to a factor 0 in that for any given sufficiently large m 0, we can construct f and x such that M = Mx, = m 0 + O and fn M + 2/7eγ M x/0 If the minimum in 32 occurs for y = y 0 then one might expect that fn looks roughly like n iy 0, so that the mean-value of fn should be around size x iy 0 / + iy 0 / + y 0 In fact we proved that if the minimum in 32 with T = log x is attained at y = y 0 then fn x + y 0 + log log x+2 2 π log x 2 π Taking fn = n iy 0 we see that this is best possible in terms of y 0 though, in this case M = 0 Next we proved a hybrid bound between the last two results, obtaining, for M = Mx, log x, fn M + eγ M + x + y 2 0 log log x log x 2 3 The right hand side of 3b has size e M / + y 0, implying that there is little room to reduce the bound in Theorem 2b In fact for any given α and m 0 we can determine f, such that M = m 0 + O, y 0 = α and the bound here is too big by at most a constant factor The minimum in 32 can be unwieldy to determine, so it is desirable to get similar decay estimates in terms of p x Refp/p or equivalently F However, in light of case II above, this is only possible if we have some additional information on f, since the p Refp/p may diverge while the absolute value of the mean value may converge One can avoid case II altogether by insisting that all fp lie in some closed convex subset D of the unit disc U this is a natural restriction for many applications, such as when f is a Dirichlet character of a given order, as in Halász [Hal, Hal2], Hall and Tenenbaum [HT], and Hall [Hl2] The result of Hall is the most general, perhaps qualitatively definitive To describe it we require some information on the geometry of D: Throughout we let D be a closed, convex subset of U with D, and define ν = νd = max δ D Reδ For α [0, ] define 33 α = 2π max 2π 0 δ D Re δα e iθ dθ, which is a continuous, increasing, convex function of α Note that 0 = λd/2π, where λd is the length of the boundary of D Define κ = κd to be the largest value of α [0, ] such that α, which exists since 0 When 0 D, Hall showed that κd = 0 only when D = U, and κd = only when D = [0, ] He also proved that κd min, 0 min, 0 νd λd 2π Moreover κdνd for all D, with equality holding if and only if D = [0, ]

23 OTTAWA NOTES 23 Theorem Hall Let D be a closed, convex subset of U with D, and define κd as above Let f be a multiplicative function with fn and fp D for all primes p Then 34 fn D exp κd Refp x p p x Hall proved that the constant κd here is optimal for every D, in that it cannot be replaced by any larger value One would also like to bound how averages of multiplicative functions vary, for example that 35 fn w x x /w fn log 2w β, log x for all w x, with as large an exponent β as possible; the only problem is that this is not true as the ubiquitous example fn = n iα reveals On the other hand Elliott proved that the absolute value of the mean of a given multiplicative function does vary slowly, and using the method here one can improve his result to: 36 x fn w x /w 2 fn log 2w π log x log + log x log 2w log log x log x 2 3 for w x/0 and any multiplicative function f with fn for all n If the minimum in 32 occurs when y = 0 then we can remove the absolute value signs here, and have a result like the one proposed, 35 In the special case that fn is non-negative we improved the 2/π to /π, see [GS6] An application of such an estimate, as Hildebrand [Hi3] observed, is to extend slightly the range of validity of Burgess famous character sum estimate For characters χ with cubefree conductor q, one can show that n N χn = on for N > q/4 o rather than N > q /4+o Our proofs are based on the following key Proposition, which we establish by a variation of Halász method Proposition 3 For x 3, T, and F s = F x s := p x for any complex number s with Res > 0, we have fn 2 x log x 0 x 2α 2α + fp p s + fp2 p 2s + max F + α + iy y T dα + O T log log x + log x

24 24 ANDREW GRANVILLE Sketch of deduction of the Halász-Montgomery theorem Whenever 0 < α < we have z α = α π α 2 +ξ z iξ dξ + O α 2 T We easily deduce that T T α log x max F + α + iy L + O where L := max y T T log x F + iy, y 2T a bound we use in the range α /L log x For larger α we have F + α + iy ζ + α /α + O Substituting these estimates into Proposition 3, it is now a manipulation of some integrals to show that fn L log eγ x L O T If f is completely multiplicative then, by Mertens theorem, log log x + log x F + iy = e γ log x + O fp p +iy p p x = e γ log x + O exp Refp k p iky kp k, p x k so that L e γ M + O/ log x, and the result follows If f is multiplicative then the result follows similarly after noting that since fp k for all k + fp p +iy + fp2 p 2+2iy + fp p +iy + 2 pp, 32 Integral delay equations a model for mean values Proofs of Proposition 3 as in [GS4] can be rather complicated and seem unmotivated It is our intention to give the proof of an analogous result about integral delay equations which can be modified to give a proof of Proposition 3 We now discuss what the connection is with integral delay equations, before getting more precise later Wirsing [Wrs] observed that questions on mean-values of multiplicative functions can be reformulated in terms of solutions to a related integral equation We formalized this connection precisely in our paper [GS] and we now recapitulate the salient details Throughout we suppose that f is a multiplicative function with fn for all n Two classes of multiplicative functions with non-zero mean values are easily dealt with in the literature, those with fp = for all of the small primes p, and those with fp = for all of the large primes p: An example with fp = for all of the small primes p It is known that there are ρuy u integers up to y u which only have prime factors y the y-smooth or

25 OTTAWA NOTES 25 y-friable integers Here ρu = for u, obviously, and ρu = /u ρu tdt 0 for u > In general suppose that fp = for all primes p y Define χu = χ f u = ϑy u p y u fp log p so that χt is a measurable function with χt for all t and χt = for t Let σu be defined as σu = for 0 u, and 37 σu = u for u > Then y u u 0 χtσu tdt u fn = σu + O n y u log y + y u In the y-smooth numbers example we had fp = if p y and fp = 0 if p > y so that χt = for t and 0 for t >, so that 37 gives us ρu = /u ρu tdt 0 In fact whenever χ : 0, C is measurable function with χt = for 0 t and χt for all t, there is a unique solution σu to 37 which is continuous, with σu for all u An example with fp = for all of the large primes p Let P be a set of primes y where y = x o It is known from sieve theory that the number of integers up to x with no prime factors from the set P is p P /p x In general suppose that fp = for all primes p y, x] We saw in 3b that fn Θf, x = Θf, y = + fp + fp2 x p p p 2 + ; p y In our example, f is totally multiplicative with fp = 0 if p P, and fp = otherwise What about f without such helpful structure? By Halász s theorem we see that the mean value 0 unless there exists α such that f α p := fp/p iα is mostly close to In that case, by partial summation and 3b and the discussion after, we deduce that fn xiα + iα f α n, and we will study this latter mean value We know that p x in particular, there exists extremely small ɛ > 0 such that x ɛ2 <p x ɛ Ref α p p Ref α p p is bounded and, < ɛ Now let gp = for all p x ɛ and gp = f α p otherwise, and hp = f α p for all p x ɛ2, and hp = otherwise By an inclusion-exclusion argument it is easy to show that f α n = gn hn + Oɛ, x x x

26 26 ANDREW GRANVILLE so now we can proceed to determine the mean value of f, using what we have discussed above about multiplicative functions like g that are on all the small primes, and those like h that are on all the large primes What we have proved is that if the mean value of f is not too small in absolute value, for some α < log x, times an Euler product, times the solution to an integral delay equation like 37 This is a strong version of the Structure Theorem of [GS] Although this is, we believe, the first such formal statement in the literature, such ideas have been used implicitly for a long time certainly it had been recognized that in many problems, the extreme cases are easily modeled by an integral delay equation Note that for any particular α, an admissible value for the Euler product, and an admissible χ, it is not difficult to construct examples f that come close to these values Hence from our model we can construct arithmetic examples then it can be written as a product, of x iα +iα Typically the xiα +iα and the Euler product are easy to determine, but the solution to the integral delay equation is difficult to determine The advantage of the structure theorem is that it allows the researcher to move away from the rather difficult arithmetic issues and focus instead on a cleaned up pure analysis question about integral delay equations I prefer to try to explain most of the proofs here in these terms however, as one referee wrote, it is usual to do these kind of calculations in rough and then translate the argument to the arithmetic setting, which is often quite challenging To start with let me justify Halász s theorem in this way Formally I am going in circles since we used Halász s theorem to prove the Structure Theorem, but in fact the proof that we now sketch can be re-written with multiplicative functions see [GS4] though with several additional complications 33 An integral delay equation version of Proposition 3 By a typical sieve iteration argument one can easily show that 38a σu = + where 38b I j u; χ = t,,t j t ++t j u j I j u; χ, j! j= χt t χt j t j dt dt j In fact if χt R for all t then we have the inclusion-exclusion relations, σ 2k+ u σu σ 2k u where 2 k j σ k u := + I j u; χ j! The Laplace transform of a function f : [0, C is defined by Lf, s = j= 0 fte st dt 2 Looking back at our papers particularly Proposition 36 of [GS] we did not, but should have, proved that σ 2k+ u is non-decreasing in k, and that σ 2k u is non-increasing in k