PRIMES IN PRIME NUMBER RACES

Transcription

1 PRIMES IN PRIME NUMBER RACES JARED DUKER LICHTMAN, GREG MARTIN, AND CARL POMERANCE Astract. The study of the relative size of the rime counting function πx and the logarithmic integral lix has led to a wealth of results over the ast century. One such result, due to Ruinstein and Sarna and conditional on the Riemann hyothesis RH and a linear indeendence hyothesis LI on the imaginary arts of the zeros of ζs, is that the set of real numers x for which πx > lix has a well-defined logarithmic density = A natural rolem, as yet unconsidered, is to examine the actual rimes in this race. We rove, assuming RH and LI, that the logarithmic density of rimes for which π > li relative to the rime numers is equal to the logarithmic density = mentioned aove. A ey element in our roof is a result of Selerg on the normal distriution of rimes in short intervals. We also extend such results to a road class of rime numer races, including the Mertens race etween <x / and e γ and the Zhang race etween x / and /. These latter results resolve a question of the first and third author from a revious aer, leading to further rogress on a 988 conjecture of Erdős on rimitive sets.. Introduction In the early twentieth century it was noticed that while the rime-counting function πx and the logarithmic integral function lix = x dt/ log t are satisfyingly close together for 0 all values of x where oth had een comuted, lix always seemed to e slightly larger than πx. It was a reathrough when Littlewood [] roved that in fact the sign of lix πx changes infinitely often as x. We still do not now a secific numerical value of x for which this difference is negative, ut the smallest such value is susected to e very large, near see [2] and susequent refinements. We do now that πx < lix for all 2 x 0 9, thans to calculations of Büthe [3]. Another imortant develoment concerning this race etween πx and lix was the aer of Ruinstein and Sarna [5]. Assuming some standard conjectures aout the zeros of the Riemann zeta-function, namely the Riemann hyothesis and a linear indeendence hyothesis on the zeros of ζ + it, they showed that the logarithmic density δπ of the 2 set. Π := {x R : πx > lix} exists and is a ositive numer.2 δπ = = Date: Setemer 9, Mathematics Suject Classification. Primary A05, N05; Secondary B83, M26. Key words and hrases. rime numer race, Cheyshev s ias, Mertens roduct formula, logarithmic density, limiting distriution, rimitive set, rimitive sequence.

2 Here, given a set M R, the logarithmic density of M is defined as usual as dt δm := lim x t, t M [,x] rovided the limit exists. Since πx counts rimes, it is natural to consider the actual rimes in the race: What can e said aout the set of rimes for which π > li? We define the discrete logarithmic density of a set M R relative to the rime numers as x M δ M := lim x log x M if the limit exists. Due to the artial summation formula = x + t log 2 t we see that if the modified limit x M.3 δ M := lim x 2 x M, t M exists, then it is equal to δ M. The converse does not hold in general, since δ M might not exist even if δ M does. For examle, let P e the set of all rimes etween 2 2! and 2 2!, and let P = P. Then δ P = /2 ut δ P does not exist. We shall find it more convenient to deal with δ M in our roofs elow. We also let δ and δ denote the exression on the right-hand side of equation.3 with lim relaced y lim su and lim inf, resectively. Our general hilosohy is that the rimes are reasonaly randomly distriuted; in articular, there seems to e no reason for the rimes to consire to lie in the set of real numers Π any more or less often than exected. In this aer we rove that there is no such consiracy; more recisely we rove, under the same two assumtions as Ruinstein and Sarna, that δ Π = see Theorem 2.2. Moreover, we rove similar results comaring the logarithmic density of a set of real numers to the relative logarithmic density of the rimes lying in that set for a numer of other rime races, some of which have not een considered efore see Theorems 4. and 5.6. Finally we remar that our aroach alies equally to rime races involving residue classes. To do this one would relace Selerg s theorem on almost all short intervals containing the right numer of rimes with a result of Kououlooulos [9, Theorem.] which does the same for rimes in a residue class to a fixed modulus. 2. A ey roosition For a naturally occurring set M of real numers for which δm exists, it is natural to wonder how δ M comares to δm. We rove the two densities are equal in the case of sets of the form 2. M a f = {x: fx > a} 2 dt,

3 for functions f that are suitaly nice. Proosition 2.. Consider a function f : R R satisfying the following two conditions: a For all real numers a >, there exists x 0 = x 0 a, such that for all x x 0, if fx > a then fz > for all z [x, x + x /3 ]; and similarly for the function fx. The function f has a continuous logarithmic distriution function: for all a R, the set M a f has a well-defined logarithmic density δm a f, and the ma a δm a f is continuous. Then for every real numer a, the relative density δ M a f exists and is equal to δm a f. It is worth noting that the assumtions and conclusion of the theorem imly that the relative density ma a δ M a f is also continuous; in articular, ties have density 0, meaning that δ {x: fx = a} = 0. Thus there is no difference etween considering fx > a and fx a in the situations we investigate. Recall the Linear Indeendence hyothesis LI, which asserts that the sequence of numers γ n > 0 such that ζ 2 + iγ n = 0 is linearly indeendent over Q. Theorem 2.2. Let the set Π and the numer = δπ e defined as in equations. and.2, resectively. Assuming RH and LI, the discrete logarithmic density of Π relative to the rimes is δ Π =. Proof of Theorem 2.2 via Proosition 2.. Consider the normalized error function E π x = log x πx lix, x and note that Π = {x: πx > lix} = M 0 E π. It thus suffices to show that E π satisfies conditions a and of Proosition 2.. Consider any numer z [x, x + x /3 ]. We have πz πx x /3 and liz lix x /3 trivially, and hence E π z E π x 2 /x /6. Since the right-hand side tends to 0, this inequality easily imlies condition a of the theorem. Moreover, condition namely the fact that E π has a continuous limiting logarithmic distriution is a consequence under RH and LI of the wor of Ruinstein and Sarna: first, they estalish a formula for the Fourier transform of this limiting logarithmic distriution see [5, equation 3.4 and the aragrah following]. They then argue that this Fourier transform is raidly decaying see [7, Section 2.3] for a more exlicit version of their method. From this they conclude that the distriution itself is continuous and indeed much more, namely that it corresonds to an analytic density function see [5, Remar.3]. The continuity of the limiting logarithmic distriution of E π can e deduced from a sustantially weaened version of LI: indeed, we only require the imaginary art of one nontrivial zero of ζs to not e a rational linear comination of other such imaginary arts see [4, Theorem 2.22]. In the next section we rove Proosition 2., which will comlete the roof of Theorem Proof of Proosition 2. We egin with some notation. For any interval I of real numers, let πi denote the numer of rimes in I. For any ositive real numer y, define the half-oen interval Iy := y, y + y /3 ]. Define an increasing sequence of real numers recursively y y = and 3

4 y + = y + y /3 for, and let I := Iy = y, y + ]. We have thus artitioned R > = = I into a disjoint union of short half-oen intervals. Lemma 3.. For any fixed real numer α > 2 3, we have = y α. Proof. For any U, the numer of integers such that y [U, 2U is at most U 2/3, since the length of each corresonding interval I is at least U /3. Therefore y α = y α 2 j 2/3 2 j α = 2, 2/3 α = j=0 : y [2 j,2 j+ j=0 since we assumed α > 2 3. Given ɛ > 0, we say that an interval Iy is ɛ-good if y/3 πiy log y ɛy/3 log y, and otherwise we say that Iy is ɛ-ad. Selerg [6] showed that there exists a set S ɛ R whose natural density equals for which 3. πy + y /3 πy y/3 log y for all y S ɛ ; in other words, for any ɛ > 0, the set of real numers y for which Iy is ɛ-ad has density 0. To aly [6] it is only imortant to realize that /3 > 9/77. This theorem has een susequently imroved. From Huxley [8], one has almost all intervals y, y+y /6+ɛ ] containing the right numer of rimes, in this regard see [9,.3]. Our next lemma shows that ɛ-ad intervals among the I are also sarse. Lemma 3.2. For each ɛ > 0, the union of the ɛ-ad intervals I has natural density 0, and hence logarithmic density 0. Proof. For every, define J := y, y + ɛ 4 y/3 ]. Suose that is chosen so that I is an ɛ-ad interval. Note that for all y J, the intervals Iy and I = Iy have nearly the same numer of rimes; more recisely, 3.2 πiy πi = π y +, y + y /3 ] π y, y], since the rimes in the larger interval y, y + ] cancel in the difference. By Titchmarsh s inequality [3, equation.2], we have πi 2h/ log h for all intervals I of length h > ; and since 2h/ log h is an increasing function of h for h > e, we deduce that for any interval I of length at most ɛ 3 y/3, πi 2 ɛ 3 y/3 log ɛ 3 y/3 y /3 < ɛ 2 log y when is sufficiently large in terms of ɛ. This deduction assumed that the length of I exceeds e, ut the final inequality is trivial for large when the length of I is at most e. In articular, oth intervals on the right-hand side of equation 3.2 have length at most 4

5 ɛ 3 y/3 when is sufficiently large, from which we see that πiy πi ɛ 2 y/3 / log y. Consequently, since I is ɛ-ad, we conclude that y/3 πiy log y πi y/3 log y πiy πi y /3 log y y/3 log y ɛy/3 y /3 y /3 ɛ + o > ɛ log y 2 log y 3 log y when is sufficiently large where the mean value theorem was used in the middle inequality. In other words, we have shown that I eing ɛ-ad imlies that Iy is ɛ -ad for all y J 3. Let J e the disjoint union of all the intervals J, where ranges over those ositive integers for which I is ɛ-ad. By the result of Selerg descried aove, the set of ɛ -ad real 3 numers which contains J has density 0, so J [, x] has measure ox. But this measure is at least ɛ times the measure of the union of all ɛ-ad intervals I 4 ; hence, the union of these intervals elow x also has measure ox, which comletes the roof. Proof of Proosition 2.. For the sae of simlicity, we areviate M a f to M a during this roof. Let ɛ and η e ositive arameters, and let B ɛ denote the union of all ɛ-ad intervals of the form I, so that B ɛ has logarithmic density 0 y Lemma 3.2. Suose that Iy is any ɛ-good interval. Since dt/t = y+y /3 dt/t = log + y 2/3 = Iy y y 2/3 + Oy 4/3, we see that 3.3 Iy log y y πiy + ɛy 2/3 = + ɛ Iy dt t + Oy 4/3, where the second inequality used the ɛ-goodness of Iy. On the other hand, even if Iy is an ɛ-ad interval, Titchmarsh s inequality still yields 3.4 log y y πiy log y y /3 y logy /3 dt y 2/3 t. Iy By condition a of the roosition, there exists a ositive integer C deending on a and η such that if is a rime in an interval I with > C, then the inequality f > a imlies that fz > a η for all z I. In articular, every I containing a rime with f > a is either a suset of B ɛ or else is an ɛ-good interval contained in M a η, so that + x f>a I M a η [,x] I is ɛ-good I I B ɛ [,x] Iy I. Using equation 3.3 for the terms in the first sum and equation 3.4 for the second sum, we otain the uer ound dt + ɛ M a η [,x] t + O y 4/3 dt + O y x B ɛ [,x] t x f>a dt + ɛ + O + o M a η [,x] t y Lemma 3. and the fact that B ɛ has logarithmic density 0. 5

6 Therefore we have δ M a = lim su x lim su x x f>a + ɛ M a η [,x] dt t + o = + ɛδm a η since δm a η exists y condition of the roosition. Similarly, the rimes in M a that are contained in ɛ-good intervals I M a form a suset of all rimes in M a. Then for a lower ound, it suffices to consider the ɛ-good intervals in M a, which y a simle comutation gives the ound δ M a ɛδm a. Since these ounds hold for all ɛ > 0, we see that 3.7 δm a δ M a δ M a δm a η. Finally, y condition the ma η δm a η is continuous, so since η > 0 was aritrary we conclude that δ M a = δ M a = δm a as desired. 4. The Mertens race In 874, Mertens roved three remarale and related results on the distriution of rime numers. His third theorem asserts that e γ as x, <x where γ is the Euler Mascheroni constant. The Mertens race etween e γ and this roduct of Mertens is mathematically analagous to the race etween lix and πx. Recent analysis of Lamzouri [0] imlies, conditionally on RH and LI, that the normalized error function E M x = log x log γ <x ossesses the exact same limiting distriution as that of E π x = log x lix πx x that aeared in the roof of Theorem 2.2. We say a rime is Mertens if E M > 0. It can e checed that the first 0 8 odd rimes are Mertens. The first and third authors have shown [2, Theorem.3], assuming RH and LI, that the lower relative logarithmic density of the Mertens rimes exceeds.995. Alying Proosition 2. to the Mertens race y choosing fx = E M x leads immediately to the following imrovement. Theorem 4.. Assuming RH and LI, the Mertens rimes have relative logarithmic density, where was defined in equation.2. 6

7 Proof. Consider any numer z [x, x + x /3 ]. We have log log z log /x y the mean value theorem and log n < x 2/3, x <z x <z x n<x+x /3 and so E M z E M x /x /6. The fact that this uer ound tends to 0 easily imlies condition a of Proosition 2.. Finally, condition is satisfied y a similar argument as in the roof of Theorem 2.2, using wor of Lamzouri [0] on the limiting logarithmic distriution of E M x. Theorem 4. has an interesting alication to the Erdős conjecture for rimitive sets. A suset of the integers larger than is rimitive if no memer divides another. Erdős [5] roved in 935 that the sum of /a log a for a running over a rimitive set A is universally ounded over all choices for A. Some years later in a 988 seminar in Limoges, he ased if this universal ound is attained for the set of rime numers. If we define fa = /a log a and fa = a A fa, and let PA denote the set of rimes that divide some memer of A, then this conjecture is seen to e equivalent to the following assertion. Conjecture 4.2 Erdős. For any rimitive set A, we have fa fpa. The Erdős conjecture remains oen, ut rogress has een made in certain cases. Say a rime is Erdős-strong if fa f for any rimitive set A such that each memer of A has as its least rime factor. By artitioning the elements of A into sets A y their smallest rime factor, it is clear that the Erdős conjecture would follow if every rime is Erdős-strong. The first and third authors [2, Corollary 3.0.] roved that every Mertens rime is Erdős-strong. In articular, the Erdős conjecture holds for any rimitive set A such that, for all a A, the smallest rime factor of a is Mertens. In [2] it was conjectured that all rimes are Erdős-strong. Since 2 is not a Mertens rime, it would e great rogress just to e ale to rove that 2 is Erdős-strong. Nevertheless, Theorem 4. imlies the following corollary. Corollary 4.3. Assuming RH and LI, the lower relative logarithmic density of the Erdősstrong rimes is at least. In articular, the Erdős conjecture holds for all rimitive sets whose elements have smallest rime factors in a set of rimes of lower relative logarithmic density at least. 5. The Zhang race By the rime numer theorem, one has the asymtotic relation x as x, and y insection one further has 5. x for a large range of x. Beyond its aesthetic aeal, this inequality arises quite naturally in the study of rimitive sets. Indeed, Z. Zhang [7] used a weaened version of 5. to rove 7

8 Conjecture 4.2 for all rimitive sets whose elements have at most 4 rime factors, which reresented the first significant rogress in the literature. Call a rime q Zhang if the inequality 5. holds for x = q. From comutations in [2], the first 0 8 rimes are all Zhang excet for q = 2, 3. Following some ideas of earlier wor of Erdős and Zhang [6], the first and third authors have shown [2, Theorem 5.] that Conjecture 4.2 holds for any rimitive set A such that every memer of PA is Zhang. We wish to find the density of N, the set of real numers for which the Zhang inequality 5. holds. Note that x N if and only if the normalized error E Z x := x 5.2 log 2 x x is nonnegative. To show the density of N exists we follow the general lan laid out y Lamzouri [0], who roved analogous results for the Mertens race, with some imortant modifications. 5.. Exlicit formula for E Z x. First we relate the sum over rimes, x /, to the corresonding series over rime owers, n x Λn/n log2 n, in the following lemma. Lemma 5.. For all x >, E Z x = Λn x log 2 n log 2 x + + O. n n x Proof. Our first ste is to convert the sum over rimes to rime owers, via 5.3 = Λn n log 2 n 2. x n x >x 2 The rime numer theorem gives that πy = y/ log y + Oy/ log 2 y, so for any y 2, 2 = πy y 2 log y + 2 log t + y y t 3 log 2 πt dt t = y log 2 y + O 2 log t + y log 3 + y y t 2 log 3 + O dt t log t = y log 2 y + 2 y log 2 y + O y log 3 = y y log 2 y + O y log 3. y In articular, taing y = x, >x 4 2 = x log 2 x + O For the larger owers of rimes, we have < n < x / + >x n>x / 8 x log 3 x. dt x t / x + x 2/3 x /

9 uniformly for 3, and thus >x x 2/3 2 x 2/3. Inserting the estimates 5.4 and 5.5 into equation 5.3 then yields = Λn n log 2 n x n x x log 2 x + O x log 3, x which imlies the statement of the lemma. By integrating twice, we relate our series Λn/n log 2 n to the series Λn/n a = ζ /ζa, which is more amenale to contour integration. This leads to the following exlicit formula for E Z x over the zeros of ζs, analogous to [0, Proosition 2.]. Proosition 5.2. Unconditionally, for any real numers x, T 5, E Z x = x ρ /2 x ρ + O + T log2 xt + Imρ <T where ρ runs over the nontrivial zeros of ζs. Imρ <T x Reρ /2, Imρ 2 Proof. Our starting oint is a tool from Lamzouri, namely [0, Lemma 2.4]: for any real numers a > and x, T 5, Λn = ζ x a a + n a ζ a x ρ a ρ a n<x Imρ <T x a + x a Λn + O T 4 a + log 2 x + log2 T Then integration with resect to a gives for any >, Λn n log n = Λn da n a where n<x n<x = log ζ + x a a da Imρ <T E x + x 4 T + + log2 T log 2 + x T Integrating once again with resect to, we have Λn n log 2 n = Λn n log n d n<x 5.6 = n<x log ζ d + x a a 9 n da d Imρ <T + T n na+/ x ρ a ρ a da + E, Λn n +/ log n.. x ρ a ρ a da d + E 2,

10 where E 2 x + 4 T log 2 x + + log2 T log 3 + Λn x T n +/ log 2 n n x + + log2 T T log 3 + Λn x T n log 2 n x + T n + log2 T log 3 x, since n Λn/n log2 n. The first term on the right-hand side of equation 5.6 can e written as 5.7 log ζ d = Λn n log n d = n n Λn n log 2 n, where the Fuini Tonelli theorem justifies the interchange of summation and integration since all terms are nonnegative. The second term on the right-hand side of equation 5.6 evaluates to 5.8 x a a da d = = x a a d da a x a da = x a =, where the interchange of integrals is again justified y the Fuini Tonelli theorem. The doule integral inside the series on the right-hand side of equation 5.6 is evaluated using a similar calculation: x ρ a ρ a a da d = ρ a xρ a da = x ρ a da ρ x ρ a ρ a da. The first integral comes out to x ρ /, while for the second, integrating y arts twice gives x ρ a xρ ρ da = ρ a + x ρ 2ρ x ρ a ρ log 2 + x log 2 x ρ a da. 3 Letting u = a, we have a = + u/ so the latter integral ecomes 2ρ log 2 x x ρ a 2ρ xρ da = ρ a 3 log 3 x Note that ρ u/ Imρ for all u R, so we deduce 2ρ x ρ a log 2 x ρ a da 3 xreρ Imρ 2 log 3 x. 0 0 e u ρ u/ 3 du.

11 Thus we have 5.9 x ρ x Reρ ρ log 2 x + O Imρ 2 log 3 x x ρ x Reρ = ρ log 2 x + O Imρ 2 log 3. x x ρ a xρ x ρ da d = ρ a + The calculations 5.7, 5.8, and 5.9 transform equation 5.6 into n<x and thus n x Λn n log 2 n = n Λn n log 2 n = log 2 x Λn n log 2 n + O x + T Imρ <T + O log 2 x x ρ ρ Imρ <T + log2 T log 3 x x ρ ρ x + + log2 T/ log 3 x T + log 3 x + log 3 x Imρ <T The roosition now follows uon comaring this formula to Lemma 5.. Imρ <T x Reρ Imρ 2 x Reρ. Imρ 2 If we assume the Riemann hyothesis we otain the following corollary, analogous to [0, Corollary 2.2]. Corollary 5.3. Assume RH, and let + iγ run over the nontrivial zeros of ζs with γ > 0. 2 Then, for any real numers x, T 5 we have E Z x = 2 Re x iγ x 5.0 /2 + iγ + O + T log2 xt, 0<γ<T Proof. By the Riemann von Mangoldt formula, x iγ γ 2 γ <T γ <T so the corollary now follows from Proosition 5.2. γ 2, 5.2. Density Results. Since the exlicit formula for the Zhang rimes in Corollary 5.3 is exactly the same as that of the Mertens rimes given y Lamzouri uon noting a tyo in [0, Corollary 2.2], namely, that E M x = + should read E M x =, the analysis therein leads to the following results. Recall that N is the set of real numers for which the Zhang inequality 5. holds, and that E Z x is defined in equation 5.2. Theorem 5.4. Assume RH. Then 0 < δn δn <.

12 Moreover, E Z x ossesses a limiting distriution µ N, that is, x lim fe Z t dt = ft dµ N t x for all ounded continuous functions f on R. 2 Proosition 5.5. Assume RH and LI. Let Xγ e a sequence of indeendent random variales, indexed y the ositive imaginary arts of the non-trivial zeros of ζs, each of which is uniformly distriuted on the unit circle. Then µ N is the distriution of the random variale 5. Y = 2 Re Xγ. γ>0 /4 + γ 2 R Theorem 5.6. Assume RH and LI. Then δn exists and equals. Hence y Proosition 2., the relative logarithmic density of the Zhang rimes is. These results are comletely analogous to Theorems. and.3 and Proositions 4. and 4.2 from [0]. Before moving on, we note a further consequence of the fact that E M x and E Z x ossess the same exlicit formula, namely that the symmetric difference of Mertens rimes and Zhang rimes has relative logarithmic density 0. Corollary 5.7. Assume RH and LI. Then we have δs = δ S = 0 for the symmetric difference S = S S 2, where S = {x : E M x > 0 E Z x} and S 2 = {x : E Z x > 0 E M x}. Proof. Tae η > 0. Comining [0, Corollary 2.2] with Corollary 5.3 and letting T tend to infinity, we find that EM x E Z x 5.2 = O. Let c e the imlied constant in equation 5.2. Thus for all x e c/η, if E M x > 0 then E Z x > η. This means that δ S δ {x : E Z x > 0} δ {x : E Z x > η} = δ{x : E Z x > 0} δ{x : E Z x > η}, which tends to 0 as η 0 y continuity, using Proosition 2. and Theorem 5.4. Since this holds for all η > 0, we conclude that δ S = 0. Interchanging the roles of E M and E Z roves δ S 2 = 0, and thus δ S = δ S S 2 = 0. A similar argument simler even, without the aeal to Proosition 2. shows that δs = 0. We also remar, however, that the analogous argument does not wor for E π. This is ecause the relevant series over nontrivial zeros is ρ xρ /ρ for E M and E Z, while for E π it is ρ xρ /ρ. Assuming RH, this amounts to the oservation that the two series γ xiγ / /2 + iγ and γ xiγ //2 + iγ are not readily comarale for a given x even though, y symmetry, oth do ossess the same limiting distriution, which exlains the aearance of δπ = in results on the Mertens and Zhang races. 2

13 The analogous rolem of determining the density of the symmetric difference etween the Mertens/Zhang rimes and the li-eats-π rimes is an interesting rolem for further investigation; it would resumaly roceed y examining the two-dimensional limiting distriution of the ordered air of normalized error terms, and understanding how the correlations of the two functions summands imacts the two-dimensional limiting distriution. 6. Other series over rime numers Before concluding our analysis, we remar that similar considerations aly more generally to series of the form α +, where Z and α R. The asic aroach is to first relate the sum of interest to the corresonding sum over rime owers via log α = n Λn n α log n m,m 2 m + mα. The next ste is to emloy an exact formula relating the sum over rime owers to series over zeros of ζs. For examle, von Mangoldt s exact formula states that 6. n x Λn = x ζ ζ 0 ρ x ρ ρ + x 2m 2m m rovided x is not a rime ower. The aove formula naturally generalizes to any real exonent α. Namely, one may rove y Perron s formula c.f. [0, Lemma 2.4] that Λn n = x α α α ζ ζ α x ρ α ρ α + x 2m α 6.2 2m α. n x ρ m rovided x is not a rime ower, and α is neither nor a negative even integer. Note that when α >, we have ζ /ζα = n Λn/nα so we may simlify the aove as 6.3 n x Λn n α = x α α + ρ x ρ α ρ α m x 2m α 2m α. To gain factors of log n in the numerator, we differentiate with resect to α. Secifically, since d/dα[x c α /c α] = + /c αx c α /c α for any c C, y induction one may show d [ x c α ] x = c α dα c α c α + O This imlies, for all, Λn log n α n n x = x α α + ρ Similarly for integration, we have α x ρ α ρ α x 2m α + O. 2m α m x c β c β dβ = lixc α = + O/ xc β c β, so an induction argument will estalish the exact formula for negative integers. 3

14 From here, all that remains is to analyze the sum over nontrivial zeta zeros. Assuming RH, it suffices to consider the series γ xiγ //2 α + iγ. Further assuming LI, this series has a limiting distriution, which may e comuted exlicitly as in [, 5]. Acnowledgments We are grateful for a helful discussion with Dimitris Kououlooulos. The first-named author thans the office for undergraduate research at Dartmouth College, and is currently suorted y a Churchill Scholarshi at the University of Camridge. The second-named author was suorted in art y a National Sciences and Engineering Research Council of Canada Discovery Grant. The second- and third-named authors are grateful to the Centre de Recherches Mathématiques for their hositality in May, 208 when some of the ideas in this aer were discussed. References. A. Aary, N. Ng, and M. Shahai, Limiting distriutions of the classical error terms of rime numer theory. Q. J. Math , C. Bays and R. J. Hudson, A new ound for the smallest x with πx > lix, Math. Com , no. 23, J. Büthe, An analytic method for ounding ψx, Math. Com , no. 32, L. Devin, Cheyshev s ias for analytic L-functions, rerint. htt://front.math.ucdavis.edu/ P. Erdős, Note on sequences of integers no one of which is divisile y any other, J. London Math. Soc , P. Erdős and Z. Zhang, Uer ound of /a i log a i for rimitive sequences, Proc. Amer. Math. Soc , D. Fiorilli and G. Martin, Inequities in the Shans Rényi rime numer race: an asymtotic formula for the densities, J. reine angew. Math , M. N. Huxley, On the difference etween consecutive rimes, Invent. Math , D. Kououlooulos, Primes in short arithmetic rogressions, Int. J. Numer Theory 205, Y. Lamzouri, A ias in Mertens roduct formula, Int. J. Numer Theory, 2 206, J. E. Littlewood, Sur la distriution des nomres remiers, Comtes Rendus de l Acad. Sci. Paris 58 94, J. D. Lichtman and C. Pomerance, The Erdős conjecture for rimitive sets, Proc. Amer. Math. Soc., in ress. 3. H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematia, , I. Richards, On the normal density of rimes in short intervals, J. Numer Theory, 2 980, M. Ruinstein and P. Sarna, Cheyshev s ias, Exeriment. Math , A. Selerg, On the normal density of rimes in small intervals, and the difference etween consecutive rimes, Arch. Math. Naturvid, , Z. Zhang, On a conjecture of Erdős on the sum n /, J. Numer Theory 39 99, 4 7. DPMMS, Centre for Mathematical Sciences, University of Camridge, Wilerforce Road, Camridge CB3 0WB, UK address: jdl65@cam.ac.u Deartment of Mathematics, University of British Columia, Room 2, 984 Mathematics Road, Vancouver, BC, Canada V6T Z2 address: gerg@math.uc.ca Deartment of Mathematics, Dartmouth College, Hanover, NH address: carl.omerance@dartmouth.edu 4