Simultaneous Distribution of the Fractional Parts of Riemann Zeta Zeros

Transcription

1 Simultaneous Distribution of the Fractional Parts of Riemann Zeta Zeros Kevin Ford, Xianchang Meng, and Alexandru Zaharescu arxiv: v2 [math.n] 1 Sep 2016 Abstract In this paper, we investigate the simultaneous distribution of the fractional parts of {α 1 γ, α 2 γ,, α n γ}, where n 2, α 1, α 2,, and α n are fixed distinct positive real numbers and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta-function. 1 Introduction and statement of results Let α be a fixed positive real number, and γ run over the imaginary parts of the zeros of the Riemann zeta-function. We are interested in the distribution of the fractional parts {αγ}. Rademacher [9] was the first to consider this problem and he conjectured that, for a certain specific type of α, there should be a predominance of terms which fulfill {αγ} 1/2 < 1/4. Since the fractional parts are uniformly distributed modulo 1, as proved by Hlawka [6] in 1975, any discrepancy must be very subtle. he first and third authors uncovered in [3] this delicate inequity in the fractional parts and not only proved Rademacher correct, but also gave a much more precise measure for this phenomenon. Let = R/Z be the torus. Since the fractional parts {αγ} are uniformly distributed mod 1 for any fixed α, for all continuous functions h : C, we have hαγ = N hudu + on, as, where N denotes the number of zeros 0 < γ. We know that [11], heorem 9.4 N = log e + Olog, 1. he first and third authors [3] showed that, for a large class of functions h : C, as, 1 hαγ N hudu = hug α udu + o1, 2010 Mathematics Subject Classification: Primary 11M26; Secondary, 11K38 Key words: Riemann zeta-function: zeros, fractional parts 1

2 where g α u is a function depending on the form of α. From this result, we can see that the right hand side is close to a constant, and thus the discrepancy of the set {hαγ : 0 < γ } is of order O 1. In [4], the first author, Soundararajan, and the third author established log connections between the discrepancy of this set, Montgomery s pair correlation function and the distribution of primes in short intervals. In the present paper, our goal is to generalize the results from [3] to the case of simultaneous distribution of fractional parts {α 1 γ}, {α 2 γ},, {α n γ}, where α 1,..., α n are distinct, positive real numbers. As we will see below, a new phenomenon involving Diophantine approximation appears in the higher dimensional case n 2. For n 2, consider a function hx defined on n, which has Fourier expansion hx = m Z n c m e im x, and assume that B 1 c m, 1.1 m + 1 for some constant B > n + 2, and is the sup-norm. Let α R n. Suppose that there is a full-rank r n r n matrix M = b ij with integer entries, gcdb i,1, b i,2,, b i,n = 1 for all 1 i r, and such that Mα = P, 1.2 where P = a 1 log p 1,, ar log p r q 1 q r for some integers a i, q i and distinct primes p i i = 1,, r. Among all possible such matrices, we take one with maximal r, in which case the above conditions ensure that M is uniquely determined. We then define { } g α x = 1 r log p j R p a j k 2 j e 2kq jπib j x π = 1 π j=1 r j=1 k 1 a j 2 log p j pj cosq j b j x 1 p a j j a j, pj cosq j b j x + 1 where b j = b j,1, b j,2,, b j,n for 1 j r. If no such matrix exists, we simply set hen we have the following. g α x = 0 x n. 1.4 heorem 1 Let n 2. Assume hx satisfies condition 1.1. hen, if α R n satisfies for all m Z n and some fixed constant C, we have 1 lim hγα N h n m α > C e m 1.5 = hg α. n 2

3 Remark 1. By a theorem of Khintchine [8] and heorem 2 in [7], the set of such α has full Lebesgue measure. Remark 2. It is possible to prove a similar conclusion with a larger class of test functions h, using more complicated Fourier analysis techniques, along the lines of what was done in [3], or by assuming the Riemann Hypothesis. For the special case when r = n and RankM = n, 1.5 will always be satisfied as a consequence of Baker s heorem [1], and we have the following. Corollary 1.1 If hx satisfies condition 1.1 and α satisfies condition 1.2 for r = n, we have 1 lim hγα N h = hg α. n n In general, the conclusion of heorem 1 may fail if the Diophantine condition 1.5 fails; for details see the Remarks at the conclusion of Section 3. For the case n = 2, the behavior of hγα is determined by the behavior of the convergents pn q n of the continued fraction of ξ := α 1 α 2. Even if 1.5 fails, which occurs when some q n+1 are very large compared to q n, we show that there are long intervals of -values on which we may draw the conclusion of the previous theorem. heorem 2 Assume 1.1 and fix any ɛ > 0. Let U α = n=1 [q1+ɛ n, e qb ɛ n ]. hen, 1 lim hα 1 γ, α 2 γ N h = hg α. U α 2 2 Finally, we show some graphs comparing the limiting density function g α with numerical computations obtained from the first 100 million zeros of the Riemann zeta-function which were kindly provided by omás Oliveira e Silva, see [10]. Define My 1, y 2 ; = M α y 1, y 2 ; := 1 N 1 y 1 y 2. {α 1 γ}<y 1,{α 2 γ}<y 2 ake = , then N = Denote = 1. We partition [0, 1 [0, into 2 small rectangles with side length. For some values of α 1, α 2, we plot the values of DM := 1 My 2 1 +, y 2 + My 1 +, y 2 My 1, y My 1, y 2 for each small rectangle [y 1, y 1 + [y 2, y 2 +. Example 1. Let α 1 and α 2 satisfy α 1 + α 2 = log 2, α 1 α 2 = log 3 4π. Figure 1.1 shows the graph of g α. Figure 1.2 shows the values of DM, the graph of which is a close approximation of g α. 3

4 Example 2. Figure 1.3 and Figure 1.4 show the analogous graphs for the case when α 1 and α 2 satisfy 2α 1 + α 2 = log 5, 2α 1 + 3α 2 = log 7. Figure 1.1: Graph of g α Figure 1.2: Values of DM 4

5 Figure 1.3: Graph of g α Figure 1.4: Values of DM 2 Proof of heorem 1 We need Lemma 1 from [3], which is an extension of a famous formula of Landau. Let ρ denote a generic nontrivial zero of the Riemann zeta-function, and denote γ = Iρ. Lemma 1 Let x, > 1, and denote by n x the nearest prime power to x. hen x ρ = Λn x e i logx/nx 1 + O x log 2 log 2 2x +, i logx/n x log x where if x = n x the first term is Λnx. Next, we extract from the proof of 3.8 in [3] the following estimate for difference between two sums over zeros, one with the real parts all normalized to be equal to 1/2. 5

6 Lemma 2 For 1 < x exp{ log 50 log log }, we have x iγ log2 x log + log Proof of heorem 1. By heorem 2 in [7], we know that the set of α s under the condition 1.5 has full Lebesgue measure. Write the Fourier expansion of hx as hx = c m e im α = m c m e im α + O c m, 2.2 m J m >J where J log is a parameter to be chosen later. By our assumption 1.1, one can see that the error term above is O1/J B n. hus, by 2.2, we have hγα = c m e iγm α N + O = N m J h + 2R c m J B n N e iγm α + O. J B n he second equality is a consequence of the identity c m = c m. Let x m = e m α. Hence, by 1.1 and Lemma 2, we have for large and J log that log log 2 hγα N h = 2R N c m m + O J B n log Write Applying Lemma 1, we get I = 1 +O = 1 I = c m m. c m Λ sin log xm xm log xm xm c m log 2 2x m + log 2 xm log x m c m Λ sin log xm + O e OJ log xm log xm 6

7 Here we used our assumption 1.5 that log x m = m α C e J. Now let J = log. By 2.3, 2.4 and B > n + 2, we get hγα N h = π R c m Λ sin log xm + o. 2.5 xm log xm Since Λnxm xm is bounded, the first sum is absolutely and uniformly convergent in under the assumption 1.1. And the terms with x m tend to zero as. hus, by 2.5, as, 1 hγα N h = 1 π R c m Λx m xm + o Since Λx m = 0 unless x m is prime power, by 1.4, when g α = 0, the sum in the right hand side of 2.6 is also 0. If α satisfies 1.2 with M being maximal maximal r, then the only terms with = x m that is, with x m being a prime power are those with m being a multiple of some row of M. Hence, recalling the definition 1.3 of g α, as, the right hand side of 2.6 becomes { r } 1 π R log p j p a j k 2 j c kqj m, j=1 k 1 which equals n hg α by 1.3. So in all cases, we have 1 lim hγα N h = hg α. n n herefore, we get the conclusion of this theorem. Proof of Corollary 1.1. For the special form of α when r = n, we can solve the above linear equations for α, and find that each α i is a linear combination of these log p i s with rational coefficients. Hence, by Baker s theorem [1], for such α, there exist constants D = Dα and µ, such that, for all m Z n, D m α m µ hus, such α satisfies condition 1.5 for some constant C. he conclusion follows from heorem 1. 7

8 3 Proof of heorem 2 We may assume that α 1 /α 2 is positive and irrational, else the problem reduces to the n = 1 case. Assume a function hx, y is defined on the two dimensional torus. By 2.3, we have hα 1 γ, α 2 γ N h = 2R m, l J mα 1 +lα 2 >0 c m,l m,l +O N J B 2 + log, 3.1 where J log is a parameter to be chosen later. We now break the double sum into two pieces, over those pairs m, l corresponding to large x m,l and those pairs m, l corresponding to small x m,l. Fix an arbitrary positive constant C and denote { E J := m, l Z 2 C : 0 < max{ m, l } J, mα 1 + lα 2 > min e, α } 2 max{ m, l } 2m, 1, 4π F J := Write { m, l Z 2 : 0 < max{ m, l } J, 0 < mα 1 + lα 2 min I E = m,l E J c m,l Applying Lemma 1 and using B > 4, we get m,l, I F = m,l F J c m,l C e, α 2 max{ m, l } 2m, 1 4π m,l. I E = 1 c m,lλn sin log xm,l xm,l,l xm,l log x m,l m,l E J,l +O c m,l xm,l log 2 log 2 2x m,l + xm,l log x m,l m,l E J = 1 c m,lλ,l xm,l m,l E J Here we have used the fact that for m, l E J, sin log xm,l,l log x m,l = mα 1 + lα 2 e J. }. log x m,l,l + O e OJ log By standard facts on Diophantine approximation, we know that if m, l F J then α 1 + l α 2 m < 1 2m, 2 and hence all elements of F J are of the form q n, p n, where p n /q n is a convergent of the continued fraction of α 1 α 2. As q n grows at least exponentially and q n J for q n, p n F J, there are Olog J = Olog log elements in F J. Let n denote the largest n such that q n, p n F J. 8

9 By elementary properties of continued fractions Chapter X, [5], or Corollary 1.4, [2], we have α 2 < q j α 1 p j α 2 < α q j + q j+1 q j+1 hus, for all j except possibly j = n, we have q j+1 J and from Lemma 1 we get writing x j = x qj, p j c qj, p j j n j = 1 c Λn qj, pj xj xj j n sin log xj n xj log x + Olog 3 j n xj = Olog 3, 3.4 where we have used that n xj = 2. We adopt the convention that the term j = n is included in the sum above if q n +1 J. In fact, the error term from Lemma 1 is acceptable, namely o, if q n +1 = o/ log as well. Now let J = log and assume that log 1/B log log < q n J. We revert to the sum over x iγ q n, p n. By 2.1 and the the trivial bound for xiγ q n, p n, we have c qn, p n q n, p n log q B n = o. Combined with 3.2 and 3.4, we see that if a F J is empty or if b there is some convergent of α 1 /α 2 with q n log 1/B log log, o/ log ], then hα 1 γ, α 2 γ = π R +N mα 1 +lα 2 >0 0<max{ m, l } J c m,l Λ,l sin log xm,l,l xm,l log x m,l,l h + o. 3.5 Note that b is satisfied if is large enough and n=1 [q1+ɛ n, e qb ɛ n ]. hen, similar to the proof of 2.6 in the proof of heorem 1, we deduce that 1 lim hα 1 γ, α 2 γ N h = hg α. U α hus, heorem 2 follows. Remarks. Suppose that there is a very long gap between convergents, say q n is small and q n +1 is very large. Put x = x qn, p n and assume that log x log. Using the known estimate N = log + Olog, we get e t x iγ = x it log dt + Olog 2 0 = log/ ei log x 1 i log x 9 + O.

10 In the range 1/ log x, there is thus a term in the sum 3.1 of order N c qn, p n, which may be larger than. Acknowledgement. Research of the first author and second author is partially supported by NSF grants DMS and DMS he authors would like to thank the helpful comments of the referee. References [1] A. Baker, ranscendental number theory, Cambridge Mathematical Library 2nd ed., Cambridge University Press, [2] Y. Bugeaud, Approximation by algebraic numbers, Cambridge University Press 2004 [3] K. Ford, A. Zaharescu. On the distribution of imaginary parts of zeros of the Riemann zeta function. J. reine angew. Math. 579, [4] K. Ford, K. Soundararajan, A. Zaharescu. On the distribution of imaginary parts of zeros of the Riemann zeta function, II. Math. Ann : [5] G. H. Hardy, E. W. Wright, An introduction to the theory of numbers, 5th ed., Oxford, 1979 [6] E. Hlawka, Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktionen zusammenhängen. Sitzungsber. Osterr. Akad. Wiss, Math.-Naturnw. Kl. Abt. II 184, [7] R. S. Kemble, A Groshev theorem for small linear forms, Mathematika, , [8] A. Ya. Khintchine, Einige Sätze über Ketternbrüche, mit Anwendungen auf die heorie der diophantischen Approximationen, Math. Ann., , [9] H. Rademacher, Fourier analysis in number theory, Symposium on harmonic analysis and related integral transforms: Final technical report, Cornell Univ., Ithica, N.Y. 1956, 25 pages; also in: H. Rademacher, Collected Works, pp [10] omás Oliveira e Silva, webpage, [11] E. C. itchmarsh, he heory of the Riemann Zeta-function, 2nd ed. rev. by D. R. Heath-Brown, Clarendon Press, Oxford 1986 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA KF: ford@math.uiuc.edu, XM: xmeng13@illinois.edu, Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA and Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest RO-70700, Romania. AZ: zaharesc@math.uiuc.edu 10