On the Uniform Distribution of Certain Sequences

Transcription

1 THE RAANUJAN JOURNAL, 7, 85 92, 2003 c 2003 Kluwer Academic Publishers. anufactured in The Netherlands. On the Uniform Distribution of Certain Sequences. RA URTY murty@mast.queensu.ca Department of athematics, Queen s University, Kingston, K7L 3N6, Ontario, Canada KOTYADA SRINIVAS srini@mast.queensu.ca Department of ath & Stats., Queen s University, Jeffery Hall, Kingston, Ontario, K7L 3N6, Canada In memory of Robert A. Rankin Received January 9, 2002; Accepted arch 22, 2002 Abstract. We investigate the uniform distribution of the sequence n α as n ranges over the natural numbers and α is a fixed positive real number which is not an integer. We then apply this in conjunction with the Linnik-Vaughan method to study the uniform distribution of the sequence p α as p ranges over the prime numbers. Key words: exponential sums, uniform distribution, the Linnik-Vaughan method 2000 athematics Subject Classification: Primary K06, L20, L07. Introduction In this paper, we will investigate the uniform distribution of the sequence {n α }, where n ranges over natural numbers and {p α }, where p ranges over prime numbers. We will focus our attention on 0 <α<, though we will make remarks for α>as well. The sequence {n α } has been investigated in the literature [4], though no explicit error terms have been written down. We will do so below in Theorem. Then we will apply Linnik-Vaughan method to obtain estimates for e 2πipαθ. p x Such sums for θ = a/q rational and α = /2 have arisen recently in the work [3], where an interesting connection is made between sharp estimates for such sums and the absence of zeros of Ls, f where f is a Hecke eigenform, on a certain segment of the real line close to the edge of the critical strip. We now elucidate the precise nature of the results we prove. For a real number x, let [x] denote the integral part of x; let {x} =x [x] be the fractional part of x or the residue of x modulo. Permanent address: The Institute of athematical Sciences, Tharamani P.O., Chennai-600 3, India. srini@ imsc.ernet.in

2 86 URTY AND SRINIVAS Let ω = x n, n =, 2,...be a given sequence of real numbers. For a positive integer N and a subset E of I = [0,, let the counting function AE; N; ω be defined as the number of terms x n, n N, for which {x n } E. The sequence x n, n =, 2...is said to be uniformly distributed modulo in short u.d. mod if for every sub-interval E of I,wehave AE; N; ω lim = E. N N In other words, x n is u.d. mod if every half open sub-interval of I eventually gets its proper share of fractional parts. There is a deep connection between the theory of u.d. mod and the estimation of exponential sums as envisaged by Weyl, which we mention below. Weyl s Criterion see page 7 of [4] says that the sequence x n, n =, 2,...is u.d. mod if and only if N e 2πihx n = on for all integers h 0. n= Remark. Using Weyl s criterion it is easy to show that the sequence nθ, n =, 2,...is u.d. mod whenever θ is irrational and is not u.d. mod if θ is rational number. In this paper we shall investigate the distribution of the fractional parts of the sequence n α for α>0 not an integer. ore precisely we prove the following Theorem. a Let SN = N n= e2πinαh. Then for all integers h 0, we have SN = O h 4 N 4 logn + 0 h 2, b whenever 0 <α<2,α SN = O h 2 N α+9 2 logn + 0 h 2 whenever 2 <α<3 SN = O max h 2 k 2 N k α 2 k 2, N whenever k <α<k, k 4 is an integer As a corollary we obtain the following well-known k α 2k 2 2 k 2 logn + 0 h 2 Theorem 2. The sequence n α, n =, 2,...is uniformly distributed modulo for α> 0 not an integer.

3 ON THE UNIFOR DISTRIBUTION OF CERTAIN SEQUENCES 87 Remark. Theorem 2 can be obtained by using Fejér s theorem see page 29 of [4] or by using Van der Corput s lemma see page 7 of [4] which gives the exponent of N in the O-term for theorem a as 3/4. However, our treatment yields the exponent 5/8. Remark. It would be nice to ask if Theorem throws any light on the bounds for the exponential sum e 2πipαh. p X Following Eratosthenes sieve, one can proceed in the following way: Let z = X, Pz = p p z then the above exponential sum is O X + n X e 2πinα h d Pz,n µd. We use a sophisticated version of the above idea as exemplified by Vaughan see page 38 of [] and elaborated by the first author and Sankaranarayana in [5] and obtain the following Theorem 3. n N ne 2πinαh = O h /8 N 4+2α 6 logn + 0 h 3 uniformly in α for 0 <α<. Remark. As stated above this, exponential sum seems to come up in the recent work of Iwaniec, Luo and Sarnak [3] concerning the Siegel zeros of Hecke L-functions attached to certain eigenforms. For details see [3]. 2. Some lemmas We will estimate the sums in question using the Poisson summation formula, but with an effective version of it. We use this occasion to point out that the Poisson summation formula can be derived from the simplest case of Euler-aclaurin sum formula N f j = j= N N f t dt + f t {t} dt 2 by writing down the Fourier series for {t} and inserting this in the integral. 2

4 88 URTY AND SRINIVAS In fact, it is not hard to show that {x} 2 = 0< m emx 2πim + O min, x so that one can write the right hand side of as N f emt t 2πim + O min, dt t 0< m = N NK log f temt dt + O 0< m where f t K for all t [, N]. This proves j= Lemma. Effective Poisson Summation Formula. Let f t be differentiable on [, N] satisfying f t K. Then N f j = N NK log f temt dt + O where ex = e 2πix. 0 m We will also need the following well-known result Lemma 2. Let Fx be real, twice differentiable function in [a, b] such that F x m > 0 or F x m < 0. Then b e ifx dx 8. m Proof: See for example page 56 of [2]. a 3. Proof of the theorems Proof of Theorem : First we split the interval [, N] into dyadic intervals of the type [W, 2W ]. Clearly there are Olog N such intervals. Therefore, it is enough to estimate the sum SW, 2W = e 2πinαh. W n 2W

5 ON THE UNIFOR DISTRIBUTION OF CERTAIN SEQUENCES 89 By taking f t = e 2πinαh in lemma we obtain SW, 2W = 2W h W e 2πit αh+mt α log dt + O 0 m where is a large positive constant to be chosen later. Now we take Ft = t α h+mt in lemma 2. 2π Observe that Therefore W F t = αα t α 2 h. 2π F t C hw α 2 > 0 or F t C 2 hw α 2 < 0, 2 depending on whether h < 0orh > 0 respectively, provided 0 <α<. Here C, C 2 are positive constants which may depend on α. Hence from lemma 2, we obtain 2W e 2πit αh+mt dt = O αw α/2 h /2 3 W By 2 and 3, we obtain SW, 2W = O W α/2 h /2 h W α log + O Choosing = C 3 [ h 3/4 W 3α 2 4 ], where C 3 is a large positive constant, we obtain SW, 2W = O h /4 W 4 log W h.. This proves n N e 2πinαh = O α h /4 N 4 log Nlog N h provided 0 <α<. Remark. The estimates for SN for α>,α not an integer are obtained by Van der Corput s lemma see Theorem 5.3 of [6] and then using the exponent pair method see page 72 of [2]. Proof of Theorem 3: To prove the theorem we invoke Vaughan s method as illustrated in [] and recently elaborated in [5].

6 90 URTY AND SRINIVAS Let n denotes the usual von angoldt function defined as n = log p if n = p m for some prime p and some integer m 0, 0 otherwise. With f n = e 2πinαh,0<α<, we form the sum n f n = a n + a 2 n + a 3 n + a 4 n f n n N n N = S N + S 2 N + S 3 N + S 4 N say Here a i n s are as given in page 39 of []. We now begin our estimations of the sums S i N for i =, 2, 3, and 4. Lemma 3. S N U log U Proof: This is clear. Lemma 4. S 2 N = O h 4 U 2 V 2 N 4 logn h 3 uniformly in α for 0 <α<. Proof: S 2 N = mµd n N f n = mdr=n m U = O mdr=n m U mµd m U r N md N m md f mdr 4 h md α 4 logn h 2 = O md /2 h 4 N 4 logn h 3 m U = O h 4 U 2 V 2 N 4 logn h 3

7 ON THE UNIFOR DISTRIBUTION OF CERTAIN SEQUENCES 9 Lemma 5. Proof: S 3 N = O h 4 V 2 N 4 logn h 3 S 3 N = µd log l e 2πihnα n N ld=n = µd l e 2πihldα l N/d { N = µd dt t } e 2πihldα t l N/d dt t From Theorem, it follows that the right hand side is O N 4 d /2 h /4 logn h 3 = O h 4 N 4 V 2 logn h 3 Lemma 6. S 4 N = O h 8 N +α/8 U /4 logn h 4 + NV /2 logn h 3 Proof: where Following [], we have S 4 N = O N /2 logn 3 max U N/V = O max V j N/ V <k N/ <m 2 m N/j m N/k f mj f mk 2 Now the innermost sum on the right hand side of the above expression is evaluated as follows e 2πimα j α k α h = O h 4 j α k α 4 4 log h 2 + Omin N/j, m

8 92 URTY AND SRINIVAS Therefore, Hence V <k N/ <m 2 e 2πimα j α k α h max e 2πimα j α k α h V < j N/ V <k N/ <m 2 Thus, This proves the lemma. = O N h 4 N N + min j, = O N h 4 + O N V α/4 4 log h 2 = O h 8 N /2+α/8 /4 log h + O N /2 V /2 N α/4 4 log h 2 Proof of Theorem 3: We choose the parameter U and V as follows: U = C 4 N /2 and V = C 5 N α 4 for 0 <α<. Here C 4, C 5 denotes positive constants. With these choices, the theorem follows from Lemma 3 6. References. H. Davenport, ultiplicative Number Theory, 2nd edn., GT 74, Springer-Verlag, Berlin-Heidelberg- New York, A. Ivić, The Riemann Zeta-Function, John Wiley and Sons, New York, H. Iwaniec, W. Luo, and P. Sarnak, Low lying zeros of families of L-functions, Inst. Hautes Etudes Sci. Publ. ath., No , L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley and Sons, New York, Ram urty and A. Sankaranarayanan, Averages of exponential twists of the Liouville function, Forum ath., , no. 2, E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn., Clarendon Press, Oxford, 986.