LOG-LIKE FUNCTIONS AND UNIFORM DISTRIBUTION MODULO ONE. 1. Introduction and results

Transcription

1 uniform distribution theory DOI: 0478/udt Unif Distrib Theory 3 (08), no, 93 0 LOG-LIKE FUNCTIONS AND UNIFORM DISTRIBUTION MODULO ONE Martin Rehberg Technische Hochschule Mittelhessen, Friedberg, GERMANY ABSTRACT For a function f satisfying f(x) =o((log x) K ), K > 0, and a sequence of numbers (q n ) n, we prove by assuming several conditions on f that the sequence (αf(q n )) n n0 is uniformly distributed modulo one for any nonzero real number α This generalises some former results due to Too, Goto and Kano where instead of (q n ) n the sequence of primes was considered Communicated by Werner Georg Nowak Introduction and results Let p n be the nth prime number in ascending order and α a nonzero real number Then G o t o and K a n o [], [], as well as Too [7], proved by assuming several conditions on the function f that the sequence (αf(p n )) n is uniformly distributed modulo one Recall that a sequence (x n ) n N of real numbers is said to be uniformly distributed modulo one if for every pair α, β of real numbers with 0 α<β the proportion of the fractional parts of the x n in the interval [α, β) tends to its length in the following sense: #{ n N : {x n } [α, β)} lim = β α N N In fact G o t o, K a n o and T o o proved their results by determining a bound for the discrepancy of the considered sequence Definition Let x,, x N be a finite sequence of real numbers The number D N := D N (x,,x N ):= sup #{ n N : {x n } [α, β)} (β α) N 0 α<β 00 M a t h e m a t i c s Subject Classification: J7, K38 K e y w o r d s: Uniform Distribution, Discrepancy 93

2 MARTIN REHBERG is called the discrepancy of the given sequence If ω = (x n ) is an infinite sequence (or a finite sequence containing at least N terms), D N (ω) ismeantto be the discrepancy of the first N terms of ω It is well known that a sequence ω =(x n ) n N is uniformly distributed modulo one if and only if lim N D N (ω) = 0 Instead of the sequence of primes, which was investigated in [], [] and [7], we consider a sequence of real numbers (q n ) n satisfying <q <q < with q n as n Further,weassumethat the sequence (q n ) n satisfies Q(x) c x dt log t x (log x) k () for every positive k>, where Q(x) := q n x andc>0 is some constant Note that condition () holds for the sequence of primes (with c =),aswell as for primes in arithmetic progressions (with c = ϕ(q),whereϕ is Euler s function and q is the modulus) However, the sequence (q n ) n we consider satisfies () and for such a sequence we prove the following theorems: Theorem Let a>0, n 0 := min{n N : q n >a} and let the function f :[a, ) (0, ) satisfy the conditions (a) f is twice differentiable with f > 0, (b) x f (x) as x, (c) (log x) f (x) and x (log x) f (x) are nonincreasing for sufficiently large x, (d) f(x) =o ( (log x) K) for some K>0 as x Then, for any nonzero real constant α, the sequence ( αf(q n ) ) is uniformly n n 0 distributedmodulooneand f(q N ) (log q N ) + K qn f (q N ) + (log q N )( qn f (q N )) as N () Theorem 3 Let a>0, n 0 := min{n N : q n >a} and let the function f :[a, ) (0, ) satisfy the conditions: (a) f is twice differentiable with f > 0, (b) x f (x) as x, (c) (log x) f (x) is nonincreasing for sufficiently large x, (d) f(x) =o ( (log x) K) for some K>0 as x 94

3 LOG-LIKE FUNCTIONS Then, for any nonzero real constant α, the sequence ( αf(q n ) ) n n 0 distributedmodulooneand f(q N ) (log q N ) + K qn f (q N ) as N is uniformly (3) Theorem 4 Let a>0, n 0 := min{n N : q n >a} and let the function f :[a, ) (0, ) satisfy the conditions: (a) f is continuously differentiable, (b) xf (x) as x, (c) (log x) f (x) is monotone for sufficiently large x, (d) f(x) =o ( (log x) K) for some K>0 as x Then, for any nonzero real constant α, the sequence ( αf(q n ) ) is uniformly n n 0 distributedmodulooneand { } f(q N ) (log q N ) K +max N, q N f (4) (q N ) as N In view of [, Theorem ] it should be remarked that in Theorem 4 a replacement of conditions (a) and (b) by: (a ) f is continuously differentiable and f(x) as x, (b ) x f (x) as x, would lead to the same discrepancy estimate, where only f (q N )hastobereplaced by f (q N ) If we compare Theorem with [7, Theorem 3], one notices that the nondecreasing condition is replaced by nonincreasing It was already remarked in [6] that this replacement is necessary Applying Theorem to the function f(x) =(logx) K with an arbitrary K>, we obtain that the sequence ( (log q n ) K) is uniformly distributed modulo one n This generalises the example in [7] on the uniform distribution modulo one of the sequence ( (log p n ) K) n Note that it was proved by W i n t n e r [0] that the sequence (log p n ) n is not uniformly distributed modulo one A shorter proof can also be found in [8, Exercise 59] Similarly, the sequence (log q n ) n is not uniformly distributed modulo one if () is replaced by Q(x) x(log x) for x The proof is analogue to the one in [8] An example of a sequence satisfying Q(x) x(log x) for x is a sequence where each q n fulfills p n q n p n+ (see [3], [5]) 95

4 MARTIN REHBERG Proofs Except for the already pointed out change of the conditions nondecreasing and nonincreasing in Theorem, the theorems of this paper are generalisations of the theorems in [7] Therefore it might not be surprising that the proofs are similar However, for the sake of completeness we state the whole proofs and do not only point out changes in the reasoning In the proofs we will make use of a theorem due to E r d ös and Turán to estimate the discrepancy, which was proved by them in 940: Theorem 5 For any finite sequence x,,x N of real numbers and any positive integer m, we have ( m ) D N C m + N h e πihx n N, where C is an absolute constant h= P r o o f o f T h e o r e m 5 can be found in [4] In addition, we need the following estimates n= Lemma 6 Let F (x) and G(x) be real functions, G(x) F (x) m>0, or F (x) G(x) m<0 Then b a G(x)e if (x) dx 4 m monotone and F (x) G(x) Lemma 7 Let G(x) be a positive decreasing function (a) If F (x) < 0, F (x) 0 and G (x) F (x) is monotone, then { } b { G(x)e πif(x) dx a 4max G(x) G } (x) + max a x b F (x) a x b F (x) (b) If F (x) > 0 and F (x) 0, then b { } G(x)e πif(x) G(x) dx 4max a x b F (x) a The first Lemma can be found in [9, Lemma 43] and the second one in [, Lemma 0, Lemma 03, p5] 96

5 LOG-LIKE FUNCTIONS P r o o f o f T h e o r e m Note first that it is enough to prove (), (3) and (4) for α>0 If we replace f by αf we see that it is enough to prove these statements for α = In view of condition (b) we can assume that f (x) < 0 for sufficiently large x Further, we may assume that for x a we have f (x) < 0andthat both (log x) f (x) andx(log x) f (x) are nonincreasing for x a Toprove that the sequence (f(q n )) n n0 is uniformly distributed modulo one, it is enough to prove estimation () In view of conditions (b) and (d), the term on the right side in () surely tends to zero as N tends to infinity To obtain (), our aim is to apply Theorem 5 in the form m m + N h N, (5) h=n 0 n=n 0 e πihf(qn) where m is an arbitrary positive integer to be specified later The essential point is to estimate the exponential sum in (5) Therefore let := q n 0 +a and χ(n) be the characteristic function of the sequence (q n ) n, ie, χ(n) =ifn is a member of the sequence (q n ) n and zero otherwise Then m x χ(n) =Q(x) and integration by parts yields E(n 0,N; q n ):= = N n=n 0 e πihf(qn) = e πihf(x) d Q(x) <m q N e πihf(m) χ(m) = Q(q N )e πihf(q N ) Q( )e πihf() ( L (x)+r (x) ) d e πihf(x), where R (x) :=Q(x) L (x), L (x) :=c x dt log t for c>0 constant, x and b a := Again by integration by parts, (a,b] and L (x)de πihf(x) = L (q N )e πihf(q N ) L ( )e πihf() c R (x)de πihf(x) =πih (log x) e πihf(x) dx R (x)f (x)e πihf(x) dx 97

6 MARTIN REHBERG Therefore, E(n 0,N; q n )= (R ) (q N )e πihf(qn) R ( )e πihf() + c πih (log x) e πihf(x) dx =: I + I + I 3, R (x)f (x)e πihf(x) dx where we used R (x) =Q(x) L (x)nowweestimateeachi i (i =,, 3) individually: In view of our assumption (), we get x R dt (x) =Q(x) c log t x (6) (log x) k for every k>, implying q N I (log q N ) +K (7) for K>0asN By estimation (6) and condition (a), I 3 h q Nf(q N ), (8) (log q N ) +K so it remains to estimate I First, let us remark that (8) for N is also an estimation for (7), since f>0andh is a positive integer To estimate I we apply Lemma 7 (a) to get { } 4 I max + x q n h(log x) f (x) hx(log x) f (x) + (log q N )( hf (q N )) q N (log q N ) ( hf (q N )) as N Note that for the last estimation condition (c) is needed in terms of nonincreasing Putting our estimation for I and (8) in (5) yields m + + N(log q N )( f (q N )) Nq N (log q N ) ( f (q N )) + mq Nf(q N ) N(log q N ) +K (9) for N After comparing the first and the last term in (9), we choose [ ( m = N (log q N) +K ) ] q N f(q N ) 98

7 LOG-LIKE FUNCTIONS ( Using this together with Q(q N )=N and Q(x) = cx log x + O x (log x) ), we end up with our desired estimation () P r o o f o f T h e o r e m 3 In respect of condition (b) we can assume that f (x) > 0 for sufficiently large x Further, we may assume that for x a we have f (x) > 0 and that (log x) f (x) is nonincreasing for x a Likeinthe proof of the previous theorem, we will show that the discrepancy D N of the sequence (f(q n )) N n=n 0 tends to zero as N Since the estimations (7) and (8) for I and I 3 still hold in the considered setting, it remains to estimate I By applying Lemma 7 (b) we get { } I max x q N (h(log x) f (x)) Combining this with (7) and (8) we obtain in (5) = (h(log q N ) f (q N )) m + mq Nf(q N ) N(log q N ) +, +K (qn f (q N )) ( where we used Q(q N )=N and Q(x) = cx log x + O x (log x) )Choosing we get [ ( m = N (log q N ) +K ) ], q N f(q N ) ( ) f(qn ) + (log q N ) K (qn f (q N )), which tends to zero as N in view of (b) and (d) P r o o f o f T h e o r e m 4 With respect to condition (b) we can assume that f (x) > 0 for sufficiently large x Further, we may assume that for x a we have f (x) > 0 and that (log x)f (x) is monotone for x a As in the previous proofs, we will show that the discrepancy D N of the sequence (f(q n )) N n=n 0 tends to zero as N Since the estimations (7) and (8) for I and I 3 still hold in the considered setting, it remains to estimate I Applying Lemma 6 yields I } {, h max [log q N f (q N )] 99

8 MARTIN REHBERG Thus, together with (7) and (8) in (5), we get { m +max N, N [log q N f (q N )] m +max { N, [q N f (q N )] } + mq Nf(q N ) N (log q N ) +K } + mq Nf(q N ) N (log q N ) +K, where we used in( the second argument in the maximum that Q(q N )=N and Q(x) = cx log x + O x (log x) ) After comparing the first and the last term in the discrepancy estimate, we choose [ ( m = N (log q N) +K ) ], q N f(q N ) and get ( ) { } f(qn ) +max (log q N ) K N, [q N f (q N )] By condition (b) and (d) this tends to zero as N tends to infinity Acknowledgements I would like to thank Prof J S c h o i ß e n g e i e r for his valuable corrections and comments REFERENCES [] GOTO, K KANO, T: Uniform Distribution of Some Special Sequences, ProcJapan Acad Ser A6 (985), no 3, [] Remarks to our Former Paper Uniform Distribution of Some Special Sequences, Proc Japan Acad Ser A 68 (99), no 0, [3] GROSSWALD, E SCHNITZER, FJ: A Class of modified ζ and L-Functions, Pacific Journal of Mathematics 74 (978), no, [4] KUIPERS, L NIEDERREITER, H: Uniform Distribution of Sequences Dover Publications, New York 006 [5] MÜLLER, H: Über eine Klasse modifizierter ζ- und L-Funktionen, Arch Math 36 (98), 57 6 [6] STRAUCH, O PORUBSKÝ, Š: Distribution of Sequences: A Sampler ebook, 06 [7] TOO, Y-H: On the Uniform Distribution Modulo One of Some Log-like Sequences, Proc Japan Acad Ser A 68 (99), no 9, 69 7 [8] PARENT, D P: Exercises in Number Theory Springer, New York

9 LOG-LIKE FUNCTIONS [9] TITCHMARSH, E C: The Theory of the Riemann Zeta-Function OxfordUniversity Press, second edition, New York 986 [0] WINTNER, A: On the cyclical distribution of the logarithms of the prime numbers, Quart J Math 6 (935), no, [] ZYGMUND, A: Trigonometric Series Cambridge University Press, third edition, London 00 Received October 0, 07 Accepted February, 08 Martin Rehberg Technische Hochschule Mittelhessen, Department MND Wilhelm-Leuschner-Str Friedberg GERMANY martinrehberg@mndthmde 0