Almost uniform distribution modulo 1 and thedistribution of primes ∗†Shigeki Akiyama
Abstract
Let (an ), n = 1, 2, . . . be a sequence of real numbers which is related with number
theoretic functions such as Pn , the n-th prime. We study the distribution of thefractional parts of (an ) using the concept of ”almost uniform distribution” defined in[9]. Then we can show a generalization of the results of [2] on the convex property oflog Pn . The method may be extended as well to other oscillation problems of number
theoretical interest.

Let (an ), n = 1, 2, . . . be a sequence of real numbers and A(I, (an ), N ) be the counting
function, that is, the number of n = 1, 2, . . . , N that {an } is contained in a certain intervalI ⊂ [0, 1]. Here we denote by {an } = an − [an ], the fractional part of an . First we recall akind of generalization of the classical definition of uniform distribution modulo 1 (see [9],[3] and [8]).
Definition. The sequence (an ) is said to be almost uniformly distributed modulo 1 if
there exist a strictly increasing sequence of natural numbers (nj ), j = 1, 2, . . . and, forevery pair of a, b with 0 ≤ a < b ≤ 1,A([a, b), (an ), nj )= b − a.
j→∞njlim

For example, we define (cn ) bycn =n
21+[log2 n]∗.

11K06, 11N05 uniform distribution, distribution of primes
Only for the private copy. See Acta Math. Hungar., Vol. 78, (1–2), (1998), 39 – 44, for the exact
published version.
†1Then (cn ) is almost uniformly distributed modulo 1 but not uniformly distributed modulo
1. It is obvious that if the sequence (an ) is uniformly distributed modulo 1, then almost
uniformly distributed modulo 1. On the contrary, if
nj+1 − nj = o(nj ),then almost uniformly distributed modulo 1 implies uniformly distributed modulo 1. Using
the classical method of uniform distribution theory (see e.g. [6]), we can show the following
Proposition 1. The sequence (an ), n = 1, 2, . . . is almost uniformly distributed modulo
1 if and only if there exist a strictly increasing sequence of natural numbers (nj ), j =1, 2, . . ., such that for every real-valued continuous function on the interval [0, 1], we have
Z 1j1 X
limf ({ai }) =f (x)dx.
j→∞ nj0i=1
nProposition 2. (Weyl’s Criterion for almost uniformly distributed modulo 1)
The sequence (an ), n = 1, 2, . . . is almost uniformly distributed modulo 1 if and only ifthere exist a strictly increasing sequence of natural numbers (nj ), j = 1, 2, . . ., such thatfor every integer h, we havenj√1 X
limexp(2πh −1ai ) = 0.
j→∞ nji=1

We should mention the next generalization of Fejér’s Theorem.
Theorem 1. (Fejér’s Theorem for almost uniformly distributed modulo 1) Let(f (n)), n = 1, 2, . . . be a sequence of real numbers and ∆f (n) = f (n + 1) − f (n). If thefollowing three conditions is satisfied, then (f (n)) is almost uniformly distributed modulo
1:
1. There exists a natural number N that ∆f (n) is monotone when n ≥ N (hereafter,we say this property as ultimately monotone),
2. lim ∆f (n) = 0,n→∞3. lim sup n|∆f (n)| = ∞.
n→∞2Note that the corresponding third condition for uniformly distributed modulo 1 is:lim n|∆f (n)| = ∞.

n→∞Moreover, it is shown in [4] (see also [5]) that lim supn→∞ n|∆f (n)| = ∞ is a necessary
condition for uniformly distributed modulo 1. Concerning this fact, in [3], it is shownthat (log n) is not almost uniformly distributed modulo 1 but almost uniformly distributed
modulo 1 in the ”average” sense. It is an interesting problem to study this delicate differencebetween uniformly distributed modulo 1 and almost uniformly distributed modulo 1. We
can show the following:
Corollary 1. Let (g(n)) be a sequence of real numbers which satisfies three conditions:(C1) g(n) = o(n),(C2) The average f (n) = n1Pn

k=1 g(k) is not almost uniformly distributed modulo 1,(C3) lim sup |f (n) − g(n + 1)| = ∞.
n→∞Then ∆2 f (n) changes its sign infinitely many times. Here ∆2 f (n) = ∆(∆f (n)).
Proof. We have∆f (n) ==nX1 n+11Xg(k) −g(k)n + 1 k=1
n k=1nX
11g(n + 1) −g(k).
n+1n(n + 1) k=1(1)

This shows that lim ∆f (n) = 0. And by (1),n→∞(n + 1)∆f (n) = g(n + 1) − f (n).
Thuslim sup n|∆f (n)| = ∞.
n→∞

If ∆f (n) is ultimately monotone, then f (n) is almost uniformly distributed modulo 1,
which contradicts the assumption.
2
Let Pn be the n-th prime. Now we show
Theorem 2. (log Pn ) is not almost uniformly distributed modulo 1.
3

Proof. Let t be a real number and π(x) be the number of primes less than or equal to x.
Consider the sum over primes p:
X √Z Np≤N

3/2

p −1t =√

x −1t dπ(x).

Integrating by parts of the right hand side, by using the prime number theorem of theform:xxπ(x) =+ O( 2 ),
log x
log x
we haveX √p −1tp≤N

√

√

Z N
N 1+ −1t √x −1t
N=− −1t
dx + O( 2 )log N
3/2 log xlog N√N 1+ −1tN√=+ O( 2 ).
log N(1 + −1t) log NThus we see√1 X √−1t log pN −1t√e∼π(N ) p≤N1 + −1t

The right hand side is not zero. By using Proposition 2, we get the result.

2

Now we give a very different proof of the results of [2].
Theorem 3. ∆2 log Pn changes its sign infinitely many times.
Proof. Let g(n) = n log Pn − (n − 1) log Pn−1 and f (n) = log Pn in Corollary 1. (Here weput P0 = 1 for example.) By using Theorem 2, it suffice to show (C1) and (C3). By using
prime number theorem, we havePn − Pn−1+ log Pn−1 ,Pn−1= o(n).

g(n) ≤ n

For the condition (C3),g(n + 1) − f (n) = (n + 1)(log Pn+1 − log Pn )(n + 1)(Pn+1 − Pn )>Pn+1
Pn+1 − Pn∼.
log Pn4Here we write f ∼ g if |f /g| → 1. P. Erdös [1] was the first to obtainlim supn→∞

Pn+1 − Pn= ∞,log Pnby showing(Pn+1 − Pn )(log log log Pn )2> c > 0.
n→∞ log Pn log log Pn log log log log PnAbout the improvement of the constant c, see [7]. This completes the proof.
lim sup2

Our method to show this type of results can be generalized by a kind of ”linearity” inmany cases. To explain this, we notice
Theorem 4. Let l be a fixed positive integer, and Ci (i = 1, 2, . . . , l) be the real numbers
P
Pwith Ci 6= 0. The sequence ( l−1i=0 Ci log Pn+i ) is not almost uniformly distributed modulo
1.
Proof. First, we consider the case (C log Pn ). Without loss of generality, we may assumethat C > 0. Then we write C log Pn = logb Pn with a constant b > 1. To see the assertion,replace e with b in the proof of Theorem 2.
If l > 1, it suffice to note thatl−1
XCi log Pn+i − log Pn

i=0l−1X

Ci = o(1).

i=0This shows the assertion.

2Theorem 5. Let l be a fixed positive integer, and fi (i = 1, 2, . . . , l) be the positive real
numbers. Thenf2
fl∆2 log(Pnf1 Pn+1. . . Pn+l−1)
changes its sign infinitely many times.
Proof. PutlXg(n) = n(lXfi log Pn+i−1 ) − (n − 1)(

i=1f (n) =lXfi log Pn+i−2 )

i=1

fi log Pn+i−1 .

i=1

By using Corollary 1 and Theorem 4, in a similar manner as in the proof of Theorem 3,
we obtain the assertion. Here, we essentially used the positiveness of fi (i = 1, 2, . . . l) in
proving (C3).
25

We expect that the conditions fi > 0 (i = 1, 2, . . . l) can be dropped.
POur method is applicable to a lot of arithmetic functions g(n) such that 1/n k≤n g(k)is not almost uniformly distributed modulo 1. For example, we can show similar assertions
Pfor the divisor function d(n) = d|n 1 asn11Xd(k) = log n + (2γ − 1) + O( √ ),n k=1nwith the Euler constant γ. The proof for this case is easier, but the results do not seemwell worthy of stating here.
Acknowledgement. The author would like to thank the referee for a considerable
simplification of the proof of Theorem 2. The author is very grateful to Prof. T. Kano for
carefully reading the original manuscript and giving helpful comments. He also informedme that the sophisticated generalization of Theorem 2 was found recently by Prof. K.
Goto.

References[1] P. Erdös, On the difference of consecutive primes., Quart. J. Math. (Oxford) 6, (1935)
124-128.
[2] P. Erdös and P. Turán, On some new question on the distribution of prime numbers.,
Bull. Amer. Math. Soc. 54, (1948) 371-378.
[3] J. Chauvineau, Sur la répartition dans R et dans Qp ., Acta Arith. 14, (1968) 225-313[4] T. Kano, Criteria for uniform and weighted uniform distribution mod 1., Comment.
Math. Univ. St. Pauli 20, (1971) 83-91.
[5] P. B. Kennedy, A note on uniformly distributed sequences., Quart. J. Math. (2)7,(1959) 125-127.
[6] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. , Pure and Applied
Math., John Wiley & Sons, (1974)[7] R. A. Rankin, The difference between consecutive prime numbers. V., Proc. Edinburgh
Math. Soc. (2) 13 (1962/63) 331-332.
[8] I. Z., Ruzsa, On the uniform and almost uniform distribution of an x mod 1., Seminaron number theory, 1982-1983 (Talence), Exp. No.20 21pp.
6

[9] I. I. Šapiro-Pyateckiı̆, On a generalization of the notion of uniform distribution offractional parts., Mat. Sbornik N.S. 30, (1952) 669-676.

Shigeki Akiyama
Faculty of Science
Niigata University
Niigata, 950-21
JAPANakiyama@math.sc.niigata-u.ac.jp

7