Almost uniform distribution modulo 1 and the distribution of primes ^{∗}^{†}

Shigeki Akiyama

Abstract

Let (a_{n}), n = 1, 2,… be a sequence of real numbers which is related with number theoretic functions such as P_{n}, the n-th prime. We study the distribution of the fractional parts of (a_{n}) 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 of log P_{n}. The method may be extended as well to other oscillation problems of number theoretical interest.

Let (a_{n}), n = 1, 2,… be a sequence of real numbers and A(I, (a_{n}), N) be the counting function, that is, the number of n = 1, 2,…, N that {a_{n}} is contained in a certain interval I ⊂ [0, 1]. Here we denote by {a_{n}} = a_{n} − [a_{n}], the fractional part of a_{n}. First we recall a kind of generalization of the classical definition of uniform distribution modulo 1 (see [9],

[3] and [8]).

Definition. The sequence (a_{n}) is said to be almost uniformly distributed modulo 1 if there exist a strictly increasing sequence of natural numbers (n_{j}), j = 1, 2,… and, for every pair of a, b with 0 ≤ a < b ≤ 1,

lim_{j→∞} \frac{A([a, b), (a_{n}), n_{j})}{n_{j}} = b − a.

For example, we define (c_{n}) by

c_{n} = \frac{n}{2^{1+[log_{2} 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.

Then (c_{n}) is almost uniformly distributed modulo 1 but not uniformly distributed modulo 1. It is obvious that if the sequence (a_{n}) is uniformly distributed modulo 1, then almost uniformly distributed modulo 1. On the contrary, if

n_{j+1} − n_{j} = o(n_{j}),

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 (a_{n}), n = 1, 2,… is almost uniformly distributed modulo 1 if and only if there exist a strictly increasing sequence of natural numbers (n_{j}), j = 1, 2,…, such that for every real-valued continuous function on the interval [0, 1], we have

lim_{j→∞} \frac{1}{n_{j}} ∑_{i=1}^{n_{j}} f(\{a_{i}\}) = ∫_{0}^{1} f(x)dx.

Proposition 2. (Weyl’s Criterion for almost uniformly distributed modulo 1) The sequence (a_{n}), n = 1, 2,… is almost uniformly distributed modulo 1 if and only if there exist a strictly increasing sequence of natural numbers (n_{j}), j = 1, 2,…, such that for every integer h, we have

lim_{j→∞} \frac{1}{n_{j}} ∑_{i=1}^{n_{j}} exp(2πh\sqrt{−1}a_{i}) = 0.

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 Δf(n) = f(n + 1) − f(n). If the following three conditions is satisfied, then (f(n)) is almost uniformly distributed modulo 1:

1. There exists a natural number N that Δf(n) is monotone when n ≥ N (hereafter, we say this property as ultimately monotone),

2. lim_{n→∞} Δf(n) = 0,

3. lim sup_{n→∞} n|Δf(n)| = ∞.

Note that the corresponding third condition for uniformly distributed modulo 1 is:

lim_{n→∞} n|Δf(n)| = ∞.

Moreover, it is shown in [4] (see also [5]) that lim sup_{n→∞} n|Δf(n)| = ∞ is a necessary condition for uniformly distributed modulo 1. Concerning this fact, in [3], it is shown that (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 difference between 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) = \frac{1}{n} ∑_{k=1}^{n} g(k) is not almost uniformly distributed modulo 1,

(C3) lim sup_{n→∞} |f(n) − g(n + 1)| = ∞.

Then Δ^{2}f(n) changes its sign infinitely many times. Here Δ^{2}f(n) = Δ(Δf(n)).

Proof. We have

Δf(n) = \frac{1}{n + 1} ∑_{k=1}^{n+1} g(k) − \frac{1}{n} ∑_{k=1}^{n} g(k)
= \frac{1}{n + 1}g(n + 1) − \frac{1}{n(n + 1)} ∑_{k=1}^{n} g(k).  (1)

This shows that lim_{n→∞} Δf(n) = 0. And by (1),

(n + 1)Δf(n) = g(n + 1) − f(n).

Thus

lim sup_{n→∞} n|Δf(n)| = ∞.

If Δf(n) is ultimately monotone, then f(n) is almost uniformly distributed modulo 1, which contradicts the assumption. □

Let P_{n} be the n-th prime. Now we show

Theorem 2. (log P_{n}) is not almost uniformly distributed modulo 1.

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:

∑_{p≤N} p^{\sqrt{−1}t} = ∫_{3/2}^{N} x^{\sqrt{−1}t}dπ(x).

Integrating by parts of the right hand side, by using the prime number theorem of the form:

π(x) = \frac{x}{log x} + O(\frac{x}{log^{2} x}),

we have

∑_{p≤N} p^{\sqrt{−1}t} = \frac{N^{1+\sqrt{−1}t}}{log N} − \sqrt{−1}t ∫_{3/2}^{N} \frac{x^{\sqrt{−1}t}}{log x} dx + O(\frac{N}{log^{2} N})

= \frac{N^{1+\sqrt{−1}t}}{(1 + \sqrt{−1}t) log N} + O(\frac{N}{log^{2} N}).

Thus we see

\frac{1}{π(N)} ∑_{p≤N} e^{\sqrt{−1}t log p} ∼ \frac{N^{\sqrt{−1}t}}{1 + \sqrt{−1}t}

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

Now we give a very different proof of the results of [2].

Theorem 3. Δ^{2} log P_{n} changes its sign infinitely many times.

Proof. Let g(n) = n log P_{n} − (n − 1) log P_{n−1} and f(n) = log P_{n} in Corollary 1. (Here we put P_{0} = 1 for example.) By using Theorem 2, it suffice to show (C1) and (C3). By using prime number theorem, we have

g(n) ≤ n\frac{P_{n} − P_{n−1}}{P_{n−1}} + log P_{n−1},
= o(n).

For the condition (C3),

g(n + 1) − f(n) = (n + 1)(log P_{n+1} − log P_{n})
> \frac{(n + 1)(P_{n+1} − P_{n})}{P_{n+1}}
∼ \frac{P_{n+1} − P_{n}}{log P_{n}} .

Here we write f ∼ g if |f/g| → 1. P. Erdös [1] was the first to obtain

lim sup_{n→∞} \frac{P_{n+1} − P_{n}}{log P_{n}} = ∞,

by showing

lim sup_{n→∞} \frac{(P_{n+1} − P_{n})(log log log P_{n})^{2}}{log P_{n} log log P_{n} log log log log P_{n}} > c > 0.

About the improvement of the constant c, see [7]. This completes the proof. □

Our method to show this type of results can be generalized by a kind of ”linearity” in many cases. To explain this, we notice

Theorem 4. Let l be a fixed positive integer, and C_{i} (i = 1, 2,…, l) be the real numbers with ∑ C_{i} ≠ 0. The sequence (∑_{i=0}^{l−1} C_{i} log P_{n+i}) is not almost uniformly distributed modulo 1.

Proof. First, we consider the case (C log P_{n}). Without loss of generality, we may assume that C > 0. Then we write C log P_{n} = log_{b} P_{n} 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 that

∑_{i=0}^{l−1} C_{i} log P_{n+i} − log P_{n} ∑_{i=0}^{l−1} C_{i} = o(1).

This shows the assertion. □

Theorem 5. Let l be a fixed positive integer, and f_{i} (i = 1, 2,…, l) be the positive real numbers. Then

Δ^{2} log(P_{n}^{f_{1}}P_{n+1}^{f_{2}} … P_{n+l−1}^{f_{l}})

changes its sign infinitely many times.

Proof. Put

g(n) = n(∑_{i=1}^{l} f_{i} log P_{n+i−1}) − (n − 1)(∑_{i=1}^{l} f_{i} log P_{n+i−2})

f(n) = ∑_{i=1}^{l} f_{i} log P_{n+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 f_{i} (i = 1, 2,… l) in proving (C3). □

We expect that the conditions f_{i} > 0 (i = 1, 2,… l) can be dropped.

Our 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 for the divisor function d(n) = ∑_{d|n} 1 as

\frac{1}{n} ∑_{k=1}^{n} d(k) = log n + (2γ − 1) + O(\frac{1}{\sqrt{n}}),

with the Euler constant γ. The proof for this case is easier, but the results do not seem well 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 informed me 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 Q_{p}., 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 a_{n}x mod 1., Seminar on number theory, 1982-1983 (Talence), Exp. No.20 21pp.

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

Shigeki Akiyama

Faculty of Science Niigata University Niigata, 950-21 JAPAN

akiyama@math.sc.niigata-u.ac.jp