An uncertainty principle for arithmetic sequences. (English) Zbl 1139.11040

In [Acta Arith. 9, 257–260 (1964; Zbl 0125.29601)], K. F. Roth established a general result on irregularities of distribution of sequences, and in [Mich. Math. J. 32, 221–225 (1985; Zbl 0569.10023)], H. Maier proved that there are small intervals containing significantly more primes than usual, and others containing significantly fewer primes than usual. Motivated by these papers and subsequent work by others on these problems, the present authors formulate a general framework for considering the irregularity of distribution of arithmetic sequences, and they apply their main results to a wealth of important and familiar examples, strengthening, improving or extending what is already known.
In describing this framework the authors show in Section 2 how to use a “matrix method” due to H. Maier to reduce the problems to one of exhibiting oscillations in the mean values of multiplicative functions. Other new technical results are established in Section 3, from which the main general results on irregularities of distribution are deduced in Section 4. Other applications are discussed in the remaining sections. The main theorems of this paper require too much notation to quote here. Instead we illustrate their power by picking out two of the consequential results of different types that are more straightforward to state.
We first turn to Theorem 1.5 of the paper. This concerns a version of the prime number theorem in small intervals. Let \(x\) be large and \(\log x\leq y\leq\exp({\beta\sqrt{\log x}\over 2\sqrt{\log\log x}})\) for a certain \(\beta> 0\). Let \(\theta(x)= \sum_{p\leq x}\log p\) and \(\Delta(x, y)={1\over y}\{\theta(x+ iy)- \theta(x)- y\}\). Then there exist numbers \(x_+\) and \(x_-\) in \((x,2x)\) such that \(\Delta(x_+, y)\geq y^{-\delta(x,y)}\), \(\Delta(x_-, y)\leq -y^{-\delta(x,y)}\), where
\[ \delta(x, y)= {1\over\log\log x} \Biggl\{\log\Biggl({\log y\over\log\log x}\Biggr)+ \log\log\Biggl({\log y\over\log\log x}\Biggr)+ O(1)\Biggr\}. \]
In particular if \(y= \exp((\log x)^\tau)\) with \(0< \tau<{1\over 2}\), then \(\delta(x, y)= \tau(1+ o(1))\); this improves a result of A. Hildebrand and H. Maier in which the range for \(\tau\) was \(0<\tau<{1\over 3}\).
Other results of this paper are given in terms of an “uncertainty principle” in the form that there is poor distribution in either arithmetic progressions with small moduli or in short intervals. We illustrate this concept with Corollary 6.3 of the paper. Let \(A\) be the set of positive integers not divisible by any prime in a set \(P\) with logarithmic density \(\alpha\in(0, 1)\), and let \(A(x)\) denote the number of elements of \(A\) that are \(\leq x\), and \(A(x; q,a)\) those elements that are also \(\equiv a\pmod q\). E. Wirsing [Acta Math. Acad. Sci. Hung. 18, 411–467 (1967; Zbl 0165.05901)] established an asymptotic formula for \(A(x)\). The expectation is that \(A(x;q,a)\sim {f_q(a)\over q\gamma_q} A(x)\), where \(\gamma_q= \prod_{p|q,p\in P} (1-{1\over p})\) and \(f_q(a)\) is a certain multiplicative function of \(a\) that is periodic \((\operatorname{mod}q)\), but there are restrictions on this as the next result shows. Let \(u= MN\) be a fixed sufficiently large number with both \(M\) and \(N\) at least 1.
Corollary 6.3 states that for each large \(x\) at least one of the following statements is true:
(i) There exists \(y\in({x\over 4},x)\) and an arithmetic progression \(a\pmod q\) with \(q\leq\exp((\log x)^{1/M})\) such that
\[ \biggl|A(y; q,a)- {f_q(a)\over q\gamma_q} A(y)\biggr| \gg_u{A(y)\over \varphi(q)}. \]
(ii) There exists \(y> {1\over 3}(\log x)^N\) and an interval \((v, v+ y)\subset({x\over 4}, x)\) such that
\[ \biggl|A(v+ y)- A(v)- y{A(v)\over v}\biggr| \gg_u y{A(v)\over v}. \]
The reader will discover cited in the paper many other different examples that illustrate the main theorems as well as extensions of previous results in the literature.


11N64 Other results on the distribution of values or the characterization of arithmetic functions
11K38 Irregularities of distribution, discrepancy
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