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Zbl 0559.10032
Elliott, P.D.T.A.
Arithmetic functions and integer products. (English)
[B] Grundlehren der Mathematischen Wissenschaften, 272. New York etc.: Springer-Verlag. XV, 461 p. DM 198.00 (1985).

This is a once usual, but now an all but extinct sort of publication: a research book. \par In 1946, Erd\H{o}s proved that an additive arithmetical function f must be a multiple of the logarithm if it is monotonic, or if $f(n+1)-f(n)\to 0$. Since then, lots of results appeared that asserted that a regular additive function (in various senses) must be equal or near to a multiple of the logarithm. E.g. Wirsing showed that if f is completely additive and $f(n+1)-f(n)=o(\log n),$ then $f(n)=c \log n;$ if $f(n+1)- f(n)=O((\log n)\sp c),$ where either $1\le c\le 6/5$ or $c\ge 3$, then $f(n)=O((\log n)\sp c).$ \par These results suggested that the proper approach would be to find a finite, effective result. The reviewer conjectured that with $\max\sb{n\le x}\vert f(n+1)-f(n)\vert =K$ always there is an L such that $\vert f(n)-L \log n\vert \le c\sb 1K$ for $n\le x\sp c$, with positive absolute constants $c,c\sb 1$. This would easily yield the above results, without the restriction on the exponent. Shortly after that, the author announced a result that was even stronger than that conjecture, in two directions: he used quadratic mean rather than the maximum, and considered more general differences of the form $f(an+b)-f(An+B).$ This is the main result of the book (Theorem 10.1). \par Let f be an additive function, a,b,A,B positive integers, $\Delta =aB+Ab\ne 0.$ Write $$ K=\max\sb{x\le y\le x\sp c}y\sp{-1}\sum\sb{x\le n\le y}\vert f(an+b)-f(An+B)\vert\sp 2. $$ With a suitable L we have $$ \sum\sb{p\sp k\le x, p \nmid aA}\vert f(p\sp k)-L \log p\sp k\vert\sp 2 p\sp{-k}\le CK, $$ where c,C depend on a,b,A,B, and L depends on f and x (which is specified in the book). This is an extremely general result: it easily implies practically everything that has so far been said on the diference of additive functions, and a lot more. The proof is very complicated (it occupies about the first 240 pages of the book), and I do not claim to have understood it completely. The main role seems to be played by the author's version of Bombieri's large sieve method. \par Now to the second part of the book, "Integer products". A seemingly natural way to attack the above problem would be the following. If $f(an+b)-f(An+B)$ is small on all or most numbers $n\le x$, then, by additivity, it is also small on numbers m representable in the form $m=((An\sb 1+B)/(an\sb 1+b))\sp{r\sb 1}...((An\sb k+B)/(an\sb k+b))\sp{r\sb k},$ with rational $r\sb j$. Now one could try to show that every integer (say up to $x\sp c)$ can be written in this form, with preferably small k and $r\sb j$. This does not work; but the author makes it work the other way, namely he is able to apply his result on additive functions to deduce the existence of such representations. E.g. (Theorem 17.1) let $R(x)=\prod\sp{h}\sb{i=1}(x+a\sb i)\sp{b\sb i}$ be a rational function with integers $a\sb i$ and $b\sb i$, and assume that the exponents $b\sb i$ are coprime. Then every positive integer n has a representation $n=\prod R(n\sb j)\sp{\epsilon\sb j}$, with $\epsilon\sb j=\pm 1$ and $n\sb j\le cn.$ \par An application to information theory is given in Chapter 20, and a central limit theorem for $f(n+1)-f(n)$ is deduced in Chapter 21. Chapter 23 supplies a collection of 108 exercises and 18 unsolved problems. Finally there is a supplement that is a sort of appendix to the author's previous monograph [Probabilistic number theory. I,II (1979; Zbl 0431.10029), (1980; Zbl 0431.10030)] briefly describing recent progress in this field. \par The book can be highly recommended to number theorists.
[I.Z.Ruzsa]

MSC 2000:: *11K65 Arithmetic functions (probabilistic number theory)
11-02 Research monographs (number theory)
11N37 Asymptotic results on arithmetic functions
00A07 Problem books
11B83 Special sequences of integers and polynomials

Keywords: multiplicative representation; quadratic mean; diference of additive functions; Bombieri's large sieve method; Integer products; application to information theory; central limit theorem; exercises; unsolved problems

Citations: Zbl 0431.10029; Zbl 0431.10030

Cited in: Zbl 1127.11063 Zbl 0929.11038 Zbl 0824.11056 Zbl 0621.10034 Zbl 0584.10026 Zbl 0581.10018 Zbl 0578.10013

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