On the Value Distribution of Arithmetic Functions

✍ Scribed by J.W Sander

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 331 KB
• Volume: 66
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2159

✦ Synopsis

In part II of a series of articles on the least common multiple, the central object of investigation was a particular integer-valued arithmetic function g 1 (n). The most interesting problem there was the value distribution of g 1 (n). We proved that the counting function card[n x: g 1 (n) d ] has order o d (x) for any fixed d. A characteristic feature of g 1 (n) is its so-called super-periodicity which will be discussed here. An integer-valued arithmetic function g(n) is called super-periodic, if there is a sequence (r j ) of positive integers with r j 2 ( j 2) such that, setting R k :=> k j=1 r j , g(rR k + j) g((r&1) R k + j) for all k 1, 1 r<r k+1 , and 1 j R k . In the present paper, we show that the above-mentioned property holds for a wide class of super-periodic functions, containing other interesting number-theoretical examples. The method is analytic and completely different from the one used in the earlier work.