Analytic Number Theory by Carl Pomerance & Michael Th. Rassias

Analytic Number Theory by Carl Pomerance & Michael Th. Rassias

Author:Carl Pomerance & Michael Th. Rassias
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham


(14)

and, more generally, that the exact counterpart of (14) holds for any d > 0, provided D and are replaced by d and , respectively. This provides, in particular, a converse (in the present one-dimensional case) to the error estimate obtained in [74] for the asymptotic second term of N ν (x) to be discussed next; see part (i) of Theorem 4.8 below, specialized to the case where N = 1 and D ∈ (0, 1).

Moreover, it is shown in [94] by means of an explicit counterexample that in the analog of (14) for the implications in one direction holds, but not in the other direction. For example, it is not true, in general, that .

Here, in Eq. (14), as well as in the sequel, given and one writes that f(x) = O(g(x)) to mean that there exists a positive constant c 2 such that | f(x) | ≤ c 2 g(x), for all x sufficiently large. (For the type of functions or sequences we will work with, we may assume that this inequality holds for all x > 0. ) We use the same classic Landau notation for sequences instead of for functions of a continuous variable.

In closing this section, we note that under mild assumptions on (about the growth of a suitable meromorphic continuation of its geometric zeta function ), it has since then been shown that within the theory of complex dimensions developed in [99–101], the characterization of Minkowski measurability obtained in [94] (Theorem 3.2 above) can be supplemented as follows (see [101], Theorem 8.15]), under appropriate hypotheses.

Theorem 3.4 (Characterization of Minkowski Measurability Revisited [94, 101]).

Let be a fractal string of Minkowski dimension D. Then, under suitable conditions on (specified in [101, Sect. 8.3]), the following statements are equivalent: (i) is Minkowski measurable.



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