Academic Journal
Asymptotic behavior of sums of multiplicative functions.
| Title: | Asymptotic behavior of sums of multiplicative functions. |
|---|---|
| Authors: | Levin, B. V., Faĭnleĭb, A. S. |
| Publisher Information: | Academy of Sciences of the Uzbek Soviet Socialist Republic - UzSSR (Akademiya Nauk UzSSR), Tashkent; ``FAN'', Tashkent |
| Subject Terms: | Asymptotic results on arithmetic functions, asymptotic behavior, sums of multiplicative functions, additive functions |
| Description: | The authors give an asymptotic formula for \(\displaystyle \sum_{n\le x} f(n)\), where \(f\) is a multiplicative function which satisfies some additional conditions. From this formula the following interesting corollaries are said to follow: 1) If \(g\) is an integer-valued additive function vanishing at primes, then for all \(k\) the number of \(n\le x\) with \(g(n) = k\) is asymptotically equal to \[ c_kx + O\left(x\exp\left(-(\log x)^{3/8 - \varepsilon}\right)\right), \] where \(c_k\) is a constant. 2) If \(g\) is a real additive function such that \[ \sum_{p\le x} \frac{e^{i\xi g(p)}}{p} \log p = \tau\left(e^{i\xi}\right)\log x + B\left(e^{i\xi}\right) + O\left(\exp\left\{-(\log x)^\alpha\right\}\right), \] \((\alpha>0\), \(\tau\left(e^{i\xi}\right)\) is of class \(C^3\) in the neighbourhood of \(\xi = 0\), \(a = \tau'(1) + \tau''(1)\) is positive) then \[ \lim_{n\to\infty} N\left\{m |
| Document Type: | Article |
| File Description: | application/xml |
| Access URL: | https://zbmath.org/3342070 |
| Accession Number: | edsair.c2b0b933574d..0e40e0d774f54a3c590616380b63ca83 |
| Database: | OpenAIRE |
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