dc.contributor.author Jennings, Emily Nicole dc.date.accessioned 2014-09-25T04:30:16Z dc.date.available 2014-09-25T04:30:16Z dc.date.issued 2014-05 dc.identifier.other jennings_emily_n_201405_ma dc.identifier.uri http://purl.galileo.usg.edu/uga_etd/jennings_emily_n_201405_ma dc.identifier.uri http://hdl.handle.net/10724/30487 dc.description.abstract Let \$sigma(n)=sum_{d|n} d\$ denote the sum-of-divisors function. Davenport  showed \$n/sigma(n)\$ has a continuous distribution function. That is, \$D(u):=lim_{xtoinfty}frac{1}{x} sum_{substack{nleq x n/sigma(n) leq u}} 1\$ exists for all \$uin[0,1]\$ and is a continuous function of \$u\$. Jennings, Pollack, and Thompson  established an analogue of Davenport's theorem. They defined the analogous distribution function as \$tilde{D}_f(u):=lim_{xtoinfty}frac{1}{S(f;x)} sum_{substack{nleq x n/sigma(n) leq u}} f(n)\$, where \$S(f;x):=sum_{nleq x} f(n)\$. They showed that for a certain class of real-valued multiplicative functions \$f\$, \$tilde{D}_f(u)\$ exists for all \$uin[ 0,1]\$ and is both continuous and strictly increasing. In this paper, we further generalize the result by replacing the function \$n/sigma(n)\$ with certain other multiplicative functions \$g(n)\$. Hence we define \$tilde{D}_{f,g}(u):=lim_{xtoinfty}frac{1}{S(f;x)} sum_{substack{nleq x g(n) leq u}} f(n)\$. We show \$tilde{D}_{f,g}(u)\$ exists for all \$u\$ in \$[0,1]\$ and is a continuous function of \$u\$. Furthermore, if \$S:={ nin mathbb{N} : f(n) >0}\$, then \$tilde{D}_{f,g}(u)\$ is strictly increasing on the interior of the closure of \$g(S)\$. dc.language eng dc.publisher uga dc.rights public dc.subject distribution function dc.subject Wirsing’s theorem dc.subject Erdos--Wintner theorem dc.subject mean values dc.title On the existence of certain distribution functions dc.type Thesis dc.description.degree MA dc.description.department Mathematics dc.description.major Mathematics dc.description.advisor Paul Pollack dc.description.committee Paul Pollack dc.description.committee Jingzhi Tie dc.description.committee Angela Gibney
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