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Title:  Zeros of PAdic LFunctions and Densities Relating to Bernoulli Numbers 
Author(s):  Sunseri, Richard Frank 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  A prime p is said to be irregular if it divides the numerator of the Bernoulli number B(,i) for some even integer i between 1 and p2. For every irregular pair (p,i) with p < 125,000 it is known that the corresponding padic Lfunction L(,p)(s,(chi)(,i)) of Kubota and Leopoldt has a unique zero ('i)(kappa) in the padic integers. For every irregular pair (p,i) with p < 1000, several padic places of ('i)(kappa) are calculated using a padic Newton's method and the formula for L(,p)(s,(chi)(,i)) derived by L. Washington. After the calculation the zeros ('i)(kappa) are transformed to zeros ('i1)(omega) of corresponding padic power series used in the study of cyclotomic fields and defined by Iwasawa. The Bernoulli numbers are also studied. Let Q(,2k) denote the denominator of the Bernoulli number B(,2k) in lowest terms, and let (delta)(,2k) denote the asymptotic density in the set of positive integers of the set of all n such that Q(,2n) = Q(,2k). It has been shown by Erdos and Wagstaff that (delta)(,2k) exists and is positive for every k. For k (LESSTHEQ) 41 specific upper and lower bounds are calculated for the (delta)(,2k), and also for general k upper and lower bounds for the (delta)(,2k) are calculated in terms of the k. The calculation makes use of the von StaudtClausen Theorem which states that for every Bernoulli number B(,2n), the denominator Q(,2n) is the product of all primes p such that p1 divides 2n. As a result of the calculation, it is shown that none of the densities is larger than (delta)(,2). 
Issue Date:  1980 
Type:  Text 
Language:  English 
Description:  210 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1980. 
URI:  http://hdl.handle.net/2142/68168 
Other Identifier(s):  (UMI)AAI8017995 
Date Available in IDEALS:  20141214 
Date Deposited:  1980 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois