University of Illinois Urbana-Champaign

Zeros of P-Adic L-Functions and Densities Relating to Bernoulli Numbers

Sunseri, Richard Frank

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Description

Title
Zeros of P-Adic L-Functions and Densities Relating to Bernoulli Numbers
Author(s)
Sunseri, Richard Frank
Issue Date
1980
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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 p-2. For every irregular pair (p,i) with p < 125,000 it is known that the corresponding p-adic L-function L(,p)(s,(chi)(,i)) of Kubota and Leopoldt has a unique zero ('i)(kappa) in the p-adic integers. For every irregular pair (p,i) with p < 1000, several p-adic places of ('i)(kappa) are calculated using a p-adic 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 ('i-1)(omega) of corresponding p-adic 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 Staudt-Clausen Theorem which states that for every Bernoulli number B(,2n), the denominator Q(,2n) is the product of all primes p such that p-1 divides 2n. As a result of the calculation, it is shown that none of the densities is larger than (delta)(,2).
Graduation Semester
1980
Type of Resource
text
Permalink
http://hdl.handle.net/2142/68168

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