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Title:  Partition asymptotics; zeros of zeta functions; and Apérylike numbers 
Author(s):  Malik, Amita 
Director of Research:  Berndt, Bruce C.; Zaharescu, Alexandru 
Doctoral Committee Chair(s):  Diamond, Harold G. 
Doctoral Committee Member(s):  Ford, Kevin 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Partitions
Arithmetic progressions Parity Asymptotics Zeros Riemann zeta function Proportion Apéry numbers 
Abstract:  PART I G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect kth powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan provided a simpler asymptotic formula in the case k = 2. In the first part of the thesis, we study the number of partitions into parts from a specific set Ak(a0; b0) :={mk : m 2 N;m _ a0 (mod b0)}, for fixed positive integers k, a0; and b0. Using the HardyLittlewood circle method, we give an asymptotic formula for the number of such partitions, thus generalizing the aforementioned results of Wright and Vaughan. We also consider the parity problem for such partitions and prove that the number of such partitions is even (odd) infinitely often, which generalizes O. Kolberg's theorem for the ordinary partition function. This material builds on the joint work with B. C. Berndt and A. Zaharescu. PART II The Riemann Hypothesis implies that the zeros of all the derivatives of the Riemann_ function lie on the critical line. Results on the proportion of zeros on the critical line of derivatives of _(s) have been investigated before by B. Conrey, and I. Rezvyakova. The percentage of zeros of _(k)(s) on the critical line approaches 100% percent as k increases. The second part of this thesis builds on the joint work with S. Chaubey, N. Robles, and A. Zaharescu. We study the zeros of combinations of derivatives of _(s). Although such combinations do not always have all their zeros on the critical line, we show that the proportion of zeros on the critical line still tends to 1. PART III The third part of this thesis focuses on the work on Apérylike numbers joint with Armin Straub. In 1982, Gessel showed that the Apéry numbers associated to the irrationality of _(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known sporadic Apérylike sequences. In several cases, we are able to employ approaches due to McIntosh, Samolvan Straten and RowlandYassawi to establish these congruences. However, for the sequences labeled s18 and (_) we require a finer analysis. As an application, we investigate modulo which numbers these sequences are periodic. In particular, we show that the AlmkvistZudilin numbers are periodic modulo 8, a special property which they share with the Apéry numbers. We also investigate primes which do not divide any term of a given Apéry like sequence. 
Issue Date:  20170713 
Type:  Text 
URI:  http://hdl.handle.net/2142/98291 
Rights Information:  Copyright 2017 Amita Malik 
Date Available in IDEALS:  20170929 
Date Deposited:  201708 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

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