Partition asymptotics; zeros of zeta functions; and Apéry-like numbers

Malik, Amita

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Description

Title

Partition asymptotics; zeros of zeta functions; and Apéry-like numbers

Author(s)

Malik, Amita

Issue Date

2017-07-13

Director of Research (if dissertation) or Advisor (if thesis)

• Berndt, Bruce C.
• Zaharescu, Alexandru

Doctoral Committee Chair(s)

Diamond, Harold G.

Committee Member(s)

Ford, Kevin

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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 Hardy-Littlewood 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 Apéry-like 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 Apéry-like sequences. In several cases, we are able to employ approaches due to McIntosh, Samol-van Straten and Rowland-Yassawi 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 Almkvist-Zudilin 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.

Graduation Semester

2017-08

Type of Resource

text

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Copyright 2017 Amita Malik

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