QSpace Collection:http://hdl.handle.net/1974/61582016-06-28T22:19:32Z2016-06-28T22:19:32ZThe Chowla Problem and its generalizationsPathak, Siddhihttp://hdl.handle.net/1974/136142015-09-17T05:11:19Z2015-09-15T04:00:00ZTitle: The Chowla Problem and its generalizations
Authors: Pathak, Siddhi
Abstract: In this thesis, we study the vanishing of certain $L$-series attached to periodic arithmetical functions. Throughout the writeup, we let $f$ be an algebraic-valued (at times even rational-valued) arithmetical function, periodic with period $q$. Define
\begin{equation*}
L(s,f) := \sum_{n=1}^{\infty} \frac{f(n)}{n^s}.
\end{equation*}
We discuss a conjecture of Sarvadaman Chowla, made in the early 1960's of the non-vanishing of $L(1,f)$ for rational-valued functions $f$. We further discuss the theorem of Baker-Birch-Wirsing that classifies all odd algebraic-valued, periodic arithmetical functions $f$ with $L(1,f)=0$. We then apply a beautiful result of Bass to give a necessary condition for even algebraic-valued, periodic arithmetical functions $f$, to satisfy $L(1,f)=0$. The sufficiency condition obtained via this method is unfortunately not very clean. This, in view of a theorem of Ram Murty and Tapas Chatterjee, completes the characterization of all algebraic-valued periodic arithmetical functions $f$ with $L(1,f) = 0$.
In the next part of the thesis, we discuss the Lerch-zeta function and its functional equation. We also deduce the transcendence of its certain special values. Keeping this in mind, we define a new $L$-series attached to periodic functions and deduce a necessary condition for the vanishing of this $L$-series at $s=1$.
Finally, we mention some research topics that were stumbled upon during the course of this study and a few partial results regarding these questions.2015-09-15T04:00:00ZThe Saddle Point Method and its Applications to Number TheorySampath, Kannappanhttp://hdl.handle.net/1974/136132015-09-16T04:58:41Z2015-09-15T04:00:00ZTitle: The Saddle Point Method and its Applications to Number Theory
Authors: Sampath, Kannappan
Abstract: In this thesis, we study the classical procedures useful in obtaining asymptotic expansions of functions defined by integrals and their
applications to number theory. The final chapter in the thesis reports on a recent joint work with Ram Murty on the theme of using arithmetic formulas
to obtain asymptotic formulas to study the Fourier coefficients of the j-function and a related
sequence j-sub-m (m ≥ 0) of modular functions.
Description: Masters' Project Report2015-09-15T04:00:00ZA survey of some methods for computation of discrete logarithms in small characteristicde Valence, Henryhttp://hdl.handle.net/1974/136122015-09-17T05:03:56Z2015-09-15T04:00:00ZTitle: A survey of some methods for computation of discrete logarithms in small characteristic
Authors: de Valence, Henry2015-09-15T04:00:00ZAssessing the impact of exposure to bisphenol A and triclosan during early pregnancy on birth weight outcomes in Canadian studiesLester, Fionahttp://hdl.handle.net/1974/136112015-09-17T05:17:03Z2015-09-15T04:00:00ZTitle: Assessing the impact of exposure to bisphenol A and triclosan during early pregnancy on birth weight outcomes in Canadian studies
Authors: Lester, Fiona
Abstract: It is of interest to know whether early pregnancy exposure to chemicals such as bisphenol-A (BPA) and triclosan (TCS) has negative impacts on birth outcomes. These chemicals are rapidly excreted from the body and therefore using a single exposure measurement for each individual, as a proxy for their average exposure, can induce large amounts of measurement error. It is well known that this will generally attenuate any exposure-outcome relationships that may exist. In this work, we consider data from two recent Canadian studies to assess the effect of BPA and TCS on low birth weight (LBW), small for gestational age (SGA) and other birth outcomes. The majority of participants in the studies have one chemical exposure measurement, while a small subset have 3 to 20 repeated chemical exposure measurements. We estimate the effect of BPA and TCS by using two regression calibration methods to handle measurement error. One method divides naive estimates obtained from an uncorrected linear or logistic exposure-outcome model by an estimated intra-class correlation coefficient (ICC). The other method calculates the best linear unbiased predictor of average exposure and uses it in place of the observed exposure in linear and logistic exposure-outcome models. Specific gravity, time of day, and time since last urine void are investigated for their ability to explain some of the variation in observed exposure within a study participant. The results suggest decreased odds of LBW (uncorrected odds ratio = 0.85, 95% confidence interval: 0.75−0.98; corrected odds ratio = 0.81, 95% confidence interval: 0.68 − 0.97) and SGA (uncorrected odds ratio = 0.87, 95% confidence interval: 0.76 − 0.98; corrected odds ratio = 0.83, 95% confidence interval: 0.70 − 0.99) birth outcomes with increased logTCS exposure, but the relationship is not highly significant. Measurement error in BPA was so severe that regression calibration results were insignificant with wide confidence intervals. We also conducted a simulation study to investigate the performance of the estimation methods. The results suggest a few options for future studies to have more power to detect the effect of BPA and TCS on birth outcomes, such as more exposure measurements for each individual, or identifying covariates that explain more within-individual variation in exposure.2015-09-15T04:00:00Z