Ramanujan's identities, Voronoi summation formula, and zeros of partial sums of zeta and L-functions

Ramanujan;Voronoi summation formula;Divisor problem;Dedekind zeta function;Dirichlet polynomial;Distribution of zeros;Hecke L-functions;Approximate functional equation;Proportion of zeros on the critical line

Roy, Arindam

关键词: Ramanujan;    Voronoi summation formula;    Divisor problem;    Dedekind zeta function;    Dirichlet polynomial;    Distribution of zeros;    Hecke L-functions;    Approximate functional equation;    Proportion of zeros on the critical line;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/88129/ROY-DISSERTATION-2015.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The focus of the first part of the thesis commences with an examination oftwo pages in Ramanujan's lost notebook, pages 336 and 335.A casual, or even more prolonged, examination of the strange formulas on these pages does not lead one to conclude that they are related to one another.Moreover, it does not appear that they have any relationships with other parts of mathematics. On page 336 in his lost notebook, Ramanujan proposes two identities.Here, it does not take a reader long to make a deduction -- the formulas are obviously wrong-- each is vitiated by divergent series.Most readers encountering such obviously false claims would dismiss them and deposit the paper on which they were written in the nearest receptacle for recycling (if they were environmentally conscientious).However, these formulas were recorded by Ramanujan.Ramanujan made mistakes, but generally his mistakes were interesting!Frequently, there were hidden truths behind his not so precise or accurate claims -- truths that were deep and influential for decades.Thus, it was difficult for us to dismiss them. We initially concentrate on only one of the two incorrect ``identities.'" This ``identity'' may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. Our identity is admittedly quite complicated, and we do not claim that what we have found is what Ramanujan originally had in mind. But there are simple and interesting special cases as well as analogues of this identity, one of which very nearly resembles Ramanujan's version. The aforementioned convergent series in Ramanujan's faulty claim is similar to one used by Vorono\"{\dotlessi}, Hardy, and others in their study of the classical Dirichlet divisor problem, and so we are motivated to study further series of this sort. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions. Although again not obvious at a first inspection, one is conjoined with the classical circle problem initiated by Gauss, while the other is associated with the Dirichlet divisor problem.Berndt, Kim, and Zaharescu have written several papers providing proofs of these two difficult formulas in different interpretations.In this thesis, We return to these two formulas and examine them in more general settings.The Vorono\"{\dotlessi} summation formula appears prominently in our study.In particular, we generalize work of Wilton and derive an analogue involving the sum of divisors function $\sigma_s(n)$.Another part of the thesis is focused on the partial sums of Dedekind zeta functions and $L$-functions attached to cusp forms. The motivation of the study of the partial sums of Dedekind zeta functions and $L$-functions attached tocusp forms arise from their approximate functional equations. The partial sums of the Dedekind zeta function of a cyclotomic field $K$ is defined by the truncated Dirichlet series\[\zeta_{K, X} (s) = \sum_{ \|\mathfrak{a}\| \leqX } \frac{1}{\|\mathfrak{a}\|^{s}},\]where the sum is to be taken over nonzero integral ideals $\mathfrak{a}$ of $K$ and $\|\mathfrak{a}\|$ denotes the absolute norm of $\mathfrak{a}$. We establish the zero-free regions for $\zeta_{K, X} (s)$ and estimate the number of zeros of $\zeta_{K, X} (s)$ up to height $T$. We consider a family of approximations of a Hecke $L$-function $L_f(s)$ attached to a holomorphic cusp form $f$ of positive integral weight with respect to the full modular group. These families are of the form \begin{align}L_f(X;s):=\sum_{n\leq X}\frac{a(n)}{n^s}+\chi_f(s)\sum_{n\leq X}\frac{a(n)}{n^{1-s}},\end{align}where $s=\sigma+it$ is a complex variable. From the approximate functional equation one sees that $L_f(X;s)$ is a good approximation to $L_f(s)$ when $X=t/2\pi$. To investigate such approximation in more general sense, we compute the $L^2$-norms of the difference of two such approximations of $L_f(s)$. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of $L_f(s)$ can be obtained from such weighted $L^2$-norms. We also obtain a vertical strips where most of the zeros of $ L_f(X;s) $ lie. We study the distribution of zeros of $L_f(X;s)$ when $X$ is independent of $t$. For $X=1,2$ we prove that all the complex zeros of $L_f(X;s)$ lie on the critical line $\sigma=1/2$. We also show that as $T\to\infty$ and$ X=T^{o(1)} $, $100\%$ of the complex zeros of $ L_f(X;s) $ up to height $T$ lie on the critical line and simple. Here by $100\%$ we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height $T$ approaches 1 as $T\to\infty$.