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First Advisor

Naomi Tanabe


The main objects of our study are L-functions, which are meromorphic functions on the complex plane that analytically continue from the series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s}, where {a_n} is a sequence of complex numbers. In particular, we are interested in two families of L-functions: ''The Dirichlet L-functions" and ''the L-functions of cusp forms." The former refers to the L-functions whose a_n's are determined by Dirichlet characters, whereas cusp forms determine the latter. We begin our study with the celebrated Riemann zeta function, the simplest Dirichlet L-function, and discuss some of its well-known properties: the Euler product, analytic continuation, functional equation, Riemann hypothesis, and Euler's formula for its critical values. Then, we generalize our exploration to the Dirichlet L-functions and point out some analogous properties to those of the Riemann zeta function. Moreover, we present our original work on computing the critical values of the Dirichlet L-function associated with the primitive character mod 4, or what is known as the Dirichlet beta function. Lastly, we establish some knowledge of the theory of modular forms and cusp forms, which are nicely-behaved modular forms, and discuss some properties of the L-functions of cusp forms.


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