Title: A Tale of Two Theta Series

Abstract: A familiar concept from an undergraduate number theory course is the product representation formula for the sum of two squares. We generalize this notion by discussing identities which connect theta series associated to various binary quadratic forms. We then give a “new” identity which connects the theta series associated to a single binary quadratic form of discriminant Δ, to a theta series associated to a subset of binary quadratic forms of discriminant Δp^2. Lastly we will give an illustrative example to show how one can use this identity to derive a Lambert series decomposition in certain cases.

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**Ahmed Ghatasheh**,**Kevin Campbell**, and**Simon Harris**were chosen as the department's Outstanding PhD Student, Outstanding Master's Student, and Outstanding Undergraduate Student, respectively. Congratulations to all.