We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that [Formula: see text] where [Formula: see text] and [Formula: see text] are Jacobi–Kronecker symbols. We prove our results using the theory of modular forms.

We find bases for the spaces M 2 Γ 0 (24), d · (d = 1, 8, 12, 24) of modular forms. We determine the Fourier coefficients of all 35 theta products ϕ[a 1 , a 2 , a 3 , a 4 ](z) in these spaces. We then deduce formulas for the number of representations of a positive integer n by diagonal quaternary quadratic forms with coefficients 1, 2, 3 or 6 in a uniform manner, of which 14 are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces E 2 Γ 0 (24), d · (d = 1, 8, 12, 24) and give their Fourier coefficients.

We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums a+pb=n σ(a)σ(b), p1a+p2b=n σ(a)σ(b) and a+p1p2b=n σ(a)σ(b) where p, p 1 , p 2 are primes.

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I have had a very fulfilling experience during my Ph.D. studies, both academically and personally. I would like to take this opportunity to thank the people who have contributed so much to this experience. I want to start by expressing my gratitude to the Turkish Ministry of Education for their generous financial support throughout my Ph.D. studies. I would also like to thank my supervisors, Dr. Ayşe Alaca and Dr. Şaban Alaca, for their academic guidance through the studies. Besides my supervisors, I would like to thank Dr. Kenneth S. Williams whose research has been a continuous inspiration for my studies. I am grateful to Dr. Shaun Cooper who, in our discussion during CNTA XIII, proposed to work on the extension of Ramanujan-Mordell formula proven in this thesis. I am also grateful to my dear friend, Dr. Fatih Deniz, for his encouragement and wise advices towards achieving my academic goals.