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Finite trigonometric sums arising from Ramanujan's theta functions
Bruce C. Berndt, Sun Kim, Alexandru Zaharescu
Two classes of finite trigonometric sums, each involving only 's, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and mo…
Exact evaluations and reciprocity theorems for finite trigonometric sums
Bruce C. Berndt, Sun Kim, Alexandru Zaharescu
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour…
Two-parameter Identities for Divisor Sums in Algebraic Number Fields
Bruce C. Berndt, Martino Fassina, Sun Kim +1
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor…
Balanced Derivatives, Identities, and Bounds for Trigonometric and Bessel Series
Bruce C. Berndt, Martino Fassina, Sun Kim +1
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier pap…
The distribution of spacings between the fractional parts of
Martino Fassina, Sun Kim, Alexandru Zaharescu
We study the distribution of spacings between the fractional parts of . For of high enough Diophantine type we prove a necessary and sufficient condition for $n^dα\mod 1,…
Sums of squares and products of Bessel functions
Bruce C. Berndt, Atul Dixit, Sun Kim +1
Let denote the number of representations of the positive integer as the sum of squares. We rigorously prove for the first time a Voronoi summation formula for $r_k…