TY - JOUR

T1 - Weighted divisor sums and Bessel function series, V

AU - Berndt, Bruce C.

AU - Kim, Sun

AU - Zaharescu, Alexandru

N1 - Funding Information:
The first author’s research was partially supported by NSA grant H98230-11-1-0200 .
Publisher Copyright:
© 2014 Elsevier Inc.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - Let r2(n) denote the number of representations of n as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ∑n≤xr2(n) is known as the circle problem. Next, let d(n) denote the number of positive divisors of n. Determining the exact order of magnitude of the error term associated with the asymptotic formula for ∑n≤xd(n) is the divisor problem. In his lost notebook, Ramanujan states without proof two identities that are associated with these two famous unsolved problems. It is natural to ask if identities exist for certain weighted sums, called Riesz sums, that generalize Ramanujan's identities. In this paper, we establish a Riesz sum identity that generalizes Ramanujan's identity linked to the divisor problem.

AB - Let r2(n) denote the number of representations of n as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ∑n≤xr2(n) is known as the circle problem. Next, let d(n) denote the number of positive divisors of n. Determining the exact order of magnitude of the error term associated with the asymptotic formula for ∑n≤xd(n) is the divisor problem. In his lost notebook, Ramanujan states without proof two identities that are associated with these two famous unsolved problems. It is natural to ask if identities exist for certain weighted sums, called Riesz sums, that generalize Ramanujan's identities. In this paper, we establish a Riesz sum identity that generalizes Ramanujan's identity linked to the divisor problem.

KW - Bessel functions

KW - Dirichlet L-series

KW - Dirichlet divisor problem

KW - Ramanujan's lost notebook

KW - Riesz sums

KW - Weighted divisor sums

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U2 - 10.1016/j.jat.2014.06.001

DO - 10.1016/j.jat.2014.06.001

M3 - Article

AN - SCOPUS:84930180744

VL - 197

SP - 101

EP - 114

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

ER -