TY - JOUR
T1 - Weighted divisor sums and Bessel function series, II
AU - Berndt, Bruce C.
AU - Kim, Sun
AU - Zaharescu, Alexandru
N1 - Funding Information:
* Corresponding author. E-mail addresses: berndt@illinois.edu (B.C. Berndt), kim.1674@math.ohio-state.edu (S. Kim), zaharesc@illinois.edu (A. Zaharescu). 1 The first author’s research was partially supported by NSA grant MDA904-00-1-0015. 2 The third author’s research was partially supported by NSF grant DMS-0901621.
PY - 2012/2/15
Y1 - 2012/2/15
N2 - On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.
AB - On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.
KW - Bessel functions
KW - Circle problem
KW - Divisor problem
KW - Fourier series
KW - Ramanujan's lost notebook
KW - Trigonometric sums
KW - Weighted divisor sums
UR - http://www.scopus.com/inward/record.url?scp=84855571287&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84855571287&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2011.10.016
DO - 10.1016/j.aim.2011.10.016
M3 - Article
AN - SCOPUS:84855571287
VL - 229
SP - 2055
EP - 2097
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
IS - 3
ER -