Weighted Divisor Sums and Bessel Function Series
نویسندگان
چکیده
منابع مشابه
Weighted Divisor Sums and Bessel Function Series
On page 335 in his lost notebook, Ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary Bessel functions. We provide the first published proof of this result. The identity yields as corollaries representations of weighted divisor sums, in particular, the summatory function for r2(n), the number of representations of the positiv...
متن کاملWeighted Divisor Sums and Bessel Function Series, Ii
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 i...
متن کاملWeighted Divisor Sums and Bessel Function Series, Iv
Abstract. One fragment (page 335) published with Ramanujan’s lost notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second,...
متن کاملWeighted divisor sums and Bessel function series, V
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≤x r2(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≤x d(n) is the divisor ...
متن کاملWeighted Divisor Sums and Bessel Function Series, Iii
On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. The two identities are intimately connected with the classical circle and divisor problems, respectively. For each of Ramanujan’s identities, there are three possible interpretations for the double series. In two earlier papers, t...
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ژورنال
عنوان ژورنال: Mathematische Annalen
سال: 2006
ISSN: 0025-5831,1432-1807
DOI: 10.1007/s00208-005-0734-3