منابع مشابه
Restricted Divisor Sums
where the function f(n) is n, n or n + n + p+1 4 , where p ≡ 3 mod 4 is a rational prime, and where dα(n) = #{d : d|n and 1 ≤ d ≤ α} for real α ≥ 1. Motivation for considering these sums comes from an expression which is derived for the class number of a quadratic field with discriminant −p, in terms of a certain restricted divisor sum. This sum is currently too difficult to estimate, in that t...
متن کاملWeighted 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 کاملRestricted sums in a field
Let Zp = Z/pZ stand for the field of all residue classes modulo prime p. In 1964 P. Erdös and H. Heilbronn (cf. [EH] and [Gu]) conjectured that for each nonempty subset A of Zp there are at least min{p, 2|A| − 3} residue classes modulo p that can be written as the sum of two distinct elements of A. This had been open for thirty years until J. A. Dias da Silva and Y. O. Hamidoune ([DH]) proved t...
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 کاملSums of divisor functions in and matrix integrals
We study the mean square of sums of the kth divisor function dk(n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q → ∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of dk(n) in terms of a lattice point count...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2002
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa101-2-2