12,224 research outputs found
Abel Summation of Ramanujan-Fourier Series
Using Abel summation the paper proves a weak form of the Wiener-Khinchin
formula for arithmetic functions with point-wise convergent Ramanujan-Fourier
expansions. The main result is that the convolution of most arithmetic
functions possessing an R-F expansion are Abel-summable to a result involving
only the Ramanujan-Fourier coefficients of the R-F expansion(s).Comment: 13 pages. No Figures. Updated the definitions and made the theorem on
uniform convergence more explici
Power partitions and a generalized eta transformation property
In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number ps(n) of partitions of a positive integer~n into s-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function Ps(q)=∏∞n=1(1−qns)−1 near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case s=1. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for ps(n) of the same general type of the one that they gave for~s=1. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of p(200) the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for p2(100000) these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying size
1/16-BPS Black Holes and Giant Gravitons in the AdS_5 X S^5 Space
We explore 1/16-BPS objects of type IIB string theory in AdS_5 * S^5. First,
we consider supersymmetric AdS_5 black holes, which should be 1/16-BPS and have
a characteristic that not all physical charges are independent. We point out
that the Bekenstein-Hawking entropy of these black holes admits a remarkably
simple expression in terms of (dependent) physical charges, which suggests its
microscopic origin via certain Cardy or Hardy-Ramanujan formula. We also note
that there is an upper bound for the angular momenta given by the electric
charges. Second, we construct a class of 1/16-BPS giant graviton solutions in
AdS_5 * S^5 and explore their properties. The solutions are given by the
intersections of AdS_5 * S^5 and complex 3 dimensional holomorphic hyperspaces
in C^{1+5}, the latter being the zero loci of three holomorphic functions which
are homogeneous with suitable weights on coordinates. We investigate examples
of giant gravitons, including their degenerations to tensionless strings.Comment: 25 pages, no figures, v2: references added, comments added in the
conclusio
A formula for the number of solutions of a restricted linear congruence
summary:Consider the linear congruence equation for , . Let denote the generalized gcd of and which is the largest with dividing and simultaneously. Let be all positive divisors of . For each , define . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on . We generalize their result with generalized gcd restrictions on and prove that for the above linear congruence, the number of solutions is where for and denotes the generalized Ramanujan sum defined by E. Cohen (1955)
The Lerch-type zeta function of a recurrence sequence of arbitrary degree
We consider the series where
satisfies a linear recurrence of arbitrary degree with integer
coefficients. Under appropriate conditions, we prove that it can be continued
to a meromorphic function on the complex -plane. Thus we may associate a
Lerch-type zeta function to a general recurrence. This
subsumes all previous results which dealt only with the ordinary zeta and
Hurwitz cases and degrees and . Our method generalizes a formula of
Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the
poles and residues of , which turn out to be polynomials in . In
addition we study the dependence of on and
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