Mathematics
https://repository.iitgn.ac.in/handle/123456789/547
Mon, 24 Sep 2018 16:17:22 GMT2018-09-24T16:17:22ZSums of squares and products of Bessel functions
https://repository.iitgn.ac.in/handle/123456789/3900
Sums of squares and products of Bessel functions
Berndt, Bruce C.; Dixit, Atul; Kim, Sun; Zaharescu, Alexandru
Letrk(n) denote the number of representations of the positive integernas thesum ofksquares. We rigorously prove for the first time a Vorono ?? summation formula forrk(n), k?2,proved incorrectly by A. I. Popov and later rediscovered by A. P. Guinand,but without proof and without conditions on the functions associated in the transformation.Using this summation formula we establish a new transformation between a series consistingofrk(n) and a product of two Bessel functions, and a series involvingrk(n) and the Gaussianhypergeometric function. This transformation can be considered as a massive generalizationof well-known results of G. H. Hardy, and of A. L. Dixon and W. L. Ferrar, as well as ofa classical result of A. I. Popov that was completely forgotten. An analytic continuationof this transformation yields further useful results that generalize those obtained earlier byDixon and Ferrar.1.IntroductionInfinite series involving arithmetic functions and Bessel functions are instrumental in study-ing some notoriously difficult problems in analytic number theory, for example, the circle andthe divisor problems. As mentioned by G. H. Hardy [19, p. 266], S. Wigert [43] was the firstmathematician to recognize the importance of series of Bessel functions in analytic numbertheory. Since then, several mathematicians have studied, and continue to study, such series,for example, with the point of view of understanding and improving the order of magnitudeof error terms associated with the summatory functions of certain arithmetic functions. Aprime tool in making the connection between a summatory function and certain series ofBessel functions is the Vorono ?? summation formula associated with the corresponding arith-metic function.Letrk(n) denote the number of representations of a positive integernas the sum ofksquares, where different signs and different orders of the summands give distinct representa-tions. The ordinary Bessel functionJ?(z) of order?is defined by [42, p. 40]J?(z) :=?Xm=0(?1)m(z/2)2m+?m!?(m+ 1 +?),|z|<?.(1.1)We record the Vorono ?? summation formula associated withr2(n), sometimes known as theHardy�Landau summation formula, in the form given in [26, p.274] (or [13, Thm. A]).Theorem 1.1.If0?? < ?andh(y)is real and of bounded variation in(?,?), thenX??n??r2(n)12(h(n?0) +h(n+ 0)) =??Xn=0r2(n)Z??h(y)J0(2??ny)dy,(1.2)2010Mathematics Subject Classification.Primary: 11E25; Secondary: 33C10, 30B40.Key words and phrases.sums of squares, Bessel functions, Vorono ?? summation formula, analyticcontinuation.1
Thu, 01 Nov 2018 00:00:00 GMThttps://repository.iitgn.ac.in/handle/123456789/39002018-11-01T00:00:00ZModuli of filtered quiver representations
https://repository.iitgn.ac.in/handle/123456789/3864
Moduli of filtered quiver representations
Amrutiya, Sanjay; Dubey, Umesh
In this note, we give a construction of the moduli space of filtered repre- sentations of a given quiver of fixed dimension vector with the appropriate notion of stability. The construction of the moduli of filtered representation uses the moduli of representation of ladder quiver. The ladder quiver is introduced using the given quiver and a linear type quiver.
Wed, 01 Aug 2018 00:00:00 GMThttps://repository.iitgn.ac.in/handle/123456789/38642018-08-01T00:00:00ZUnboundedness of Betti numbers of curves
https://repository.iitgn.ac.in/handle/123456789/3869
Unboundedness of Betti numbers of curves
Mehta, Ranjana; Saha, Joydip; Sengupta, Indranath
Bresinsky defined a class of monomial curves in A4 with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for this class. We also propose a general construction of such curves in arbitrary embedding dimension.
Wed, 01 Aug 2018 00:00:00 GMThttps://repository.iitgn.ac.in/handle/123456789/38692018-08-01T00:00:00ZZeros of partial sums of L-functions
https://repository.iitgn.ac.in/handle/123456789/3849
Zeros of partial sums of L-functions
Vatwani, Akshaa; Roy, Arindam
We consider a certain class of multiplicative functions f:N→C. Let F(s)=∑∞n=1f(n)n−s be the associated Dirichlet series and FN(s)=∑n≤Nf(n)n−s be the truncated Dirichlet series. In this setting, we obtain new Hal\'asz-type results for the logarithmic mean value of f. More precisely, we prove estimates for the sum ∑xn=1f(n)/n in terms of the size of |F(1+1/logx)| and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums FN(s).
In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field K. More precisely, we give some improved results for the number of zeros up to height T as well as new zero density results for the number of zeros up to height T, lying to the right of R(s)=σ, where σ>1/2.
Sun, 01 Jul 2018 00:00:00 GMThttps://repository.iitgn.ac.in/handle/123456789/38492018-07-01T00:00:00Z