In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.
We fix a number field and study statistical properties of the ring as varies over algebraic numbers of a fixed degree . Given , we explicitly compute the density of for which and show that this does not depend on the number field . In particular, we show that the density of for which is . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define to be a certain finite subset of
We introduce the multi-Gaussian polynomial , a multi-partition analogue of the Gaussian polynomial (also known as -binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of . We also derive a Sylvester-type identity and its application.
We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.
Following a theorem of Hayes, we give a geometric interpretation of the special value at of certain -cocycle on previously introduced by the author. This work yields three main results: an explicit formula for our cocycle at , a generalization and a new proof of Hayes’ theorem, and an elegant summation formula for the zeroth coefficient of the Ehrhart quasi-polynomial of certain triangles in .
In this paper, we provide an algorithm to detect linear congruences of , the number of MacMahon’s -rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the -rowed plane partition functions. We also show that the number of partitions at most parts always admits linear congruences.
We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.
By the circle method, an asymptotic formula is established for the number of integer points on certain hypersurfaces within multiprojective space. Using Möbius inversion and the modified hyperbola method, we prove the Manin–Peyre conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for certain smooth hypersurfaces in the multiprojective space of sufficiently large dimension.
For , let denote the reciprocal sum up to of numbers with prime factors, counted with multiplicity. In prior work, the authors obtained estimates for , extending Mertens’ second theorem, as well as a finer-scale estimate for up to error for any . In this paper, we establish the limiting behavior of the higher Mertens constants from the estimate. We also extend these results to , and we comment on the general case .
Elementary irreducibility criteria are established for where is irreducible over and is a prime. For instance, our main criterion implies that if is reducible over , then divides modulo . Among several applications, it is shown that if has coefficients in , then is irreducible over excluding a couple of obvious exceptions. As another application, it is proved that if and
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