The Ramanujan Journal最新文献

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let (sigma _otext {mex}(n)) (resp., (sigma _etext {mex}(n))) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we find Hardy-Ramanujan type asymptotic formulae for both (sigma _otext {mex}(n)) and (sigma _etext {mex}(n)). We also prove some infinite families of congruences for (sigma _otext {mex}(n)) and (sigma _etext {mex}(n)) modulo 4 and 8

In the present article, we formulate a conjectural uniform error term in the Chebotarev–Sato–Tate distribution for abelian surfaces (mathbb {Q})-isogenous to a product of not (overline{mathbb {Q}})-isogenous non-CM-elliptic curves, established by the author in Amri (Eur J Math, 2023. https://doi.org/10.1007/s40879-023-00682-5, Theorem 1.1). As a consequence, we provide a conditional direct proof to the generalized Lang–Trotter conjecture recently formulated and studied in Chen et al. (Ramanujan J, 2022).

We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin’s continued fraction algorithm for arbitrary imaginary quadratic fields.

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments (M_{2p}^textrm{r}). The latter are expressed in terms of the ({}_3 F_2) hypergeometric functions, with a simplification to the ({}_2 F_1) hypergeometric function possible for (p=0) and (p=1), allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence ({ M_{2p}^textrm{r} }_{p=0}^infty ). Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.

Let (K= mathbb {Q}(sqrt{d})) be a real quadratic field with d having three distinct prime factors. We show that the 2-class group of each layer in the (mathbb {Z}_2)-extension of K is (mathbb {Z}/2mathbb {Z}) under certain elementary assumptions on the prime factors of d. In particular, it validates Greenberg’s conjecture on the vanishing of the Iwasawa (lambda )-invariant for a new family of infinitely many real quadratic fields.

We provide an upper bound for the effective irrationality exponents of cubic algebraics x with the minimal polynomial (x^3 - tx^2 - a). In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville, in the case (|t| > 19.71 a^{4/3}). Moreover, under the condition (|t| > 86.58 a^{4/3}), we provide an explicit lower bound for the expression ||qx|| for all large (qin mathbb {Z}). These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.

In this article, we study the solutions of certain type over a totally real number field K of the Diophantine equation (x^2= By^p+Cz^p) with prime exponent p, where B is an odd integer and C is either an odd integer or (C=2^r) for (r in mathbb {N}). Further, we study the non-trivial primitive solutions of the Diophantine equation (x^2= By^p+2^rz^p) ((rin {1,2,4,5})) (resp., (2x^2= By^p+2^rz^p) with (r in mathbb {N})) with prime exponent p, over K. We also present several purely local criteria of K

Let f and g be two distinct normalized primitive Hecke–Maass cusp forms of weight zero with Laplacian eigenvalues (frac{1}{4}+u^{2}) and (frac{1}{4}+v^{2}) for the full modular group (Gamma =SL(2,mathbb {Z})), respectively. Denote by (lambda _{f}(n)) and (lambda _{g}(n)) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the non-trivial upper bounds for the sum (sum _{nin S}|lambda _{f}(n)lambda _{g}(n)|), where S is a suitable subset of (mathbb {Z}^{+}cap [1,x]) with certain properties.

Recently a new class of continued fraction algorithms, the ((N,alpha ))-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each (Nin mathbb {N}), (Nge 2) and (alpha in (0,sqrt{N}-1]). Each of these continued fraction algorithms has only finitely many possible digits. These ((N,alpha ))-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. In this paper we will show that when all digits in the digit set are co-prime with N, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for (alpha ) small enough, all rationals have an eventually periodic expansion with period 1. This happens for all (alpha ) when (N=2). We also find infinitely many matching intervals for (N=2), as well as rationals that are not contained in any matching interval.