For every nonconstant rational function , the Galois groups of the dynatomic polynomials of encode various properties of are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as varies in a particular one-parameter family of maps, namely, the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.
We study the relations of multiple -values of general level. The generating function of sums of multiple -(star) values of level with fixed weight, depth and height is represented by the generalized hypergeometric function , which generalizes the results for multiple zeta(-star) values and multiple -(star) values. As applications, we obtain formulas for the generating functions of sums of multiple -(star) values of level with height one and maximal height and a weighted sum formula for sums of multiple -(star) values of level with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman’s restricted sum formulas for multiple -(star) values of level . Some evaluations of multiple -star values of level with one–two–three indices are given.
We study the problem of determining, given an integer , the rational solutions to . For , the curve has genus and its Jacobian is isogenous to the product of three elliptic curves , , . We explicitly determine the rational points on under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all .
This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers such that and are all sums of two squares where and are two arbitrary integers, and as an immediate corollary obtain, in parametric terms, three consecutive integers that are sums of two squares. Similarly we obtain in parametric terms such that all the four integers are sums of two squares. We also find infinitely many integers such that all the five integers are sums of two squares, and finally, we find infinitely many arithmetic progressions, with common difference , of five integers all of which are sums of two squares.
Finding the Frobenius number and the genus of any numerical semigroup is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.
Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer partition classes with modulus . We also extend separable integer partition classes with modulus to overpartitions, called separable overpartition classes. We study overpartitions and the overpartition analogue of Rogers–Ramanujan identities, which are separable overpartition classes.
By assuming Vinogradov–Korobov-type zero-free regions and the generalized Ramanujan–Petersson conjecture, we establish nontrivial upper bounds for almost all short sums of Fourier coefficients of Hecke–Maass cusp forms for . As applications, we obtain nontrivial upper bounds for the averages of shifted sums involving coefficients of the Hecke–Maass cusp forms for . Furthermore, we present a conditional result regarding sign changes of these coefficients.
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