Denote by the class number of the imaginary quadratic field with prime. It is well known that for . Recently, all the solutions of the Diophantine equation with were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation in unknown integers where , , ,
We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular -values for short intervals.
Serre showed that for any integer for almost all where is the Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study over the set of integers with many prime factors. Moreover, we show that any residue class can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of
In this paper, we establish the expected order of magnitude of the th-moment of central values of the family of Dirichlet -functions to a fixed prime modulus over function fields for all real .
We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers by , and for . We show that for a given , there is at most one such that is a near-square. With the exceptions of and , any such can be a nea
We investigate the subspace of the homology of a congruence subgroup of with coefficients in the Steinberg module which is spanned by certain modular symbols formed using the units of a totally real cubic field . By Borel–Serre duality, is isomorphic to . The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology . Their span is a naturally defined subspace of . Using a computer, we study where sits between
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 .
For positive integers , let be the truncated binomial expansion of consisting of all terms of degree It is conjectured that for , the polynomial is irreducible. We confirm this conjecture when Also we show for any and
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