10 research outputs found
Special values of derivatives of -series and generalized Stieltjes constants
The connection between derivatives of for periodic arithmetical
functions at and generalized Stieltjes constants has been noted
earlier. In this paper, we utilize this link to throw light on the arithmetic
nature of and certain Stieltjes constants. In particular, if is
an odd prime greater than , then we deduce the transcendence of at least
of the generalized Stieltjes constants, , conditional on a conjecture of S. Gun, M. R. Murty and P. Rath.Comment: 11 page
Rank stability of elliptic curves in certain metabelian extensions
Fix an odd prime number , a finite abelian group and an elliptic curve
. Let be a Galois extension with
. Suppose that has rank . Given a
positive integer , let and set
. In this paper, we show that there exist infinitely
many extensions such that and the rank of remains . We also prove asymptotic lower
bound for the number of such extensions, ordered by their absolute
discriminant. This extends a result of Mazur and Rubin, who study rank
stability in cyclic extensions of prime power order.Comment: Version 1: 25 page
Linear independence of values of the -exponential and related functions
In this paper, we derive results concerning the linear independence of values
of the -exponential and certain cognate functions at algebraic arguments
when the associated modulus is a Pisot-Vijayraghavan number. As a
consequence, we deduce the irrationality of special values of these functions,
some of which were known earlier using alternate methods