arXiv subjects

Ryutaro Sekigawa

Publications and source records attributed to Ryutaro Sekigawa.

3 recordsLinked to original sources

arXiv2025

A note on the second supplementary law of rational power residue symbols

As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a/p)_q$ is defined for primes $p$ and $q$ with $p\equiv 1 \bmod q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q/p)_q = 1$, when $p$ has a special form $p = \sum_{i=0}^{q-1} m^i n^{q-1-i}$. This condition is expressed in terms of the polylogarithm $\mathrm{Li}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer's proof of Euler's conjectures for cubic residue.

math.NT↗

arXiv2021

Bijective enumerations for symmetrized poly-Bernoulli polynomials

Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.

math.CO↗

arXiv2020

The characterization of cyclic cubic fields with power integral bases

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.

math.NT↗