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Hisanobu Shinya

Publications and source records attributed to Hisanobu Shinya.

7 recordsLinked to original sources
arXiv2026

On a new approach to the Riemann hypothesis

Suppose that the Riemann hypothesis is false and $ρ_{*} = 1/2 + η_{*} + i γ_{*}$, $η_{*} > 0$, is a nontrivial zero of the Riemann $ζ$-function off the critical line. Under the negation of the Riemann hypothesis for the Riemann $ζ$-function, we establish an asymptotic relation (as $γ_{*} \to \infty$) which relates the residues of the series $\sum_{n \geq 1} Λ(n) e^{- 2πi p n } n^{-s}$ at $s =$ corresponding nontrivial zeros of some Dirichlet $L$-functions to some function, valid for any rational number of the form $p = a/b < 1$ with $b \ll \log γ_{*}$. This related function is continuous in $p$ and we mention its implication to the Riemann hypothesis.

math.GM↗
arXiv2011

A note on gaps

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_ε(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + ε}$, where $ε> 0$ is arbitrary and fixed, and by $π(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\lim_{x \to \infty} N_ε(x)/π(x) = 1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.

math.GM↗
arXiv2011

On an arithmetical approach to the Riemann hypothesis

In the paper, we first prove a sufficient condition for the Riemann hypothesis which involves the order of magnitude of the partial sum of the Liouville function. Then we show a formula which is curiously related to the proved sufficient condition.

math.GM↗