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Hirotaka Kobayashi

Publications and source records attributed to Hirotaka Kobayashi.

10 recordsLinked to original sources
arXiv2026

On Hardy's $Z$-function and its derivatives associated with the extended Selberg class

Hardy's $Z$-function $Z(t)$ is a real-valued function of the real variable $t$, and whose zeros correspond exactly to the zeros of the Riemann zeta-function on the critical line. In 2012, K. Matsuoka showed that for every non-negative integer $k$, there exists a $T=T(k)>0$ such that $Z^{(k+1)}(t)$ has exactly one zero between consecutive zeros of $Z^{(k)}(t)$ for $t\ge T$ under the Riemann Hypothesis. In this paper, we extend Matsuoka's theorem to $L$-functions in extended Selberg class.

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arXiv2024

Counting relatively prime pairs of palindromes

For a given base $g\ge2$, a positive integer is called a palindrome if its base $g$ expansion reads the same backwards as forwards. In this paper, we give an asymptotic formula for the number of relatively prime pairs of palindromes of a fixed odd length and of any base $g\ge2$, which solves an open problem proposed by Banks and Shparlinski (2005).

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arXiv2023

Mean-square values of the Riemann zeta function on arithmetic progressions

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when $a$ is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of $e^{πk/a}$ for positive integers $k$.

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arXiv2021

On a generalisation of the Riemann $ξ$-function

It is known that we can construct the meromorphic function $Z_k(s)$ associated with the higher derivative of Hardy's $Z$-function. In this paper, we introduce the entire function derived from $Z_k(s)$, a generalisation of the Riemann $ξ$-function and prove some properties.

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arXiv2020

Transcendence of values of the iterated exponential function at algebraic points

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \] converges to an algebraic number is approximated by $(e-1/e)φ(k)$. Here $φ(k)$ denotes Euler's totient function.

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arXiv2018

On the discrete mean of the derivative of Hardy's $Z$-function

Update: This result was obtained by Milinovich with a better error term. He used $ζ'(s)$, but we considered $Z'(t)$. We corrected a typo in the main theorem. We consider the sum of the square of the derivative of Hardy's $Z$-function over the zeros of Hardy's $Z$-function. If the Riemann Hypothesis is true, it is equal to the sum of $|ζ'(ρ)|^2$, where $ρ$ runs over the zeros of the Riemann zeta-function. In 1984, Gonek obtained an asymptotic formula for the sum. In this paper we prove a sharper formula.

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