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Tomokazu Onozuka

Publications and source records attributed to Tomokazu Onozuka.

26 records · Page 2Linked to original sources
arXiv2019

Bowman-Bradley type theorem for finite multiple zeta values in $\mathcal{A}_2$

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings $\{1,3,\ldots,1,3\}$ and $\{2,\ldots,2\}$ is a rational multiple of a power of $π^2$. Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring $\mathcal{A}_1$ vanish. In this paper, we partially lift Saito-Wakabayashi's theorem from $\mathcal{A}_1$ to $\mathcal{A}_2$. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in $\mathcal{A}_2$ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.

math.NT↗
arXiv2018

Zeros of a polynomial of $ζ^{(j)}(s)$

We give results on zeros of a polynomial of $ζ(s),ζ'(s),\ldots,ζ^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate the number of zeros whose imaginary part is in $(1,T)$. Finally, we study the distribution of the real part and the imaginary part of zeros, respectively.

math.NT↗
arXiv2017

The Graph Ramsey Number $R(F_\ell,K_6)$

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that \[ R(F_\ell,K_n)=2\ell(n-1)+1 \] for $\ell\ge n\ge3$, where $F_\ell$ is the join of $K_1$ and $\ell K_2$. In this paper, we prove that this conjecture is true for the case $n=6$.

math.CO↗
arXiv2016

On the $a$-points of the derivatives of the Riemann zeta function

We prove three results on the $a$-points of the derivatives of the Riemann zeta function. The first result is a formula of the Riemann-von Mangoldt type; we estimate the number of the $a$-points of the derivatives of the Riemann zeta function. The second result is on certain exponential sum involving $a$-points. The third result is an analogue of the zero density theorem. We count the $a$-points of the derivatives of the Riemann zeta function in $1/2-(\log\log T)^2/\log T<\Re s<1/2+(\log\log T)^2/\log T$.

math.NT↗
arXiv2016

Laurent series expansions of multiple zeta-functions of Euler-Zagier type at integer points

We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the Mellin-Barnes integral formula and the harmonic product formulas. The Mellin-Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence.

math.NT↗
arXiv2016

The multiple Dirichlet product and the multiple Dirichlet series

First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.

math.NT↗