arXiv subjects

Takashi Miyagawa

Publications and source records attributed to Takashi Miyagawa.

10 recordsLinked to original sources

arXiv2026

Laurent series expansions for the Barnes multiple zeta function

We study the Laurent coefficients of the Barnes multiple zeta function with complex parameters in a common open half-plane and a fixed holomorphic determination of the logarithm. At the highest pole, we derive explicit limit formulae for every regular Laurent coefficient in terms of finite multiple sums and logarithmic correction terms. At the lower possible poles, we establish all-order relations with the Taylor coefficients at the origin and their parameter derivatives. We also determine the large-order behavior by subtracting all principal parts. The remaining function is entire, so Cauchy's estimate separates explicit residue contributions from a remainder that decays faster than any fixed geometric rate. The neighboring residues yield the limiting even and odd subsequences, with vanishing residues and cancellations accounted for. The constant-term cases recover known finite-part representations; the main focus is their extension to higher Laurent coefficients.

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arXiv2026

On the Laurent series expansions of the Barnes double zeta function

We investigate the Laurent series expansions of the Barnes double zeta function $ζ_2(s,α;v,w)$ at $s=1$ and $s=2$. We derive explicit limit representations for the Laurent coefficients, which may be regarded as analogues of the Euler--Stieltjes constants in the Barnes setting. In particular, we obtain formulas in terms of finite double sums with explicit correction terms, including a new limit representation for the constant term at $s=1$. We also establish relations between the Laurent coefficients at $s=1$ and the Taylor coefficients at $s=0$, as well as relations between the coefficients at $s=1$ and $s=2$. Furthermore, we study the asymptotic behavior of these coefficients as the order tends to infinity and show that they approach simple closed-form expressions with exponentially decaying error terms. Finally, we investigate arithmetic properties of certain constant terms in the Laurent expansions. We establish transcendence results for some infinite families of special values, including a family at $s=1$ for which every member is transcendental and a family at $s=2$ for which all but at most one are transcendental.

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arXiv2026

Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems

For the Lindelöf Hypothesis concerning the Riemann zeta function $ζ(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindelöf Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise upper bounds and integral mean values of zeta-functions. In this paper, we consider an analogue of the Lindelöf Hypothesis for the Barnes multiple zeta function $ζ_r (s,a,(w_1,\dots,w_r)) = \sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty (a+m_1 w_1+\cdots+m_r w_r)^{-s} $, and establish equivalent conditions in terms of integral mean values. In particular, the situation depends essentially on the $\Q$-rank of $\langle w_1,\dots,w_r\rangle$, and it is especially interesting that phenomena peculiar to the Barnes multiple zeta function appear according to this rank.

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arXiv2026

Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function

The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental properties such as the upper bounds, the distribution of zeros, and the zero-free regions in the Riemann zeta function derive from functional equations. In this paper, we consider the functional equations for the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Additionally, by applying this functional equation and the Phragmén-Lindelöf convexity principle, we obtain some upper bounds for $ ζ_2(σ+ it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.

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arXiv2026

On the mean values of the Barnes multiple zeta function

The asymptotic behavior of the mean values of multiple zeta functions is of significant interest due to its close connection with the Riemann zeta function. In this paper, we establish asymptotic behavior of the mean square values of Barnes multiple zeta functions.

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arXiv2026

Mean values and upper bounds for the Hurwitz and Barnes multiple zeta functions

Due to their deep connection with the Riemann zeta function, the asymptotic behavior of mean values of multiple zeta functions has attracted considerable attention. In this paper, we study the mean square values of Hurwitz-type and Barnes-type multiple zeta functions. For the Hurwitz-type multiple zeta function, we establish asymptotic formulas and upper bounds for its mean square values in terms of the parameter $σ$. Our approach relies on the fact that Hurwitz-type multiple zeta functions can be expressed as linear combinations of the classical Hurwitz zeta function, which allows us to apply known results on the mean values of the latter almost directly. For the Barnes-type multiple zeta function, we show that the behavior of the mean square values depends essentially on the arithmetic structure of the parameter vector. In the case where the parameters are linearly dependent over $\Q$, we obtain asymptotic formulas analogous to the Hurwitz-type case. In contrast, for general parameters, we derive upper bounds for the mean square values from bounds for the function itself. In particular, we clarify how the order of the mean square values varies in terms of the dimension of the $\Q$-vector space spanned by the parameters of the Barnes zeta function.

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arXiv2019

Approximate functional equation and upper bounds for the Barnes double zeta-function

As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation of the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Also, applying this approximate functional equation and the van der Corput method, we obtain upper bounds for $ ζ_2(1/2 + it, α; v, w) $ and $ ζ_2(3/2 + it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.

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arXiv2017

Mean values of the Barnes double zeta-function

In the study of order estimation of the Riemann zeta-function $ ζ(s) = \sum_{n=1}^\infty n^{-s} $, solving Lindelöf hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied. Furthermore, the theory of the mean values is also noted in the double zeta-functions, and the mean values of the Euler-Zagier type of double zeta-function and Mordell-Tornheim type of double zeta-function were studied. In this paper, we prove asymptotic formulas for mean square values of the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $ with respect to $ \mathrm{Im}(s) $ as $ \mathrm{Im}(s) \rightarrow + \infty $.

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arXiv2017

Approximate functional equations for the Hurwitz and Lerch zeta-functions

As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunkštis, A. Laurinčikas, and J. Steuding (in [1]) proved the Riemann-Siegel type of the approximate functional equation for the Lerch zeta-function $ ζ_L (s, α, λ) = \sum_{n=0}^\infty e^{2πi n λ}(n + α)^{-s} $. In this paper, we prove another type of approximate functional equations for the Hurwitz and Lerch zeta-functions. R. Garunkštis, A. Laurinčikas, and J. Steuding (in \cite{GLS2}) obtained the results on the mean square values of $ ζ_L (σ+ it, α, λ) $ with respect to $ t $. We obtain the main term of the mean square values of $ ζ_L (1/2 + it, α, λ) $ using a simpler method than their method in [2].

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arXiv2015

Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and L-functions

Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced.In this paper, similarly we consider generalized multiple zeta-functions and $ L $-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases.We prove the meromorphic continuation to the multi-dimensional complex space, and give the results on possible singularities.

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