Takashi Nakamura

activity

20102026

most citedA complete Riemann zeta distribution and the Riemann hypothesis

22 citations · 53 across the 28 of their papers we have counts for

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23 papers · 1 filter

math.NT2026

Double $L$ -values of Negative Weight Admitting Series Representations

Daiki Iju, Takashi Nakamura

In this paper, we construct double $L$ -values of negative weight that admit series representations. In our construction, we use the Thue-Morse sequence, which appears in combinator…

math.NT2023

A probabilistic interpretation for central zeros of $L$ -functions in the Selberg class

Takashi Nakamura, Masatoshi Suzuki

We show that central zeros of $L$ -functions in the Selberg class have a probabilistic interpretation by stating an equivalence condition of the Riemann hypothesis for the $L$ -funct…

math.NT2023

On infinitely divisible distributions related to the Riemann hypothesis

Takashi Nakamura, Masatoshi Suzuki

For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann…

math.NT2022

Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility

Takashi Nakamura

Let $a (1) > 0$ , $a (n) \geq 0$ for $n \geq 2$ and $a (n) = O (n^{ε})$ for any $ε > 0$ , and put $Z (σ + i t) := \sum_{n = 1}^{\infty} a (n) n^{- σ - i t}$ where $σ, t \in {\mathbb{R}…

math.NT2020

Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$ -functions

Takashi Nakamura

In this paper, we give Dirichlet series with periodic coefficients that have Riemann's functional equation and real zeros of Dirichlet $L$ -functions. The details are as follows. Le…

math.NT2020★ 5 cited

The values of zeta functions composed by the Hurwitz and periodic zeta functions at integers

Takashi Nakamura

For $s \in C$ and $0 < a < 1$ , let $ζ (s, a)$ and $Li_{s} (e^{2 π ia})$ be the Hurwitz and periodic zeta functions, repectively. For $0 < a \leq 1/2$ , put $Z(s,a) := ζ(s,…

math.NT2020

On a basis for Euler-Zagier double zeta functions with non-positive components

Hideki Murahara, Takashi Nakamura

For a non-negative integer $N$ , let $Z_{N} := \sum_{c = 0}^{N} Q \cdot ζ (- c, s + c)$ , where the right-hand side is the vector space spanned by the Euler-Zagier double z…

math.NT2020★ 1 cited

On Lerch's formula and zeros of the quadrilateral zeta function

Takashi Nakamura

Let $0 < a \leq 1/2$ and define the quadrilateral zeta function by $2 Q (s, a) := ζ (s, a) + ζ (s, 1 - a) + Li_{s} (e^{2 π ia}) + Li_{s} (e^{2 π i (1 - a)})$ , where $ζ (s, a)$ is the Hurwitz…

math.NT2019

On zeros of bilateral Hurwitz and periodic zeta and zeta star functions

Takashi Nakamura

In this paper, we show the following; (1) The periodic zeta function $Li_{s} (e^{2 π ia})$ with $0 < a < 1/2$ or $1/2 < a < 1$ does not vanish on the real line. (2) All real zeros of…

math.NT2019★ 3 cited

Functional equation and zeros on the critical line of the quadrilateral zeta function

Takashi Nakamura

For $0 < a \leq 1/2$ , we define the quadrilateral zeta function $Q (s, a)$ using the Hurwitz and periodic zeta functions and show that $Q (s, a)$ satisfies Riemann's functional equation…

math.NT2017

Zeros and the functional equation of the quadrilateral zeta function

Takashi Nakamura

In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z (s, a) := ζ (s, a) + ζ (s, 1 - a)$ with $1/4 \leq a \leq 1/2$ are on only the non-positive even integers ex…

math.NT2016

Zeros of polynomials of derivatives of zeta functions

Takashi Nakamura

Let $P_{s} \in D_{s} [X_{0}, X_{1}, \dots, X_{l}]$ be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane $\Re (s…

math.NT2015

Real zeros of Hurwitz-Lerch zeta functions in the interval $(- 1, 0)$

Takashi Nakamura

For $0 < a \leq 1$ , $s, z \in C$ and $0 < ∣ z ∣ \leq 1$ , the Hurwitz-Lerch zeta function is defined by $Φ (s, a, z) := \sum_{n = 0}^{\infty} z^{n} (n + a)^{- s}$ when $σ := ℜ (s) > 1$ . In t…

math.NT2015

Selberg's orthonormality conjecture and joint universality of L-functions

Yoonbok Lee, Takashi Nakamura, Łukasz Pańkowski

In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove join…

math.NT2015

Joint universality for Lerch zeta-functions

Yoonbok Lee, Takashi Nakamura, Łukasz Pańkowski

For $0 < α, λ \leq 1$ , the Lerch zeta-function is defined by $L (s; α, λ)$ $:= \sum_{n = 0}^{\infty} e^{2 π iλn} (n + α)^{- s}$ , where $σ > 1$ . In this paper, we prove joint universality for Lerch…

math.NT2015

Value distribution for the derivatives of the logarithm of $L$ -functions from the Selberg class in the half-plane of absolute convergence

Takashi Nakamura, Łukasz Pańkowski

In the present paper, we show that, for every $δ > 0$ , the function $(lo g L (s))^{(m)}$ , where $m \in N \cup {0}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty…

math.NT2014★ 3 cited

Real zeros of Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta functions

Takashi Nakamura

Let $0 < a \leq 1$ , $s, z \in C$ and $0 < ∣ z ∣ \leq 1$ . Then the Hurwitz-Lerch zeta function is defined by $Φ (s, a, z) := \sum_{n = 0}^{\infty} z^{n} (n + a)^{- s}$ when $σ := ℜ (s) > 1$ .…

math.NT2013

Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions

Takashi Nakamura

In this paper, we give Hurwitz zeta distributions with $0 < σ \neq = 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $ζ (σ, a)$ does not…

math.NT2012★ 8 cited

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

Takashi Nakamura, Łukasz Pańkowski

In this paper, we show that any polynomial of zeta or $L$ -functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant a…

math.NT2012

Corrigendum for "The generalized strong recurrence for non-zero rational parameters" Archiv der Mathematik 95 (2010), 549-555

Takashi Nakamura, Łukasz Pańkowski

In the present paper, we prove that self-approximation of $lo g ζ (s)$ with $d = 0$ is equivalent to the Riemann Hypothesis. Next, we show self-approximation of $lo g ζ (s)$ with respe…

math.NT2010★ 1 cited

Simple proof of the functional relation for the Lerch type Tornheim double zeta function

Takashi Nakamura

In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation…

math.NT2010★ 1 cited

The generalized strong recurrence for non-zero rational parameters

Takashi Nakamura

The strong recurrence is equivalent to the Riemann hypothesis. On the other hand, the generalized strong recurrence holds for any irrational number. In this paper, we show the gene…

math.NT2010

A simple proof of the generalized strong recurrence for any non-zero parameter

Takashi Nakamura

The strong recurrence is equivalent to the Riemann hypothesis. In the present paper, we give a simple proof of the generalized strong recurrence for all non-zero parameters.

Double L-values of Negative Weight Admitting Series Representations

A probabilistic interpretation for central zeros of L-functions in the Selberg class

On infinitely divisible distributions related to the Riemann hypothesis

Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility

Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet L-functions

The values of zeta functions composed by the Hurwitz and periodic zeta functions at integers

On a basis for Euler-Zagier double zeta functions with non-positive components

On Lerch's formula and zeros of the quadrilateral zeta function

On zeros of bilateral Hurwitz and periodic zeta and zeta star functions

Functional equation and zeros on the critical line of the quadrilateral zeta function

Zeros and the functional equation of the quadrilateral zeta function

Zeros of polynomials of derivatives of zeta functions

Real zeros of Hurwitz-Lerch zeta functions in the interval (−1,0)

Selberg's orthonormality conjecture and joint universality of L-functions

Joint universality for Lerch zeta-functions

Value distribution for the derivatives of the logarithm of L-functions from the Selberg class in the half-plane of absolute convergence

Real zeros of Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta functions

Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

Corrigendum for "The generalized strong recurrence for non-zero rational parameters" Archiv der Mathematik 95 (2010), 549-555

Simple proof of the functional relation for the Lerch type Tornheim double zeta function

The generalized strong recurrence for non-zero rational parameters

A simple proof of the generalized strong recurrence for any non-zero parameter

Double $L$ -values of Negative Weight Admitting Series Representations

A probabilistic interpretation for central zeros of $L$ -functions in the Selberg class

Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$ -functions

Real zeros of Hurwitz-Lerch zeta functions in the interval $(- 1, 0)$

Value distribution for the derivatives of the logarithm of $L$ -functions from the Selberg class in the half-plane of absolute convergence