Irreducibility of truncations of the Catalan generating function
We prove that, for every positive integer $n$, the polynomial $C_0+C_1x+\cdots+C_nx^n$ is irreducible over $\mathbb{Q}$, where $C_k$ denotes the $k$th Catalan number.
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arXiv subjects
Shin-ichiro Seki
Publications and source records attributed to Shin-ichiro Seki.
A note on Zlobin's note
We prove several results concerning the arithmetic nature of multiple zeta values related to Zlobin's 2006 note. Our proofs are based on Zagier's theorem and Lindemann's theorem.
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On the $\mathcal{A}$-transcendence of a Champernowne-type constant
Let $p_n$ denote the $n$-th prime. We prove that for every nonzero polynomial $f(x)\in\mathbb{Z}[x]$, there exist infinitely many positive integers $n$ such that $p_n\nmid f(n)$.
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Some results on naive transcendence in the ring of integers modulo infinitely large primes
This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal{A}$. In the ring $\mathcal{A}$, one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results of Anzawa-Funakura and Luca-Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as $\log_{\mathcal{A}}(2)$ under the ABC conjecture.
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Diamond lift of Hirose--Sato's formula involving the Hoffman basis
In this paper, we give a new proof of Hirose--Sato's formula for the expansion of $ζ(\{2\}^{a_1-1},3,\dots,\{2\}^{a_r-1},3,\{2\}^{c-1},1,\{2\}^{b_1},\dots, 1,\{2\}^{b_s})$ in the Hoffman basis, using the drop 1 relation.
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The $\mathbb{Z}$-module of multiple zeta values is generated by ones for indices without ones
We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $ζ(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified multiple harmonic sums that partially satisfy the relations among multiple zeta values and to determine the structure of the space generated by them.
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On the length over which $k$-Göbel sequences remain integers
We prove that the sequence $(N_k)_k$, where each $N_k$ is defined as the smallest positive integer $n$ for which the $n$th term $g_{k,n}$ of the $k$-Göbel sequence is not an integer, is unbounded.
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On finite analogues of Euler's constant
We introduce and study finite analogues of Euler's constant in the same setting as finite multiple zeta values. We define a couple of candidate values from the perspectives of a ``regularized value of $ζ(1)$'' and of Mascheroni's and Kluyver's series expressions of Euler's constant using Gregory coefficients. Moreover, we reveal that the differences between them always lie in the $\mathbb{Q}$-vector space spanned by 1 and values of a finite analogue of logarithm at positive integers.
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A note on non-integrality of the $(k,l)$-Göbel sequences
The $(k,l)$-Göbel sequences defined by Ibstedt remain integers for the first (in some cases, many) terms, but for selected values of $(k,l)$, computations show that the terms eventually stop being integers. It is still unresolved whether the integrality of these sequences breaks down for all $k, l\geq 2$. In this article, we prove the non-integrality for a specific class of $(k,l)$ values. Our proof is based on geometric arguments related to the distribution of quadratic residues modulo a prime.
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A proof of the extended double shuffle relation without using integrals
We present a new proof of the extended double shuffle relation for multiple zeta values which notably does not rely on the use of integrals. This proof is based on a formula recently obtained by Maesaka, Watanabe, and the author.
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A discretization of the iterated integral expression of the multiple polylogarithm
Recently, Maesaka, Watanabe, and the third author discovered a phenomenon where the iterated integral expressions of multiple zeta values become discretized. In this paper, we extend their result to the case of multiple polylogarithms and provide two proofs. The first proof uses the method of connected sums, while the second employs induction based on the difference equations that discrete multiple polylogarithms satisfy. We also investigate several applications of our main result.
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Regular primes, non-Wieferich primes, and finite multiple zeta values of level $N$
We introduce finite multiple zeta values of general level and discuss the relationship between the non-zeroness of these values and regular or non-Wieferich primes. Because it's challenging to prove the infinitude of these types of primes, we suggest tackling several related problems more promptly.
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Deriving two dualities simultaneously from a family of identities for multiple harmonic sums
We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values.
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Constellations in prime elements of number fields
Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.
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Multivariable connected sums and multiple polylogarithms
We introduce the multivariable connected sum which is a generalization of Seki-Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno's relations for multiple polylogarithms.
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A note on $\mathcal{F}_n$-multiple zeta values
For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid in $\mathcal{F}_n$-multiple zeta values for both the case where $\mathcal{F}=\mathcal{A}$ and $\mathcal{F}=\mathcal{S}$. In particular, the Bowman-Bradley type theorem and sum formulas for $\mathcal{S}_2$-multiple zeta values are proved.
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Truncated $t$-adic symmetric multiple zeta values and double shuffle relations
We study a refinement of the symmetric multiple zeta value, called the $t$-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the $t$-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding $t$-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for $t$-adic symmetric multiple zeta values, first proved by Jarossay. In order to prove the shuffle relation, we develop the theory of truncated $t$-adic symmetric multiple zeta values associated with $2$-colored rooted trees. Finally, we discuss a refinement of Kaneko-Zagier's conjecture and the $t$-adic symmetric multiple zeta values of Mordell-Tornheim type.
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Connectors
Recently, the author and Yamamoto invented a new proof of the duality for multiple zeta values. The technique is applicable in other series identities. In this article, we exhibit such proofs for some series identities.
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