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Journal articles on the topic 'Zeta'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 July 2024

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1

Лауринчикас, Антанас, and Antanas Laurinčikas. "О совместной универсальности дзета-функций Римана и Гурвица." Matematicheskie Zametki 111, no. 4 (2022): 551–60. http://dx.doi.org/10.4213/mzm13259.

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В 2007 г. Г. Мишу доказал теорему универсальности о совместном приближении пары аналитических функций сдвигами $(\zeta(s+i\tau),\zeta(s+i\tau,\alpha))$ дзета-функции Римана и дзета-функции Гурвица с трансцендентным параметром $\alpha$. В статье получена аналогичная теорема о приближении сдвигами $$ (\zeta_{u_N}(s+ikh_1),\zeta_{u_N}(s+ikh_2,\alpha)),\qquad k\in\mathbb{N}\cup\{0\},\quad h_1,h_2>0, $$ где $\zeta_{u_N}(s)$ и $\zeta_{u_N}(s,\alpha)$ - абсолютно сходящиеся ряды Дирихле и при $N\to\infty$ в среднем стремятся к $\zeta(s)$ и $\zeta(s,\alpha)$ соответственно. Библиография: 11 названий.

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2

Зудилин, Вадим Валентинович, and Wadim Valentinovich Zudilin. "Одно из чисел $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ иррационально." Uspekhi Matematicheskikh Nauk 56, no. 4 (2001): 149–50. http://dx.doi.org/10.4213/rm427.

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3

Зудилин, Вадим Валентинович, and Wadim Valentinovich Zudilin. "Одно из восьми чисел $\zeta(5),\zeta(7),…,\zeta(17),\zeta(19)$ иррационально." Matematicheskie Zametki 70, no. 3 (2001): 472–76. http://dx.doi.org/10.4213/mzm759.

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4

Balčiūnas, Aidas, Mindaugas Jasas, and Audronė Rimkevičienė. "A DISCRETE VERSION OF THE MISHOU THEOREM RELATED TO PERIODIC ZETA-FUNCTIONS." Mathematical Modelling and Analysis 29, no. 2 (March 26, 2024): 331–46. http://dx.doi.org/10.3846/mma.2024.19502.

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In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts $\zeta_{u_N}(s+ikh_1; \ga)$ and $\zeta_{u_N}(s+ikh_2, \alpha; \gb)$ of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence $\ga$, and the periodic Hurwitz zeta-function, respectively. We suppose that $u_N\to\infty$ and $u_N\ll N^2$ as $N\to\infty$, and the set $\{(h_1\log p:\! p\in\! \PP), (h_2\log(m+\alpha): m\in \NN_0), 2\pi\}$ is linearly independent over $\QQ$.

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5

Lambert, Joseph B., Lourdes A. Salvador, and Charlotte L. Stern. "The .zeta.(Zeta) effect of tin." Journal of Organic Chemistry 58, no. 20 (September 1993): 5428–33. http://dx.doi.org/10.1021/jo00072a027.

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6

Hernández-Eliseo, Yessica, Josué Ramírez-Ortega, and Francisco G. Hernández-Zamora. "Toeplitz operators on two poly-Bergman-type spaces of the Siegel domain $ D_2 \subset \mathbb{C}^2 $ with continuous nilpotent symbols." AIMS Mathematics 9, no. 3 (2024): 5269–93. http://dx.doi.org/10.3934/math.2024255.

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<abstract><p>We studied Toeplitz operators acting on certain poly-Bergman-type spaces of the Siegel domain $ D_{2} \subset \mathbb{C}^{2} $. Using continuous nilpotent symbols, we described the $ C^* $-algebras generated by such Toeplitz operators. Bounded measurable functions of the form $ \tilde{c}(\zeta) = c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2}) $ are called nilpotent symbols. In this work, we considered symbols of the form $ \tilde{a}(\zeta) = a(\text{Im}\, \zeta_1) $ and $ \tilde{b}(\zeta) = b(\text{Im}\, \zeta_2 -|\zeta_1|^{2}) $, where both limits $ \lim\limits_{s\rightarrow 0^+} b(s) $ and $ \lim\limits_{s\rightarrow +\infty} b(s) $ exist, and $ a $ belongs to the set of piece-wise continuous functions on $ \overline{\mathbb{R}} = [-\infty, +\infty] $ and with one-sided limits at $ 0 $. We described certain $ C^* $-algebras generated by such Toeplitz operators that turned out to be isomorphic to subalgebras of $ M_n(\mathbb{C}) \otimes C(\overline{\Pi}) $, where $ \overline{\Pi} = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+ $ and $ \overline{\mathbb{R}}_+ = [0, +\infty] $.</p></abstract>

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7

Lagarias, Jeffrey C., and Wen-Ching Winnie Li. "The Lerch zeta function I. Zeta integrals." Forum Mathematicum 24, no. 1 (January 2012): 1–48. http://dx.doi.org/10.1515/form.2011.047.

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8

Kurokawa, Nobushige, Masato Wakayama, and Yoshinori Yamasaki. "Milnor–Selberg zeta functions and zeta regularizations." Journal of Geometry and Physics 64 (February 2013): 120–45. http://dx.doi.org/10.1016/j.geomphys.2012.10.015.

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9

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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10

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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11

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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12

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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13

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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14

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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15

Lai, Li, and Li Zhou. "At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational." Publicationes Mathematicae Debrecen 101, no. 3-4 (October 1, 2022): 353–72. http://dx.doi.org/10.5486/pmd.2022.9252.

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16

Beasley, Bruce. "Zeta Hercules." Antioch Review 49, no. 2 (1991): 252. http://dx.doi.org/10.2307/4612368.

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17

Kurokawa, Nobushige, and Masato Wakayama. "Zeta extensions." Proceedings of the Japan Academy, Series A, Mathematical Sciences 78, no. 7 (September 2002): 126–30. http://dx.doi.org/10.3792/pjaa.78.126.

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18

Kuwata, Kazuo. "Protein Zeta." Seibutsu Butsuri 43, supplement (2003): S75. http://dx.doi.org/10.2142/biophys.43.s75_1.

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19

Bostock, David. "METAPHYSICS ZETA." Classical Review 52, no. 2 (September 2002): 258–59. http://dx.doi.org/10.1093/cr/52.2.258.

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20

Harrington, Robert S., Geoffery G. Douglass, and Charles E. Worley. "Zeta Cancri." International Astronomical Union Colloquium 135 (1992): 321–22. http://dx.doi.org/10.1017/s0252921100006679.

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AbstractZeta Cancri is a multiple system in which the well-observed orbit is an inner, or close one. Three-body coupling effects are an order of magnitude more significant for the close orbit in a hierarchical system, and they must be included. An exact numerical integration is very easy and requires no approximations. Inclusion of these effects improves the residuals for this system.

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21

Lewis, Richard. "Zeta malfunction." New Scientist 198, no. 2654 (May 2008): 20. http://dx.doi.org/10.1016/s0262-4079(08)61085-4.

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22

Kurokawa, Nobushige, and Masato Wakayama. "Zeta Regularizations." Acta Applicandae Mathematicae 81, no. 1 (March 2004): 147–66. http://dx.doi.org/10.1023/b:acap.0000024207.37694.3b.

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23

Ohno, Yasuo, and Wadim Zudilin. "Zeta stars." Communications in Number Theory and Physics 2, no. 2 (2008): 325–47. http://dx.doi.org/10.4310/cntp.2008.v2.n2.a2.

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24

Yıldırım, C. Yalçın. "A note on $\zeta ”(s)$ and $\zeta ”’(s)$." Proceedings of the American Mathematical Society 124, no. 8 (August 1, 1996): 2311–14. http://dx.doi.org/10.1090/s0002-9939-96-03755-0.

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25

Ihara, Kentaro, Jun Kajikawa, Yasuo Ohno, and Jun-ichi Okuda. "Multiple zeta values vs. multiple zeta-star values." Journal of Algebra 332, no. 1 (April 2011): 187–208. http://dx.doi.org/10.1016/j.jalgebra.2010.12.029.

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26

Yamamoto, Shuji. "Interpolation of multiple zeta and zeta-star values." Journal of Algebra 385 (July 2013): 102–14. http://dx.doi.org/10.1016/j.jalgebra.2013.03.023.

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27

KELLIHER, JAMES P., and RIAD MASRI. "Analytic continuation of multiple Hurwitz zeta functions." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 3 (November 2008): 605–17. http://dx.doi.org/10.1017/s0305004107001028.

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AbstractWe use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to $\mathbb{C}^{d}$, to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of $\zeta_{d}(s;\theta)$.

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28

He, Yuan, and Zhuoyu Chen. "Some Weighted Sum Formulas for Multiple Zeta, Hurwitz Zeta, and Alternating Multiple Zeta Values." Journal of Mathematics 2021 (May 13, 2021): 1–16. http://dx.doi.org/10.1155/2021/6672532.

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We perform a further investigation for the multiple zeta values and their variations and generalizations in this paper. By making use of the method of the generating functions and some connections between the higher-order trigonometric functions and the Lerch zeta function, we explicitly evaluate some weighted sums of the multiple zeta, Hurwitz zeta, and alternating multiple zeta values in terms of the Bernoulli and Euler polynomials and numbers. It turns out that various known results are deduced as special cases.

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29

Dyall, Kenneth G. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the actinides Ac–Lr." Theoretical Chemistry Accounts 117, no. 4 (October 11, 2006): 491–500. http://dx.doi.org/10.1007/s00214-006-0175-4.

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30

Gomes, André S. P., Kenneth G. Dyall, and Lucas Visscher. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the lanthanides La–Lu." Theoretical Chemistry Accounts 127, no. 4 (January 23, 2010): 369–81. http://dx.doi.org/10.1007/s00214-009-0725-7.

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31

Chan, V., TK Chan, ST Liang, A. Ghosh, YW Kan, and D. Todd. "Hydrops fetalis due to an unusual form of Hb H disease." Blood 66, no. 1 (July 1, 1985): 224–28. http://dx.doi.org/10.1182/blood.v66.1.224.224.

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Abstract The occurrence of Hb H hydrops fetalis is reported for the first time. The mother has zeta-alpha thalassemia 1 (zeta zeta alpha alpha/----) and the father has non-deletion alpha thalassemia [zeta zeta alpha alpha/zeta zeta (alpha alpha)T]. The complete deletion of the zeta alpha cluster on one chromosome was confirmed by quantitation of alpha and zeta gene numbers, the normal alpha and zeta gene patterns arising from the remaining normal chromosome, and the decreased alpha/beta globin chain ratio of 0.57. The non-deletion alpha thalassemia defect could only be identified by the imbalanced alpha/beta globin chain ratio of 0.65 in the presence of normal gene numbers and patterns. The newborn was markedly anemic, unlike those with classical Hb H disease, because the non-deletion alpha thalassemia defect is more severe than alpha thalassemia 2. The decreased zeta genes during fetal life might have additional deleterious effects. In this family, the distinct BamHI restriction fragment length polymorphism in the hypervariable region of the zeta genes may be used for future prenatal diagnosis.

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32

Chan, V., TK Chan, ST Liang, A. Ghosh, YW Kan, and D. Todd. "Hydrops fetalis due to an unusual form of Hb H disease." Blood 66, no. 1 (July 1, 1985): 224–28. http://dx.doi.org/10.1182/blood.v66.1.224.bloodjournal661224.

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The occurrence of Hb H hydrops fetalis is reported for the first time. The mother has zeta-alpha thalassemia 1 (zeta zeta alpha alpha/----) and the father has non-deletion alpha thalassemia [zeta zeta alpha alpha/zeta zeta (alpha alpha)T]. The complete deletion of the zeta alpha cluster on one chromosome was confirmed by quantitation of alpha and zeta gene numbers, the normal alpha and zeta gene patterns arising from the remaining normal chromosome, and the decreased alpha/beta globin chain ratio of 0.57. The non-deletion alpha thalassemia defect could only be identified by the imbalanced alpha/beta globin chain ratio of 0.65 in the presence of normal gene numbers and patterns. The newborn was markedly anemic, unlike those with classical Hb H disease, because the non-deletion alpha thalassemia defect is more severe than alpha thalassemia 2. The decreased zeta genes during fetal life might have additional deleterious effects. In this family, the distinct BamHI restriction fragment length polymorphism in the hypervariable region of the zeta genes may be used for future prenatal diagnosis.

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33

TANAKA, Hidekazu. "GAMMA FACTORS OF ZETA FUNCTIONS AS ABSOLUTE ZETA FUNCTIONS." Kyushu Journal of Mathematics 74, no. 2 (2020): 441–49. http://dx.doi.org/10.2206/kyushujm.74.441.

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34

Komori, Yasushi, Kohji Matsumoto, and Hirofumi Tsumura. "Multiple zeta values and zeta-functions of root systems." Proceedings of the Japan Academy, Series A, Mathematical Sciences 87, no. 6 (June 2011): 103–7. http://dx.doi.org/10.3792/pjaa.87.103.

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35

Watanabe, Tsuguturo. "Efficient Computation of the Plasma Dipersion Function .ZETA.(.ZETA.)." Kakuyūgō kenkyū 65, no. 5 (1991): 556–80. http://dx.doi.org/10.1585/jspf1958.65.556.

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36

Weng, Lin, and Don Zagier. "Higher-rank zeta functions andSLn-zeta functions for curves." Proceedings of the National Academy of Sciences 117, no. 12 (March 9, 2020): 6398–408. http://dx.doi.org/10.1073/pnas.1912501117.

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In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

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37

Sarnak, P. "Quantum Chaos, Symmetry, and Zeta functions, II: Zeta Functions." Current Developments in Mathematics 1997, no. 1 (1997): 145–59. http://dx.doi.org/10.4310/cdm.1997.v1997.n1.a4.

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38

Sasaki, Yoshitaka. "Zeta Mahler measures, multiple zeta values and L-values." International Journal of Number Theory 11, no. 07 (October 21, 2015): 2239–46. http://dx.doi.org/10.1142/s1793042115501006.

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The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

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39

Andrea, J. E., and M. P. Walsh. "Identification of a brain-specific protein kinase Cζ pseudogene (ΨPKCζ) transcript." Biochemical Journal 310, no. 3 (September 15, 1995): 835–43. http://dx.doi.org/10.1042/bj3100835.

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Protein kinase C (PKC), a widely-distributed enzyme implicated in the regulation of many physiological processes, consists of a family of at least twelve isoenzymes which differ in tissue distribution, subcellular localization, regulatory properties, etc. In addition to this heterogeneity at the protein level, we identify here for the first time a PKC zeta pseudogene (psi PKC zeta) transcript, specifically expressed in the brain, which is identical with PKC zeta except for sequence divergence within the first variable domain (V1). The authenticity of this unique V1 sequence (V1′) in mRNA was confirmed by RNase protection and reverse transcriptase PCR (RT-PCR) analysis. When translated in-frame with PKC zeta, a stop codon is located 28 amino acids towards the N-terminus of the divergence point and the intervening sequence lacks an expected initiating methionine. psi PKC zeta is non-functional in terms of protein synthesis since Western blotting with an antibody directed against the C-terminus of PKC zeta failed to reveal a protein smaller than PKC zeta, and synthetic psi PKC zeta RNA failed to support protein synthesis in a translation system in vitro. PCR amplification of rat genomic DNA demonstrated lack of an intron at the junction between V1′ and the first constant domain (the V1′-C1 border), and genomic DNA Southern blot analysis using PKC zeta and psi PKC zeta-specific probes indicated that they have different loci. psi PKC zeta, therefore, is not derived from the PKC zeta gene by alternative splicing, but rather is the product of a distinct gene. In Northern blot analysis, brain PKC zeta mRNA was identified as a low-abundance 3.1 kb transcript, while the abundant 2.5 and 4.7 kb mRNAs previously reported to encode PKC zeta are, in fact, psi PKC zeta transcripts. Analysis of rat brain, heart, lung, liver, kidney and skeletal muscle revealed psi PKC zeta mRNA only in brain. PKC zeta transcripts were most abundant in lung and kidney (2.7 and 4.7 kb mRNAs), correlating with the tissue profile of PKC zeta immunoreactivity in Western blots. Probes complementary to the common V5 and C1 domains detected both PKC zeta and psi PKC zeta transcripts. Interestingly, the C1 probe also detected an abundant novel 1.75 kb mRNA in brain and heart, suggesting the existence of an additional PKC zeta-related species. This work, therefore, also emphasizes the importance of careful choice of oligonucleotide and cDNA probes to study PKC zeta mRNA.

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40

Tekrony, Amy, and David Cramb. "Determination of the mobility of amine- and carboxy-terminated fluospheres and quantum dots by capillary electrophoresis." Canadian Journal of Chemistry 94, no. 4 (April 2016): 430–35. http://dx.doi.org/10.1139/cjc-2015-0349.

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The pharmacokinetics of nanoparticle (NP) theranostics can, in principle, be predicted based on NP size and zeta potential. Zeta potentials are typically measured using bench top zetasizer instruments, which calculate zeta potential based on mobility data collected from solutions in a small sample cell. However, correlations between zeta potentials measured by zetasizer instruments and those calculated from mobilities determined by instruments designed for capillary electrophoresis may not be direct. To that end, mobilities of a variety of NPs were determined by a capillary electrophoresis and used to calculate zeta potentials based on Henry’s equation. The calculated zeta potentials were then compared to zeta potentials measured directly from a zetasizer. It was found that absolute values of the two methods differed, but the relative zeta potential trends per particle type were similar. These trends were demonstrated by data that showed that the zeta potentials measured using a zetasizer correlated highly with zeta potentials determined by capillary electrophoresis.

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41

Dyall, Kenneth G. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 5d elements Hf?Hg." Theoretical Chemistry Accounts 112, no. 5-6 (November 16, 2004): 403–9. http://dx.doi.org/10.1007/s00214-004-0607-y.

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42

Dyall. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 5d elements Hf–Hg." Theoretical Chemistry Accounts 112, no. 5-6 (2004): 403. http://dx.doi.org/10.1007/s00214-004-607-y.

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43

Dyall, Kenneth G. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 4d elements Y–Cd." Theoretical Chemistry Accounts 117, no. 4 (October 12, 2006): 483–89. http://dx.doi.org/10.1007/s00214-006-0174-5.

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44

Dyall, Kenneth G. "Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 6d elements Rf–Cn." Theoretical Chemistry Accounts 129, no. 3-5 (February 24, 2011): 603–13. http://dx.doi.org/10.1007/s00214-011-0906-z.

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45

Shores, E., V. Flamand, T. Tran, A. Grinberg, J. P. Kinet, and P. E. Love. "Fc epsilonRI gamma can support T cell development and function in mice lacking endogenous TCR zeta-chain." Journal of Immunology 159, no. 1 (July 1, 1997): 222–30. http://dx.doi.org/10.4049/jimmunol.159.1.222.

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Abstract Fc epsilonRI gamma (Fc gamma) is a member of the zeta family of signal transducing molecules that function as components of both the TCR and Fc receptors (FcR). While the majority of thymocytes and T cells express TCRs containing zeta-chain homodimers, certain unique populations of T cells express TCRs that contain both zeta and Fc gamma. To examine the ability of Fc gamma to substitute for zeta-chain in T cell development and function, we introduced a transgene encoding Fc gamma into mice made genetically deficient for zeta-chain (zeta(e)-/-). Analysis of thymocyte development in zeta(e)-/-;Fc gamma Tg mice demonstrated that Fc gamma was able to support the maturation of both gammadelta TCR+ and alphabeta TCR+ T cells. However, positive selection of alphabeta TCR+ thymocytes was less efficient in zeta(e)-/-;Fc gamma Tg mice than in zeta(e)-/- mice reconstituted with zeta-chain. This difference may be due to the fact that Fc gamma contains a single immunoreceptor tyrosine-based activation motif (ITAM) whereas zeta-chain contains three ITAMs. Interestingly, the peripheral T cells that develop in zeta(e)-/- mice reconstituted with Fc gamma are functional and respond to TCR-specific stimuli. These data suggest that Fc gamma and zeta are interchangeable in their ability to mediate T cell development and function, however zeta-chain is more efficient at promoting positive selection and T cell maturation. The difference in efficiency between zeta and Fc gamma may be responsible in part for the unusual developmental and functional properties of T cells that constitutively express Fc gamma as a signaling component of their TCRs.

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46

Vaitekhovich, Tatyana S. "Biharmonic Green function of a ring domain." MATHEMATICA SCANDINAVICA 106, no. 2 (June 1, 2010): 267. http://dx.doi.org/10.7146/math.scand.a-15137.

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A biharmonic Green function of a circular ring domain $R=\{z\in \mathsf {C}: 0<r<|z|<1\}$ is found in the form 26741 \widehat{G}_{2}(z,\zeta)=|\zeta-z|^{2}G_{1}(z,\zeta)+\widehat{h}_{2}(z,\zeta), 26741 where $G_{1}(z,\zeta)$ is the harmonic Green function of the ring $R$, and $\widehat{h}_{2}(z,\zeta)$ is a specially constructed biharmonic function.

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47

Quan, Junjie. "Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $." AIMS Mathematics 7, no. 1 (2021): 288–93. http://dx.doi.org/10.3934/math.2022019.

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<abstract><p>In a recent paper <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>, Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.</p></abstract>

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48

MACHIDE, TOMOYA. "WEIGHTED SUMS WITH TWO PARAMETERS OF MULTIPLE ZETA VALUES AND THEIR FORMULAS." International Journal of Number Theory 08, no. 08 (September 19, 2012): 1903–21. http://dx.doi.org/10.1142/s1793042112501084.

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A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coefficients in the rational polynomial ring generated by the two parameters, and obtain some identities for weighted sums of multiple zeta values.

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49

Vivier, E., P. Morin, Q. S. Tian, J. Daley, M. L. Blue, S. F. Schlossman, and P. Anderson. "Expression and tyrosine phosphorylation of the T cell receptor zeta-subunit in human thymocytes." Journal of Immunology 146, no. 4 (February 15, 1991): 1142–48. http://dx.doi.org/10.4049/jimmunol.146.4.1142.

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Abstract Recent evidence suggests that the zeta-subunit of the TCR complex plays a critical role in transducing signals initiated by the Ag receptor heterodimer. Because thymic maturation involves specific interactions between the TCR complex and thymic stromal cells, the zeta-subunit has been postulated to also play a role in this process. To assess the potential for zeta to contribute to thymocyte maturation, we have used an anti-zeta mAb (TIA-2) to quantitate its expression in mature (CD3bright) and immature (CD3dim and CD3-) populations of human thymocytes. Using both flow cytometric and immunoblotting analysis, we found that the relative expression of TCR-zeta varied directly with the surface expression of CD3. Importantly, TCR-zeta was detected in the majority of CD3- thymocytes, indicating that its expression precedes the surface appearance of CD3:TCR. In thymocytes, TCR-zeta was found to be constitutively phosphorylated on tyrosine residues. The relative expression of phospho-zeta varied directly with the maturational stage of the thymocyte, with the mature (CD3bright), single positive cells accounting for most of the phospho-zeta found in the human thymus. The expression of phospho-zeta could be significantly increased by activating thymocytes with mAb reactive with either CD3 or CD2. These results suggest that TCR-zeta is functionally linked to the major thymocyte activation receptors.

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50

Akahori, Jiro, Norio Konno, and Iwao Sato. "Absolute Zeta fountions for Zeta functions of quantum cellular automata." Quantum Information and Computation 23, no. 15&16 (December 2023): 1261–74. http://dx.doi.org/10.26421/qic23.15-16-1.

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Our previous work delt with the zeta function for the interacting particle system (IPS) including quantum cellular automaton (QCA) as a typical model in the study of ``IPS/Zeta Correspondence". On the other hand, the absolute zeta function is a zeta function over $\mathbb{F}_1$ defined by a function satisfying an absolute automorphy. This paper proves that a new zeta function given by QCA is an absolute automorphic form of weight depending on the size of the configuration space. As an example, we calculate an absolute zeta function for a tensor-type QCA, and show that it is expressed as the multiple gamma function. In addition, we obtain its functional equation by the multiple sine function.

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