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Journal articles on the topic 'Mathematics ; Number theory ; Analytic number theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Zaharescu, Alexandru. "Book Review: Analytic number theory." Bulletin of the American Mathematical Society 43, no. 02 (2006): 273–79. http://dx.doi.org/10.1090/s0273-0979-06-01084-6.

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2

Maslov, V. P. "Analytic number theory and disinformation." Mathematical Notes 100, no. 3-4 (2016): 568–78. http://dx.doi.org/10.1134/s0001434616090285.

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3

Zong, Chuanming. "Analytic Number Theory in China." Mathematical Intelligencer 32, no. 1 (2009): 18–25. http://dx.doi.org/10.1007/s00283-009-9087-1.

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4

Lucht, Lutz G. "CONTRIBUTIONS TO ABSTRACT ANALYTIC NUMBER THEORY." Quaestiones Mathematicae 24, no. 3 (2001): 309–22. http://dx.doi.org/10.1080/16073606.2001.9639220.

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5

CHEN, WEIYAN. "Analytic number theory for 0-cycles." Mathematical Proceedings of the Cambridge Philosophical Society 166, no. 1 (2017): 123–46. http://dx.doi.org/10.1017/s0305004117000767.

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AbstractThere is a well-known analogy between integers and polynomials over 𝔽q, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorisation of effective 0-cycles on an arbitrary connected varietyVover 𝔽q, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles are typically Poisson distributed.

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6

Vardi, Ilan. "Integrals, an Introduction to Analytic Number Theory." American Mathematical Monthly 95, no. 4 (1988): 308. http://dx.doi.org/10.2307/2323562.

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7

Munshi, Ritabrata. "Analytic number theory in India during 2001–2010." Indian Journal of Pure and Applied Mathematics 50, no. 3 (2019): 719–38. http://dx.doi.org/10.1007/s13226-019-0351-6.

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8

Gelbart, S. "Book Review: Advanced analytic number theory: L-functions." Bulletin of the American Mathematical Society 45, no. 01 (2007): 169–76. http://dx.doi.org/10.1090/s0273-0979-07-01158-5.

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9

Karatsuba, A. A. "The Hilbert-Kamke problem in analytic number theory." Mathematical Notes of the Academy of Sciences of the USSR 41, no. 2 (1987): 155–61. http://dx.doi.org/10.1007/bf01138339.

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10

Hall, Richard R. "DUALITY IN ANALYTIC NUMBER THEORY (Cambridge Tracts in Mathematics 122)." Bulletin of the London Mathematical Society 30, no. 3 (1998): 318–19. http://dx.doi.org/10.1112/s0024609397224094.

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11

Schwarz, Wolfgang. "JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY, AND THE THEORY OF ARITHMETICAL FUNCTIONS." Quaestiones Mathematicae 24, no. 3 (2001): 273–90. http://dx.doi.org/10.1080/16073606.2001.9639218.

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12

Ivić, Aleksandar. "Some applications of Laplace transforms in analytic number theory." Novi Sad Journal of Mathematics 45, no. 1 (2015): 31–44. http://dx.doi.org/10.30755/nsjom.dans14.02.

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13

Knopfmacher, John. "RECENT DEVELOPMENTS AND APPLICATIONS OF ABSTRACT ANALYTIC NUMBER THEORY." Quaestiones Mathematicae 24, no. 3 (2001): 291–307. http://dx.doi.org/10.1080/16073606.2001.9639219.

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14

Harman, G. "INTRODUCTION TO ANALYTIC NUMBER THEORY (Translations of Mathematical Monographs 68)." Bulletin of the London Mathematical Society 21, no. 6 (1989): 602–4. http://dx.doi.org/10.1112/blms/21.6.602.

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15

Maslov, V. P., S. Yu Dobrokhotov, and V. E. Nazaikinskii. "Volume and entropy in abstract analytic number theory and thermodynamics." Mathematical Notes 100, no. 5-6 (2016): 828–34. http://dx.doi.org/10.1134/s0001434616110225.

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16

Halter-Koch, Franz, and Richard Warlimont. "Analytic number theory in formations based on additive arithmetical semigroups." Mathematische Zeitschrift 215, no. 1 (1994): 99–128. http://dx.doi.org/10.1007/bf02571701.

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17

Shkredov, Il'ya D. "Fourier analysis in combinatorial number theory." Russian Mathematical Surveys 65, no. 3 (2010): 513–67. http://dx.doi.org/10.1070/rm2010v065n03abeh004681.

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18

Borwein, Peter B. "Chebyshev polynomials: From approximation theory to algebra and number theory." Journal of Approximation Theory 66, no. 3 (1991): 353. http://dx.doi.org/10.1016/0021-9045(91)90038-c.

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19

Charafi, A. "Chebyshev polynomials—From approximation theory to algebra and number theory." Engineering Analysis with Boundary Elements 9, no. 2 (1992): 190. http://dx.doi.org/10.1016/0955-7997(92)90065-f.

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20

Shiu, Peter. "Euler's contribution to number theory." Mathematical Gazette 91, no. 522 (2007): 453–61. http://dx.doi.org/10.1017/s0025557200182099.

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Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

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21

Schochet, Steven. "The mathematical theory of low Mach number flows." ESAIM: Mathematical Modelling and Numerical Analysis 39, no. 3 (2005): 441–58. http://dx.doi.org/10.1051/m2an:2005017.

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22

Baeth, Nicholas, Vadim Ponomarenko, Donald Adams, et al. "Number theory of matrix semigroups." Linear Algebra and its Applications 434, no. 3 (2011): 694–711. http://dx.doi.org/10.1016/j.laa.2010.09.028.

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23

Daili, N. "Analytic densities in number theory – Part III: Extensions to Epstein’s zeta-function." Journal of Interdisciplinary Mathematics 10, no. 5 (2007): 697–713. http://dx.doi.org/10.1080/09720502.2007.10700526.

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24

Newman, Charles M., and Wei Wu. "Constants of de Bruijn–Newman type in analytic number theory and statistical physics." Bulletin of the American Mathematical Society 57, no. 4 (2019): 595–614. http://dx.doi.org/10.1090/bull/1668.

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25

Aistleitner, Christoph, and Christian Elsholtz. "The Central Limit Theorem for Subsequences in Probabilistic Number Theory." Canadian Journal of Mathematics 64, no. 6 (2012): 1201–21. http://dx.doi.org/10.4153/cjm-2011-074-1.

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Abstract Let (nk)k≥1 be an increasing sequence of positive integers, and let f (x) be a real function satisfyingIf the distribution ofconverges to a Gaussian distribution. In the casethere is a complex interplay between the analytic properties of the function f , the number-theoretic properties of (nk)k≥1, and the limit distribution of (2).In this paper we prove that any sequence (nk)k≥1 satisfying lim contains a nontrivial subsequence (mk)k≥1 such that for any function satisfying (1) the distribution ofconverges to a Gaussian distribution. This result is best possible: for any ε > 0 there exists a sequence (nk)k≥1 satisfying lim such that for every nontrivial subsequence (mk)k≥1 of (nk)k≥1 the distribution of (2) does not converge to a Gaussian distribution for some f.Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

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26

Barner, David. "In defense of intuitive mathematical theories as the basis for natural number." Behavioral and Brain Sciences 31, no. 6 (2008): 643–44. http://dx.doi.org/10.1017/s0140525x0800558x.

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AbstractThough there are holes in the theory of how children move through stages of numerical competence, the current approach offers the most promising avenue for characterizing changes in competence as children confront new mathematical concepts. Like the science of mathematics, children's discovery of number is rooted in intuitions about sets, and not purely in analytic truths.

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27

Kauffman, Robert M., and Mayumi Sakata. "Generalized Eigenfunction Expansions for Spectral Multiplicity One and Application in Analytic Number Theory." Rocky Mountain Journal of Mathematics 40, no. 1 (2010): 243–77. http://dx.doi.org/10.1216/rmj-2010-40-1-243.

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28

Jorgenson, Jay, and Serge Lang. "Extension of analytic number theory and the theory of regularized harmonic series from Dirichlet series to Bessel series." Mathematische Annalen 306, no. 1 (1996): 75–124. http://dx.doi.org/10.1007/bf01445243.

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29

Fieker, Claus, and Yinan Zhang. "An application of the -adic analytic class number formula." LMS Journal of Computation and Mathematics 19, no. 1 (2016): 217–28. http://dx.doi.org/10.1112/s1461157016000097.

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We propose an algorithm to verify the$p$-part of the class number for a number field$K$, provided$K$is totally real and an abelian extension of the rational field$\mathbb{Q}$, and$p$is any prime. On fields of degree 4 or higher, this algorithm has been shown heuristically to be faster than classical algorithms that compute the entire class number, with improvement increasing with larger field degrees.

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30

Hajir, Farshid, and Christian Maire. "On the Invariant Factors of Class Groups in Towers of Number Fields." Canadian Journal of Mathematics 70, no. 1 (2018): 142–72. http://dx.doi.org/10.4153/cjm-2017-032-9.

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AbstractFor a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

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31

Shi, Zhiping, Liming Li, Yong Guan, Xiaoyu Song, Minhua Wu, and Jie Zhang. "Formalization of the Complex Number Theory in HOL4,." Applied Mathematics & Information Sciences 7, no. 1 (2013): 279–86. http://dx.doi.org/10.12785/amis/070135.

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32

Daili, N. "About binomial densities: some applications in number theory." Journal of Discrete Mathematical Sciences and Cryptography 13, no. 5 (2010): 465–72. http://dx.doi.org/10.1080/09720529.2010.10698308.

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33

Ames, W. F. "Number theory, trace formulas and discrete groups." Mathematics and Computers in Simulation 31, no. 6 (1990): 601. http://dx.doi.org/10.1016/0378-4754(90)90087-y.

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34

Pomerance, Carl, and Eric Bach. "Analytic Methods in the Analysis and Design of Number-Theoretic Algorithms." Mathematics of Computation 48, no. 177 (1987): 441. http://dx.doi.org/10.2307/2007905.

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35

Hall, Richard R. "TEN LECTURES ON THE INTERFACE BETWEEN ANALYTIC NUMBER THEORY AND HARMONIC ANALYSIS (Regional Conference Series in Mathematics 84)." Bulletin of the London Mathematical Society 28, no. 5 (1996): 540–41. http://dx.doi.org/10.1112/blms/28.5.540.

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36

Harrison, John. "Formalizing an Analytic Proof of the Prime Number Theorem." Journal of Automated Reasoning 43, no. 3 (2009): 243–61. http://dx.doi.org/10.1007/s10817-009-9145-6.

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37

Elaydi, Saber. "Difference equations in combinatorics, number theory, and orthogonal polynomials." Journal of Difference Equations and Applications 5, no. 4-5 (1999): 379–92. http://dx.doi.org/10.1080/10236199908808198.

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38

Ivic, Aleksandar. "Convolutions and mean square estimates of certain number-theoretic error terms." Publications de l'Institut Math?matique (Belgrade) 80, no. 94 (2006): 141–56. http://dx.doi.org/10.2298/pim0694141i.

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We study the convolution function C[f(x)]:=\int_1^x f(y)f\Bigl(\frac xy\Bigr)\frac{dy}y When f(x) is a suitable number-theoretic error term. Asymptotics and upper bounds for C[f(x)] are derived from mean square bounds for f(x). Some applications are given, in particular to |\zeta(\tfrac12+ix)|^{2k} and the classical Rankin--Selberg problem from analytic number theory.

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39

Hildebrand, Adolf, and Gérald Tenenbaum. "On a class of differential-difference equations arising in number theory." Journal d'Analyse Mathématique 61, no. 1 (1993): 145–79. http://dx.doi.org/10.1007/bf02788841.

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40

Maslov, V. P. "New Formulas Related to Analytic Number Theory and Their Applications in Statistical Physics." Theoretical and Mathematical Physics 196, no. 1 (2018): 1082–87. http://dx.doi.org/10.1134/s0040577918070127.

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41

Forrester, Peter J. "Octonions in random matrix theory." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (2017): 20160800. http://dx.doi.org/10.1098/rspa.2016.0800.

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The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N =2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N =3. We then proceed to consider the matrix structure X † X , when X has random octonion entries. Analytic results are obtained from N =2, but are observed to break down in the 3×3 case.

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42

CHAMBERT-LOIR, ANTOINE. "THE THEOREM OF JENTZSCH–SZEGŐ ON AN ANALYTIC CURVE: APPLICATION TO THE IRREDUCIBILITY OF TRUNCATIONS OF POWER SERIES." International Journal of Number Theory 07, no. 07 (2011): 1807–23. http://dx.doi.org/10.1142/s1793042111004691.

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A theorem of Jentzsch–Szegő describes the limit measure of a sequence of discrete measures associated to zeroes of a sequence of polynomials in one variable. Following the presentation by Andrievskii and Blatt in [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] we extend this theorem to compact Riemann surfaces and to analytic curves in the sense of Berkovich over ultrametric fields, using classical potential theory in the former case, and Baker/Rumely, Thuillier's potential theory on analytic curves in the latter case. We then apply this equidistribution theorem to the question of irreducibility of truncations of power series with coefficients in ultrametric fields. Résumé français: Le théorème de Jentzsch–Szegő décrit la mesure limite d'une suite de mesures discrètes associée aux zéros d'une suite convenable de polynômes en une variable. Suivant la présentation que font Andrievskii et Blatt dans [Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics (Springer-Verlag, New York, 2002)] on étend ici ce résultat aux surfaces de Riemann compactes, puis aux courbes analytiques sur un corps ultramétrique. On donne pour finir quelques corollaires du cas particulier de la droite projective sur un corps ultramétrique à l'irréductibilité des polynômes-sections d'une série entière en une variable.

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43

Hall, R. R. "Ω theorems for the complex divisor function". Mathematical Proceedings of the Cambridge Philosophical Society 115, № 1 (1994): 145–57. http://dx.doi.org/10.1017/s030500410007198x.

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This paper is a sequel to [6] and concerns the complex divisor functionwhich has had a number of applications in the analytic theory of numbers. Thus Hooley's Δ-function [8] defined bysatisfies the inequalityand Erdös' τ+-function, defined bysatisfies

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44

Tou, Erik R. "Asymptotic counting theorems for primitive juggling patterns." International Journal of Number Theory 15, no. 05 (2019): 1037–50. http://dx.doi.org/10.1142/s1793042119500568.

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The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

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45

Bourgain, J. "Problems of almost everywhere convergence related to harmonic analysis and number theory." Israel Journal of Mathematics 71, no. 1 (1990): 97–127. http://dx.doi.org/10.1007/bf02807252.

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46

Haddad, L., C. Helou, and J. Pihko. "Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem." International Journal of Mathematics and Mathematical Sciences 2005, no. 23 (2005): 3767–80. http://dx.doi.org/10.1155/ijmms.2005.3767.

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We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.

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47

Schaller, Paul Schmutz. "GROUPS ACTING ON HYPERBOLIC SPACES: HARMONIC ANALYSIS AND NUMBER THEORY (Springer Monographs in Mathematics)." Bulletin of the London Mathematical Society 31, no. 6 (1999): 754–55. http://dx.doi.org/10.1112/s0024609399306159.

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48

Cabbolet, Marcoen J. T. F. "A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory." Axioms 10, no. 2 (2021): 119. http://dx.doi.org/10.3390/axioms10020119.

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It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

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49

Mo, Hongming. "A New Evaluation Methodology for Quality Goals Extended by D Number Theory and FAHP." Information 11, no. 4 (2020): 206. http://dx.doi.org/10.3390/info11040206.

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Evaluation of quality goals is an important issue in process management, which essentially is a multi-attribute decision-making (MADM) problem. The process of assessment inevitably involves uncertain information. The two crucial points in an MADM problem are to obtain weight of attributes and to handle uncertain information. D number theory is a new mathematical tool to deal with uncertain information, which is an extension of evidence theory. The fuzzy analytic hierarchy process (FAHP) provides a hierarchical way to model MADM problems, and the comparison analysis among attributes is applied to obtain the weight of attributes. FAHP uses a triangle fuzzy number rather than a crisp number to represent the evaluation information, which fully considers the hesitation to give a evaluation. Inspired by the features of D number theory and FAHP, a D-FAHP method is proposed to evaluate quality goals in this paper. Within the proposed method, FAHP is used to obtain the weight of each attribute, and the integration property of D number theory is carried out to fuse information. A numerical example is presented to demonstrate the effectiveness of the proposed method. Some necessary discussions are provided to illustrate the advantages of the proposed method.

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50

Yaremko, Natalia N., and Maria V. Glebova. "Correctness of determining the degree of a real number with a rational exponent." Science and School, no. 2, 2020 (2020): 165–75. http://dx.doi.org/10.31862/1819-463x-2020-2-165-175.

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The article considers correctness of definition for the root of the n-th degree from a real number and the degree of a real number with an arbitrary real exponent. The article analyzes the definitions of these concepts in different school textbooks and university textbooks, and discusses the differences and contradictions that arise. The article solves exponential and power-exponential equations depending on those approaches that were chosen by the authors. The solution to the problem of correctly defining these concepts lies outside the school mathematics course and goes into the theory of analytic functions. The authors of the article suggest ways that, in their opinion, can be implemented when teaching mathematics.

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