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Dissertations / Theses on the topic 'Rieman's Zeta Function'

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Published: 4 June 2021
Last updated: 13 February 2022

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1

Dauguet, Simon. "Généralisations du critère d’indépendance linéaire de Nesterenko." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112085/document.

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Cette thèse s'inscrit dans le prolongement du résultat d'Apéry donnant l'irrationalité de ζ (3) et de celui de Ball-Rivoal prouvant qu'il existe une infinité d'entiers impairs en lesquels la fonction zêta de Riemann prend des valeurs irrationnelles. Un outil crucial dans la démonstration de Ball-Rivoal est le critère d'indépendance linéaire de Nesterenko, qui a été généralisé par Fischler et Zudilin pour exploiter sous des hypothèses très restrictives la présence de diviseurs communs aux coefficients des formes linéaires. Une généralisation ultérieure due à Fischler s'applique lorsqu'on dispose d'approximations simultanées des nombres réels en question (et non plus de combinaisons Z-linéaires petites de ces nombres).Dans cette thèse, on améliore ce dernier résultat en affaiblissant considérablement les hypothèses sur les diviseurs. On démontre aussi un critère d'indépendance linéaire analogue, dans l'esprit de celui de Siegel. Dans une autre partie en commun avec Zudilin, on construit, en utilisant des identités hypergéométriques, des approximations simultanées de ζ (2) et ζ (3) qui permettent de démontrer en même temps l'irrationalité de ces deux nombres. En appliquant essentiellement le critère démontré précédemment, on en déduit une minoration des combinaisons Z-linéaires de 1, ζ 2) et ζ (3), sous des hypothèses de divisibilité très fortes sur les coefficients (si bien que l'indépendance linéaire sur Q de ces trois nombres est toujours conjecturale)
This Ph.D. thesis lies in the path opened by Apéry who proved the irrationality of ζ(3) andalready followed by Ball-Rivoal who proved that there are infinitely many odd integers at which Riemann zeta function takes irrational values. A fundamental tool in the proof of Ball-Rivoal is Nesterenko’s linear independence criterion. This criterion has been generalized by Fischler and Zudilin to use common divisors of the coefficients of linear forms, under some restrictive assumptions. Then Fischler gave another generalization for simultaneous approximations (instead of small Z-linear combinations).In this Ph.D. thesis, we improve this last result by greatly weakening the assumption on thedivisors. We prove also an analogous linear independence criterion in the spirit of Siegel. Inanother part joint with Zudilin, we construct simultaneous linear approximations to ζ(2) and ζ(3) using hypergeometric identitites. These linear approximations allow one to prove at thesame time the irrationality of ζ(2) and that of ζ(3). Then, using a criterion from the previouspart, we deduce a lower bound on Z-linear combinations of 1, ζ(2) and ζ(3), under somestrong divisibility hypotheses on the coefficients (so that the Q-linear independence of thesethree numbers still remains an open problem)
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2

Kalpokas, Justas. "Discrete moments of the Riemann zeta function and Dirichlet L-functions." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2012. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2012~D_20121119_130728-97328.

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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems that concern the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions. In analytic number theory one of the main investigation objects is the Riemann zeta function. The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function lie on the critical line. In the thesis we investigate value distribution of the Riemann zeta function on the critical line. To do so we use the curve of the Riemann zeta function on the critical line. A problem connected to the curve asks the question whether the curve is dense in the complex plane. We prove that the curve expands to all directions on the complex plane. A separete case of the main result can be stated as follows Riemann zeta function has infinetly many negative values on the critical line and they are unbounded.
Analizinė skaičių teorija yra skaičių teorijos dalis, kuri, naudodama matematinės analizės ir kompleksinio kintamojo funkcijų tyrimo metodus, sprendžia uždavinius susijusius su sveikaisiais skaičiais. Manoma, kad analizinės skaičių teorijos pradžią žymi Dirichlet eilučių ir Dirichlet L-funkcijų taikymai. Vienas iš pagrindinių analizinės skaičių teorijos tyrimo objektų yra Riemann’o dzeta funkcija. Riemann’o hipotezė teigia, kad visi netrivialieji nuliai yra ant kritinės tiesės. Disertacijoje nagrinėjamas Riemann’o dzeta funckijos reikšmių pasiskirstymas ant kritinės tiesės. Tam pasitelkiama Riemann’o dzeta funkcijos kreivė. Svarbus klausimas susijęs su kreive yra ar ši kreivė yra visur tiršta kompleksinių skaičių plokštumoje. Disertacijoje įrodoma, kad kreivė plečiasi į visas puse kompleksinių skaičių plokštumoje. Atskiras disertacijos pagrindinio rezultato atvejis gali būti formuluojamas taip – Riemann’o dzeta funkcija ant kritinės tiesės įgyja be galo daug neigiamų reikšmių, kurios yra neaprėžtos.
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3

Reyes, Ernesto Oscar. "The Riemann zeta function." CSUSB ScholarWorks, 2004. https://scholarworks.lib.csusb.edu/etd-project/2648.

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The Riemann Zeta Function has a deep connection with the distribution of primes. This expository thesis will explain the techniques used in proving the properties of the Rieman Zeta Function, its analytic continuation to the complex plane, and the functional equation that the the Riemann Zeta Function satisfies.
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4

Merrill, Katherine J. "Ramanujan's Formula for the Riemann Zeta Function Extended to L-Functions." Fogler Library, University of Maine, 2005. http://www.library.umaine.edu/theses/pdf/MerrillKJ2005.pdf.

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5

Juchmes, Franziska. "Zeta Functions and Riemann Hypothesis." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-32363.

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In this thesis the zeta functions in analytic number theory are stud-ied. The distribution of primes and the connection between primes andzeta functions are discussed. Numerical results for linear combinationsof zeta functions are presented. These functions have a symmetric dis-tribution of zeros around the critical line.
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6

Šimėnas, Raivydas. "Riemann'o hipotezės Speiser'io ekvivalentas." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2012~D_20140704_171541-67476.

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A. Speiser'is parodė, kad Riemann'o hipotezė yra ekvivalenti tam, kad Riemann'o dzeta funkcijos išvestinė neturi netrivialių nulių į kairę nuo kritinės tiesės. Kiekybinis šio fakto rezultatas buvo pasiektas N. Levinsono ir H. Montgomerio. Šie rezultatai buvo apibendrinti daugeliui dzeta funkcijų, kurioms tikimasi, kad Riemann'o hipotezė galioja. Šiame darbe mes apibendriname Speiser'io ekvivalentą dzeta-funkcijoms. Mes tiriame sąryšį tarp netrivialių nulių išplėstinės Selbergo klasės funkcijoms ir jų išvestinėms šiame regione. Šiai klasei priklauso ir funkcijos, kurioms Riemann'o hipotezė neteisinga. Kaip pavyzdį, mes skaitiniu būdu tiriame sąryšius tarp Dirichlet L-funkcijų ir jų išvestinių tiesinių kombinacijų.
A. Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the Riemann zeta-function left of the critical line. The quantitative version of this result was obtained by N. Levinson and H. Montgomery. This result (or the quantitative version of this result proved by N. Levinson and H. Montgomery) were generalized for many zeta-functions for which the Riemann hypothesis is expected. Here we generalize the Speiser equivalent for zeta-functions. We also investigate the relationship between the on-trivial zeros of the extended Selberg class functions and of their derivatives in this region. This class contains zeta functions for which Riemann hypothesis is not true. As an example, we study the relationship between the trajectories of zeros of linear combinations of Dirichlet $L$-functions and of their derivatives computationally.
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7

Tsumura, Hirofumi, and Kohji Matsumoto. "On Witten multiple zeta-functions associated with semisimple Lie algebras I." Annales de L'Institut Fourier, 2006. http://hdl.handle.net/2237/20336.

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8

Gauthier, Paul M., and Nikolai Tarkhanov. "A covering property of the Riemann zeta-function." Universität Potsdam, 2004. http://opus.kobv.de/ubp/volltexte/2008/2668/.

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For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails.
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9

Steuding, Jörn. "On simple zeros of the Riemann zeta-function." [S.l. : s.n.], 1999. http://deposit.ddb.de/cgi-bin/dokserv?idn=95589820X.

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10

Steuding, Rasa, Jörn Steuding, Kohji Matsumoto, Antanas Laurinčikas, and Ramūnas Garunkštis. "Effective uniform approximation by the Riemann zeta-function." Department of Mathematics of the Universitat Autònoma de Barcelona, 2010. http://hdl.handle.net/2237/20429.

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11

Musser, Jason. "Higher Derivatives of the Hurwitz Zeta Function." TopSCHOLAR®, 2011. http://digitalcommons.wku.edu/theses/1093.

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The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).
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12

Mack, Rüdiger [Verfasser]. "Quantum Mechanics meets the Riemann-Zeta Function / Rüdiger Mack." München : Verlag Dr. Hut, 2011. http://d-nb.info/1010446622/34.

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13

TSUMURA, Hirofumi, and Kohji MATSUMOTO. "Functional relations among certain double polylogarithms and their character analogues." Šiauliai University, 2008. http://hdl.handle.net/2237/20444.

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14

MATSUMOTO, KOHJI, and MASANORI KATSURADA. "Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III." Cambridge University Press, 2002. http://hdl.handle.net/2237/10253.

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15

Ranorovelonalohotsy, Marie Brilland Yann. "Riemann hypothesis for the zeta function of a function field over a finite field." Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/85713.

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16

Alvites, José Carlos Valencia. "Hipótese de Riemann e física." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042012-084309/.

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Neste trabalho, introduzimos a função zeta de Riemann \'ZETA\'(s), para s \'PERTENCE\' C \\ e apresentamos muito do que é conhecido como justificativa para a hipótese de Riemann. A importância de \'ZETA\' (s) para a teoria analítica dos números é enfatizada e fornecemos uma prova conhecida do Teorema dos Números Primos. No final, discutimos a importância de \'ZETA\'(s) para alguns modelos físicos de interesse e concluimos descrevendo como a hipótese de Riemann pode ser acessada estudando estes sistemas
In this work, we introduce the Riemann zeta function \'ZETA\'(s), s \'IT BELONGS\' C \\ and present much of what is known to support the Riemann hypothesis. The importance of \'ZETA\'(s) to the Analytic number theory is emphasized and a proof for the Prime Number Theorem is reviewed. In the end, we report on the importance of \'ZETA\'(s) to some relevant physical models and conclude by describing how the Riemann Hypothesis can be accessed by studying these systems
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17

Matsumoto, Kohji. "Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions." Hindustan Book Agency & The Indian National Science Academy, 2000. http://hdl.handle.net/2237/20433.

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18

Mirjana, Vidanović. "Sumiranje redova sa specijalnim funkcijama." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2003. https://www.cris.uns.ac.rs/record.jsf?recordId=73367&source=NDLTD&language=en.

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Disertacija se bavi sumiranjem redova sa specijalnim funkcijama. Ovi redovi se posredstvom trigonometrijskih redova svode na redove sa Riemannovom zeta funkci­jom i srodnim funkcijama. U određenim slučajevima sumacione formule se mogu dovesti na takozvani zatvoreni oblik, što znači da se beskonačni redovi predstavljaju konačnim sumama. Predloženi metodi sumacije omogućavaju ubrzanje konvergencije, a mogu se primeniti i kod nekih graničnih problema matematičke fizike. Sumacione formule uključuju kao specijalne slučajeve neke formule poznate iz literature, ali i nove sume, s obzirom da su opšteg karaktera. Pomoću ovih formula sumirani su i redovi sa integralima trigonometrijskih i specijalnih funkcija.
This dissertation deals with the summation of series over special functions. Throughtrigonometric series these series are reduced to series in terms of Riemann zeta andrelated functions. They can be brought in closed form in some cases, i.e. infiniteseries are expressed as finite sums. Closed form formulas make it possible to accele­rate the convergence of some series, and have many applications in various scientificfields as well. For example, closed form solutions of the boundary value problem inmathematical physics can be obtained. Summation formulas include particular casesknown from the literature, but because of their general character one can come tonew sums. By means of these formuláis the sums of series over integrals containingtrigonometric or special functions have been found.
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19

Segarra, Elan. "An Exploration of Riemann's Zeta Function and Its Application to the Theory of Prime Distribution." Scholarship @ Claremont, 2006. https://scholarship.claremont.edu/hmc_theses/189.

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Identified as one of the 7 Millennium Problems, the Riemann zeta hypothesis has successfully evaded mathematicians for over 100 years. Simply stated, Riemann conjectured that all of the nontrivial zeroes of his zeta function have real part equal to 1/2. This thesis attempts to explore the theory behind Riemann’s zeta function by first starting with Euler’s zeta series and building up to Riemann’s function. Along the way we will develop the math required to handle this theory in hopes that by the end the reader will have immersed themselves enough to pursue their own exploration and research into this fascinating subject.
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20

Fernandez, Arran. "Analysis in fractional calculus and asymptotics related to zeta functions." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/284390.

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This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.
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21

Lau, Yuk-kam, and 劉旭金. "Some results on the mean square formula for the riemann zeta-function." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1993. http://hub.hku.hk/bib/B31211586.

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22

Lau, Yuk-kam. "Some results on the mean square formula for the riemann zeta-function /." [Hong Kong] : University of Hong Kong, 1993. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13762394.

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23

Alcántara, Bode Julio. "A conjecture about the non-trivial zeroes of the Riemann zeta function." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97185.

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Some heuristic arguments are given in support of the following conjecture: If the Riemann Hypothesis (RH) does not hold then the number of zeroes of the Riemann zeta function with real part σ >  ½ is infinite.
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24

Černigova, Sondra. "Moment problem for the periodic zeta-function." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2014~D_20141111_114553-36360.

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In the thesis, problems related to the moments of the periodic zeta-function are considered. The aim of the thesis is to obtain asymptotic formulae for some analytic objects related to the periodic zeta-function. The problems are the following: 1. To prove the Atkinson-type formula with a new error term in the critical strip for the periodic zeta-function with rational parameter. 2. To prove a mean square formula for the error term in the Atkinson-type formula on the critical line for the periodic zeta-function. 3. To prove a mean square formula for the error term in the Atkinson-type formula in the critical strip for the periodic zeta-function. 4. To obtain an asymptotic formula for the fourth power moment of the periodic zeta-function.
Disertacijos tyrimo objektas yra periodinė dzeta funkcija. Mokslinė problema - šios funkcijos momentų problema. Darbo tikslas - įrodyti asimptotines formules periodinės funkcijos momentams bei kai kuriems objektams, susijusiems su šios funkcijos momentais. Darbo uždaviniai yra šie: 1. Įrodyti Atkinsono tipo formulę su korektišku liekamuoju nariu kritinėje juostoje periodinei dzeta funkcijai su racionaliuoju parametru. 2. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje tiesėje vidurkio formulę liekamojo nario modulio kvadratui. 3. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje juostoje vidurkio formulę liekamojo nario modulio kvadratui. 4. Gauti asimptotinę formulę periodinės dzeta funkcijos ketvirtajam momentui.
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25

Henderson, Cory. "Exploring the Riemann Hypothesis." Kent State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=kent1371747196.

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26

Feiler, Cornelia [Verfasser]. "Quantum physics and number theory connected by the Riemann zeta function / Cornelia Feiler." Ulm : Universität Ulm. Fakultät für Naturwissenschaften, 2014. http://d-nb.info/1049561953/34.

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MATSUMOTO, Kohji. "An introduction to the value-distribution theory of zeta-functions." Šiauliai University, 2006. http://hdl.handle.net/2237/20445.

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28

Christ, Thomas [Verfasser], Jörn [Gutachter] Steuding, and Ramūnas [Gutachter] Garunkštis. "Value-distribution of the Riemann zeta-function and related functions near the critical line / Thomas Christ. Gutachter: Jörn Steuding ; Ramunas Garunkštis." Würzburg : Universität Würzburg, 2013. http://d-nb.info/1102825875/34.

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Lee, Kai-yuen, and 李啟源. "On the mean square formula for the Riemann zeta-function on the critical line." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44674405.

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Hughes, Christopher Paul. "On the characteristic polynomial of a random unitary matrix and the Riemann zeta function." Thesis, University of Bristol, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364865.

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Souza, Uender Barbosa de. "Sobre somas infnitas e uma forma recursiva para a soma da série Zeta (2p) de Riemann." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/5264.

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This paper presents methods to calculate some in nite sums and use the Fourier series of function f(x) = x2p with p 2 N to get results on the behavior of Zeta(2p) function Riemann, including their sum and rational multiplicity of 2p.
Neste trabalho apresentamos métodos para o cálculo de algumas somas in nitas e usamos a série de Fourier da função f(x) = x2p com p 2 N para obter resultados sobre o comportamento da função Zeta(2p) de Riemann, tais como sua soma e sua multiplicidade racional por 2p.
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Dalpizol, Luiz Gustavo. "O conjunto excepcional do problema de Goldbach." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2018. http://hdl.handle.net/10183/180946.

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Seja E(X) a cardinalidade dos números pares menores ou iguais a X que não podem ser escritos como soma de dois primos. O objetivo central desta dissertação é apresentar uma demonstração de uma estimativa para E(X) dada por Hugh L. Montgomery e Robert C. Vaughan em [22]. Mais precisamente, estabeleceremos a existência de uma constante positiva (efetivamente computável) tal que E(X) X1 ; para todo X su cientemente grande.
Let E(X) the cardinality of even numbers not exceeding X which cannot be written as a sum of two primes. The main goal of this dissertation is to present a proof of an estimate for E(X) given by Hugh L. Montgomery e Robert C. Vaughan in [22]. More precisely, we will establish the existence of a positive constant (e ectively computable) such that E(X) X1 for all su ciently large X:
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33

Gulas, Michael Allen. "Using Hilbert Space Theory and Quantum Mechanics to Examine the Zeros of The Riemann-Zeta Function." Bowling Green State University / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1594035551136634.

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Černigova, Sondra. "Momentų problema periodinei dzeta funkcijai." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2014~D_20141111_114605-33009.

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Disertacijos tyrimo objektas yra periodinė dzeta funkcija. Mokslinė problema - šios funkcijos momentų problema. Darbo tikslas - įrodyti asimptotines formules periodinės funkcijos momentams bei kai kuriems objektams, susijusiems su šios funkcijos momentais. Darbo uždaviniai yra šie: 1. Įrodyti Atkinsono tipo formulę su korektišku liekamuoju nariu kritinėje juostoje periodinei dzeta funkcijai su racionaliuoju parametru. 2. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje tiesėje vidurkio formulę liekamojo nario modulio kvadratui. 3. Įrodyti Atkinsono tipo formulės periodinei dzeta funkcijai kritinėje juostoje vidurkio formulę liekamojo nario modulio kvadratui. 4. Gauti asimptotinę formulę periodinės dzeta funkcijos ketvirtajam momentui.
In the thesis, problems related to the moments of the periodic zeta-function are considered. The aim of the thesis is to obtain asymptotic formulae for some analytic objects related to the periodic zeta-function. The problems are the following: 1. To prove the Atkinson-type formula with a new error term in the critical strip for the periodic zeta-function with rational parameter. 2. To prove a mean square formula for the error term in the Atkinson-type formula on the critical line for the periodic zeta-function. 3. To prove a mean square formula for the error term in the Atkinson-type formula in the critical strip for the periodic zeta-function. 4. To obtain an asymptotic formula for the fourth power moment of the periodic zeta-function.
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35

Wu, Dongsheng. "Eigenvalues of Differential Operators and Nontrivial Zeros of L-functions." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8729.

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The Hilbert-P\'olya conjecture asserts that the non-trivial zeros of the Riemann zeta function $\zeta(s)$ correspond (in a certain canonical way) to the eigenvalues of some positive operator. R. Meyer constructed a differential operator $D_-$ acting on a function space $\H$ and showed that the eigenvalues of the adjoint of $D_-$ are exactly the nontrivial zeros of $\zeta(s)$ with multiplicity correspondence. We follow Meyer's construction with a slight modification. Specifically, we define two function spaces $\H_\cap$ and $\H_-$ on $(0,\infty)$ and characterize them via the Mellin transform. This allows us to show that $Z\H_\cap\subseteq\H_-$ where $Zf(x)=\sum_{n=1}^\infty f(nx)$. Also, the differential operator $D$ given by $Df(x)=-xf'(x)$ induces an operator $D_-$ on the quotient space $\H=\H_-/Z\H_\cap$. We show that the eigenvalues of $D_-$ on $\H$ are exactly the nontrivial zeros of $\zeta(s)$. Moreover, the geometric multiplicity of each eigenvalue is one and the algebraic multiplicity of each eigenvalue is its vanishing order as a nontrivial zero of $\zeta(s)$. We generalize our construction on the Riemann zeta function to some $L$-functions, including the Dirichlet $L$-functions and $L$-functions associated with newforms in $\mathcal S_k(\Gamma_0(M))$ with $M\ge1$ and $k$ being a positive even integer. We give spectral interpretations for these $L$-functions in a similar fashion.
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36

Tamašauskaitė, Ugnė. "Sudėtinės funkcijos universalumas." Bachelor's thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2013~D_20130730_105103-87570.

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37

Mendes, Fernando Vasconcelos. "Quantum gate teleportation, universal entanglers and connections with the number theory." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13647.

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The present thesis can be divided in three parts: 1) Quantum gate teleportation; 2) Numerical search of universal entanglers; 3) Connections between quantum information and number theory. Regarding the quantum gate teleportation, a separability criterion of normal matrices is used to find the analytical conditions of the preservation of separability under conjugation. That analytical condition allowed to find the general formula of an element of $mathbb{C}^{4}$ Clifford group, as well to understand the role of the basis of measurement in the quantum gate teleportation protocol. Considering the searching for universal entanglers, the same separability criterion of normal matrices was used as fitness function in a computational heuristics, in prder to find good candidates for universal entanglers in $mathbb{C}^{3} otimes mathbb{C}^{4}$ and $mathbb{C}^{4} otimes mathbb{C}^{4}$ Hilbert spaces. At last, in the connection of quantum information with the number theory, it is presented the study of the preparation and entanglement of several multi-qubit quantum states based in integer sequences, and the Riemannian quantum circuit, a quantum circuit whose eigenvalues are related to the zeros of the Riemann zeta function. The existence of such circuit proves that is always possible to construct a physical system related to a finite amount of zeros.
A presente tese està dividida em trÃs partes: 1) TeleportaÃÃo de portas quÃnticas; 2) Busca numÃrica por entrelaÃadores universais; 3) ConexÃes entre a informaÃÃo quÃntica e a teoria dos nÃmeros. No que diz a teleportaÃÃo de portas quÃnticas, um critÃrio de separabilidade para matrizes normais à usada para encontrar as condiÃÃes analÃticas da preservaÃÃo da separabilidade sob conjugaÃÃo. Tais condiÃÃes analÃticas permitiram encontrar a forma geral de um elemento do grupo de Clifford em $mathbb{C}^{4}$, assim como tambÃm entender o papel da base de mediÃÃo no protocolo de teleportaÃÃo de portas quÃnticas. Considerando a busca por entrelaÃadores universais, o mesmo critÃrio de separabilidade de matrizes normais foi utilizado como funÃÃo de aptidÃo em uma heurÃstica computacional aplicada para encontrar bons candidatos a entrelaÃadores universais nos espaÃos de Hilbert de dimensÃes $mathbb{C}^{3} otimes mathbb{C}^{4}$ e $mathbb{C}^{4} otimes mathbb{C}^{4}$. Por fim, sobre as conexÃes da informaÃÃo quÃntica com a teoria dos nÃmeros, à apresentado um estudo da preparaÃÃo e entrelaÃamento de vÃrios estados quÃnticos de mÃltiplos qubits baseados em sequÃncias de nÃmeros inteiros. Apresenta-se ainda o circuito quÃntico Riemanniano, um circuito quÃntico cujos autovalores sÃo relacionados aos zeros da funÃÃo Zeta de Riemann. A existÃncia deste circuito prova que à sempre possÃvel construir um sistema fÃsico relacionado a uma quantidade finita de zeros.
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Coatney, Ryan D. "Mean Square Estimate for Primitive Lattice Points in Convex Planar Domains." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/2501.

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The Gauss circle problem in classical number theory concerns the estimation of N(x) = { (m1;m2) in ZxZ : m1^2 + m2^2 <= x }, the number of integer lattice points inside a circle of radius sqrt(x). Gauss showed that P(x) = N(x)- pi * x satisfi es P(x) = O(sqrt(x)). Later Hardy and Landau independently proved that P(x) = Omega_(x1=4(log x)1=4). It is conjectured that inf{e in R : P(x) = O(x^e )}= 1/4. I. K atai showed that the integral from 0 to X of |P(x)|^2 dx = X^(3/2) + O(X(logX)^2). Similar results to those of the circle have been obtained for regions D in R^2 which contain the origin and whose boundary dD satis fies suff cient smoothness conditions. Denote by P_D(x) the similar error term to P(x) only for the domain D. W. G. Nowak showed that, under appropriate conditions on dD, P_D(x) = Omega_(x1=4(log x)1=4) and that the integral from 0 to X of |P_D(x)|^2 dx = O(X^(3/2)). A result similar to Nowak's mean square estimate is given in the case where only "primitive" lattice points, {(m1;m2) in Z^2 : gcd(m1;m2) = 1 }, are counted in a region D, on assumption of the Riemann Hypothesis.
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39

Remeikaitė, Solveiga. "Ribinė teorema Rymano dzeta funkcijos Melino transformacijai." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20110802_165253-48345.

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Darbe pateikta funkcijų tyrimo apžvalga, svarbiausi žinomi rezultatai, suformuluota problema. Pagrindinė ribinė teorema įrodoma, taikant tikimybinius metodus, analizinių funkcijų savybes, aproksimavimo absoliučiai konvertuojančiu integralu principą.
The main limit theorem is proved using probabilistic methods, the analytical functions of the properties.
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40

Yamagishi, Shuntaro. "Moment Polynomials for the Riemann Zeta Function." Thesis, 2009. http://hdl.handle.net/10012/4218.

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In this thesis we calculated the coefficients of moment polynomials of the Riemann zeta function for k= 4, 5, 6...13 using cubic acceleration, which is an improved method from quadratic acceleration. We then numerically verified the moment conjectures. The results we obtained appear to support the conjectures. We also present a brief history of the moment polynomials by illustrating some of the important results of the field along with proofs for two of the classic results. The heuristics to find the integral moments of the Riemann zeta function is described in this thesis as well.
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41

Christ, Thomas. "Value-distribution of the Riemann zeta-function and related functions near the critical line." Doctoral thesis, 2013. https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-97763.

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The Riemann zeta-function forms a central object in multiplicative number theory; its value-distribution encodes deep arithmetic properties of the prime numbers. Here, a crucial role is assigned to the analytic behavior of the zeta-function on the so called critical line. In this thesis we study the value-distribution of the Riemann zeta-function near and on the critical line. Amongst others we focus on the following. PART I: A modified concept of universality, a-points near the critical line and a denseness conjecture attributed to Ramachandra. The critical line is a natural boundary of the Voronin-type universality property of the Riemann zeta-function. We modify Voronin's concept by adding a scaling factor to the vertical shifts that appear in Voronin's universality theorem and investigate whether this modified concept is appropriate to keep up a certain universality property of the Riemann zeta-function near and on the critical line. It turns out that it is mainly the functional equation of the Riemann zeta-function that restricts the set of functions which can be approximated by this modified concept around the critical line. Levinson showed that almost all a-points of the Riemann zeta-function lie in a certain funnel-shaped region around the critical line. We complement Levinson's result: Relying on arguments of the theory of normal families and the notion of filling discs, we detect a-points in this region which are very close to the critical line. According to a folklore conjecture (often attributed to Ramachandra) one expects that the values of the Riemann zeta-function on the critical line lie dense in the complex numbers. We show that there are certain curves which approach the critical line asymptotically and have the property that the values of the zeta-function on these curves are dense in the complex numbers. Many of our results in part I are independent of the Euler product representation of the Riemann zeta-function and apply for meromorphic functions that satisfy a Riemann-type functional equation in general. PART II: Discrete and continuous moments. The Lindelöf hypothesis deals with the growth behavior of the Riemann zeta-function on the critical line. Due to classical works by Hardy and Littlewood, the Lindelöf hypothesis can be reformulated in terms of power moments to the right of the critical line. Tanaka showed recently that the expected asymptotic formulas for these power moments are true in a certain measure-theoretical sense; roughly speaking he omits a set of Banach density zero from the path of integration of these moments. We provide a discrete and integrated version of Tanaka's result and extend it to a large class of Dirichlet series connected to the Riemann zeta-function
Die Riemannsche Zetafunktion ist ein zentraler Gegenstand der multiplikativen Zahlentheorie; in ihrer Werteverteilung liegen wichtige arithmetische Eigenschaften der Primzahlen kodiert. Besondere Bedeutung kommt hierbei dem analytischen Verhalten der Zetafunktion auf der sog. kritischen Geraden zu. Wir untersuchen in dieser Arbeit die Werteverteilung der Riemannschen Zetafunktion auf und nahe der kritischen Geraden. Wir fokusieren wir uns dabei u.a. auf folgende Punkte. TEIL I: Ein modifiziertes Universalitätskonzept, a-Stellen nahe der kritischen Geraden und eine Dichtheitsvermutung nach Ramachandra. Die kritische Gerade fungiert als natürliche Grenze für die Voroninsche Universalitätseigenschaft der Riemannschen Zetafunktion. Wir modifizieren Voronins Universalitätskonzept dahingehend, dass wir die vertikalen Translationen aus Voronins Universalitätssatz mit einer zusätzlichen Skalierung versehen. Wir untersuchen, ob durch dieses modifizierte Konzept eine abgeschwächte Universalitätseigenschaft der Riemannschen Zetafunktion um die kritschen Gerade aufrecht erhalten werden kann. Es stellt sich heraus, dass die Gestalt der Funktionen, die sich auf diese Weise durch die Zetafunktion approximieren lassen, stark von der Funktionalgleichung und der Wahl des skalierenden Faktors abhängt. Nach einem Resultat von Levinson liegen fast alle a-Stellen der Riemannschen Zetafunktion in einem trichterförmigen Bereich um die kritische Gerade. Gewisse Normalitätsargumenten sowie das Konzept der 'filling discs' erlauben uns Levinsons Resultat zu ergänzen und a-Stellen in diesem trichterförmigen Bereich aufzuspüren, die sehr nahe an der kritischen Geraden liegen. Man vermutet, dass die Werte der Riemannschen Zetafunktion auf der kritischen Geraden dicht in den komplexen Zahlen liegen. Wir nähern uns dieser Vermutung (die man oft Ramachandra zuschreibt), indem wir die Existenz gewisser Kurven nachweisen, die sich asymptotisch an die kritische Gerade anschmiegen und die Eigenschaft besitzen, dass die Werte der Zetafunktion auf diesen Kurven dicht in den komplexen Zahlen liegen. Viele unserer Ergebnisse in Teil I sind unabhängig von der Eulerproduktdarstellung der Zetafunktion und gelten allgemein für beliebige meromorphe Funktionen, die einer Funktionalgleichung vom Riemann-Typ genügen. TEIL II: Diskrete und kontinuierliche Momente. Die Lindelöf Vermutung trifft eine Aussage über das Wachstumsverhalten der Zetafunktion auf der kritischen Geraden. Nach klassischen Arbeiten von Hardy und Littlewood lässt sie sich mittels Potenzmomente der Zetafunktion rechts von der kritischen Geraden umformulieren. Tanaka konnte kürzlich nachweisen, dass die asymptotischen Formeln, die man für diese Potenzmomente erwartet in einem gewissen maßtheoretischem Sinne Gülitgkeit besitzen: grob gesprochen wird heibei eine Menge mit Banachdichte null vom Integrationsweg der Potenzmomente ausgespart. Wir stellen eine diskrete und eine integrierte Version von Tanakas Resultat zur Verfügung. Zudem verallgemeinern wir Tanakas Ergebnis auf eine große Klasse von Dirichletreihen
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42

"Theory of the generalized modified Bessel function K_{z,w}(x) and 2-adic valuations of integer sequences." Tulane University, 2017.

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acase@tulane.edu
Modular-type transformation formulas are the identities that are invariant under the transformation α → 1/α, and they can be represented as F (α) = F (β) where α β = 1. We derive a new transformation formula of the form F (α, z, w) = F (β, z, iw) that is a one-variable generalization of the well-known Ramanujan-Guinand identity of the form F (α, z) = F (β, z) and a two-variable generalization of Koshliakov’s formula of the form F (α) = F (β) where α β = 1. The formula is generated by first finding an integral J that is comprised of an invariance function Z and evaluating the integral to give F (α, z, w) mentioned above. The modified Bessel function K z (x) appearing in Ramanujan-Guinand identity is generalized to a new function, denoted as K z,w (x), that yields a pair of functions reciprocal in the Koshliakov kernel, which in turn yields the invariance function Z and hence the integral J and the new formula. The special function K z,w (x), first defined as the inverse Mellin transform of a product of two gamma functions and two confluent hypergeometric functions, is shown to exhibit a rich theory as evidenced by a number of integral and series representations as well as a differential-difference equation. The second topic of the thesis is 2-adic valuations of integer sequences associated with quadratic polynomials of the form x 2 +a. The sequence {n 2 +a : n ∈ Z} contains numbers divisible by any power of 2 if and only if a is of the form 4 m (8l+7). Applying this result to the sequences derived from the sums of four or fewer squares when one or more of the squares are kept constant leads to interesting results, that also points to an inherent connection with the functions r k (n) that count the number of ways to represent n as sums of k integer squares. Another class of sequences studied is the shifted sequences of the polygonal numbers given by the quadratic formula, for which the most common examples are the triangular numbers and the squares.
1
Aashita Kesarwani
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43

Huang, Chun-Yueh, and 黃駿岳. "Riemann Zeta Function Model for Online Data Change Detection." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/89095098996029232892.

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44

Pugh, Glendon Ralph. "The Riemann-Siegel formula and large scale computations of the Riemann zeta function." Thesis, 1998. http://hdl.handle.net/2429/9023.

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This thesis is a survey of the derivation and implementation of the Riemann-Siegel formula for computing values of Riemann's zeta function on the line s = 1/2 + it. The formula, devised by Riemann and later published by Siegel following study of Riemann's unpublished work, is the method of choice for both numerically verifying the Riemann Hypothesis and locating zeros on the critical line at large values of t. Simply stated, the Riemann Hypothesis is that all of the zeros of ζ(s) in the strip 0 < R(s) < 1 lie on the line R(s) = 1/2. Since Riemann made his conjecture in 1859, much work has been done towards developing efficient numerical techniques for verifying the hypothesis and possibly finding counter-examples. This thesis is meant to serve as a guide book for using Riemann-Siegel. It is mostly a distillation of work done in the field since Siegel's results published in the early 1930's. Computer programs and examples are included, and error bounds are discussed. The question of how and why Riemann-Siegel is used to verify the Riemann Hypothesis is examined, and a detailed Riemann Hypothesis verification example is illustrated. Finally, recent work in the field is noted. The derivation of the Riemann-Siegel formula for computing ζ(l/2 + it) is based on the saddle point method of evaluating integrals, and yields results of considerable accuracy in time t1/2. The saddle point method is an approximation technique which concentrates the "bulk" of an integral on a path through a point at which the modulus of the integrand is a maximum. [Mathematical formulae used in this abstract could not be reproduced.]
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45

Steuding, Jörn [Verfasser]. "On simple zeros of the Riemann zeta-function / von Jörn Steuding." 1999. http://d-nb.info/95589820X/34.

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46

Liu, Chih Shiuan, and 劉志璿. "The connection between the functions of Riemann zeta and Bernoulli Number." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/17154599310613619902.

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碩士
國立臺中教育大學
數學教育學系
96
This research hung over from the extended functions for the sum of powers of consecutive integers, we colleted the literatures of the related research about the functions of Riemann zeta and Bernoulli Number, both newly interpreted and predigested the properties of the functions of Riemann zeta and Bernoulli Number. Thus we built the connection between the functions of Riemann zeta and Bernoulli Number, according to \zeta(2 k)=(-1)^{k-1} 2^{2k-1} \frac{B_{2k} \pi^{2k}}{(2k)!}, \ k \in \mathbb{N},and S_{2k}^{\prime}(-1)=\frac{(-1)^{k-1} (2k)!}{2^{2k-1} (\pi)^{2k}}\zeta(2k), S_{2k+1}^{\prime}(-1)=0,Take the function of Riemann zeta as bridge, we find that S_{2k}^{\prime}(-1)=B_{2k},B_{2k}=\frac{1}{2k+1} \left \{ C_{2k}^{2k+1} S_{1}^{\prime}(-1)+ \sum_{i=1}^{k} C_{2i+1}^{2k+1} S_{2k-2i}^{\prime}(-1) \right \},where $S_k^{\prime}(x)$ denotes the first derivative of $S_k(x)$ for each positive integer $k$.
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47

若狭, 尊裕, and Takahiro Wakasa. "The explicit estimation for the argument of the Riemann zeta function on the critical line." Thesis, 2014. http://hdl.handle.net/2237/19981.

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48

Bunn, Jared Ross. "Nikolski's approach to the theorems of Beurling and Nyman regarding zeros of the Riemann [zeta]-function." 2006. http://etd.utk.edu/2006/BunnJared.pdf.

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49

Menz, Petra Margarete. "An algorithm for computing the riemann zeta function based on an analysis of Backlund’s remainder estimate." Thesis, 1994. http://hdl.handle.net/2429/5448.

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The Riemann zeta function, Ϛ(s) with complex argument s, is a widely used special function in mathematics. This thesis is motivated by the need of a cost reducing algorithm for the computation of Ϛ (s) using its Euler-Maclaurin series. The difficulty lies in finding small upper bounds, call them n and k, for the two sums in the Euler-Maclaurin series of Ϛ (s) which will compute Ϛ (s) to within any given accuracy for any complex argument s, and provide optimal computational cost in the use of the Euler-Maclaurin series. This work is based on Backlund’s remainder estimate for the Euler-Maclaurin remain- der, since it provides a close enough relationship between n, k, s, and е. We assumed that the cost of computing the Bernoulli numbers, which appear in the series, is fixed, and briefly discuss how this may influence high precision calculation. Based on our study of the behavior of Backlund’s remainder estimate, we define the ‘best’ pair (n, k), and present a reliable method of computing the best pair. Furthermore, based on a compu- tational analysis, we conjecture that there is a relationship between n and k which does not depend on s. We present two algorithms, one based on our method and the other on the conjecture, and compare their costs of finding n and k as well as computing the Euler-Maclaurin series with an algorithm presented by Cohen and Olivier. We conclude that our algorithm reduces the cost of computing Ϛ(s) drastically, and that good numerical techniques need to be applied to our method and conjecture for finding n and k in order to keep this computational cost low as well.
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50

Ouimet, Frédéric. "Extremes of log-correlated random fields and the Riemann zeta function, and some asymptotic results for various estimators in statistics." Thèse, 2019. http://hdl.handle.net/1866/22667.

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