Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Topic: Dirichlet L-series

    Note: these results are not from the primary (high quality) database.

Dirichlet character - Wikipedia, the free encyclopedia Dirichlet characters and their L-series were introduced by Dirichlet, in 1831, in order to prove Dirichlet's theorem about the infinitude of primes in arithmetic progressions. Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis. A Dirichlet L-series can be expressed as a linear combination of the Hurwitz zeta function, and thus the study of L-series can be unified through a study of the Hurwitz zeta. en.wikipedia.org /wiki/Dirichlet_character   (333 words)

Series (mathematics) - Wikipedia, the free encyclopedia Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète. Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions. en.wikipedia.org /wiki/Series_(mathematics)   (1682 words)

Dirichlet series -- Facts, Info, and Encyclopedia article In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the (Click link for more info and facts about radius of convergence) radius of convergence for (The sum of terms containing successively higher integral powers of a variable) power series. The Dirichlet series case is more complicated, though: (Click link for more info and facts about absolute convergence) absolute convergence and (Click link for more info and facts about uniform convergence) uniform convergence may occur in distinct half-planes. www.absoluteastronomy.com /encyclopedia/d/di/dirichlet_series.htm   (432 words)

PlanetMath: Dirichlet series This is version 4 of Dirichlet series, born on 2003-10-09, modified 2004-02-22. The following are the basic convergence properties of Dirichlet series. , which is called the abscissa of convergence of the Dirichlet series. planetmath.org /encyclopedia/DirichletSeries.html   (275 words)

Search Results for Series Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the Sun caused by the perturbing body. Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius. www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Series&CONTEXT=1   (15235 words)

Meandering Convergence of a Dirichlet Series This leads to the interesting "Dirichlet" series pi 1 1 1 1 1 1 1 1 1 1 -- = 1 + - - - + - + - + -- - -- - -- - -- + -- + -- +... In a previous note we discussed how Leibniz might have anticipated Euler's summation of the inverse square integers by factoring the product form of the arctan(1) series. If we let s(x) denote the partial sum of this series up to 2j+1 www.mathpages.com /home/kmath483.htm   (256 words)

Analytic Number Theory This is called a Dirichlet series with coefficients f(n), and the function F(s) is called a generating function of the coefficients. Dirichlet used arithmetic progressions to show that every arithmetic progression kn+h, where h and k are relatively prime, contains infinitely many primes. The Dirichlet product is seen by multiplying the two Dirichlet series and rearranging terms, as follows: www.risberg.ws /Hypertextbooks/Mathematics/Numbers/analytic.htm   (724 words)

Connexions - Content The Dirichlet conditions are the sufficient conditions to guarantee existence and convergence of the Fourier series or the Fourier transform. The Dirichlet conditions suggest that discontinuous signals may have a Fourier Series representation so long as there are a finite number of discontinuities. La condición de Dirichlet sugiere que las señales discontinuas pueden tener una representación de series de Fourier mientras existan un número finito de discontinuidades. cnx.rice.edu /content/similarity?objectId=m12130   (418 words)

PlanetMath: Dirichlet L-series This is version 8 of Dirichlet L-series, born on 2003-01-20, modified 2004-07-29. Cross-references: complex, Dirichlet's theorem on primes in arithmetic progression, poles, generalized Bernoulli number, integer, analytic, meromorphic, Gauss sum, gamma function, functional equation, symmetric, analytic continuation, primitive character, conductor, Riemann zeta function, trivial character, primes, identity, Euler product, positive, domain, converges absolutely, series, Dirichlet character to prove his famous Dirichlet's theorem on primes in arithmetic progression. planetmath.org /encyclopedia/DirichletLSeries.html   (169 words)

abstract The Dirichlet series consists of operators on the graded ring of adelic theta functions and can be expressed in terms of the eigenvalues of the Eisenstein series by a result of Shintani. Each term is a product of (1) an explicit monomial of Hecke and Dirichlet L-series for the specified character and weight, and (2) a theta function obtained by applying formal Dirichlet series of operators to the indexed adelic theta function. The formula is deduced from an expansion of the Eisenstein series which is closely related to the Fourier-Jacobi expansion and which has the form of an infinite sum of terms indexex by primitive adelic theta function. www.cs.brandeis.edu /~tim/Papers/Dissertation/abstract   (255 words)

Abstract from Pacific Journal of Mathematics - 208-1-7 - Haakan Hedenmalm and Eero Saksman To each Dirichlet series of the above type corresponds a ``trigonometric'' series $\sum_{n=1}^\infty a_n \chi(n)$, where $\chi$ is a multiplicative character from the positive integers to the unit circle. A Hilbert space of Dirichlet series is obtained by considering the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{-s}$ that satisfy $\sum_{n=0}^\inftya_n^2<+\infty$. An analog of Carleson's celebrated convergence theorem is obtained: Each such Dirichlet series converges almost everywhere on the critical line $\re s=\frac12$. nyjm.albany.edu:8000 /PacJ/2003/208-1-7nf.htm   (146 words)

Math 506 Extra Credit This is the Dirichlet series for the constant sequence 1. In number theory, the typical example is to construct a Dirichlet series. If you have a function defined on the integers (a sequence), then it can be useful to use these values as the coefficients of a series and use techniques from calculus to study the properties of the resulting function. www.math.ksu.edu /math506/m506xc2.html   (391 words)

dszweb2 We will also need the fact that functions defined by Dirichlet integrals, under the conditions of 1.7.6, are holomorphic (though of course this is now clear when the Dirichlet integral is obtained from a Dirichlet series as in 1.7.5). The last result shows that every Dirichlet series is essentially a case of a Dirichlet integral. We now formulate a result on convergence for Dirichlet integrals, and derive one for Dirichlet series. www.maths.lancs.ac.uk /~jameson/dszweb   (1037 words)

L-functions and elliptic curves And it is but a special case in the "Langlands program", which conjectures that members of a much broader class of Dirichlet series are meromorphic and have a functional equation. Throughout number theory and related fields, such as algebraic geometry and the theory of automorphic functions, many generalizations of the zeta function and Dirichlet series have been found useful. For many such series of interest, the corresponding function can be extended to a meromorphic function on the whole plane by a process known as "analytic continuation". www.mbay.net /~cgd/flt/flt06.htm   (2077 words)

chap15.html The study of Dirichlet series leads naturally to a proof of Dirichlet's theorem on the infinitude of primes in arithmetic progressions. The Dirichlet series are introduced to explore the multiplicative properties of these functions. Dirichlet proved that there are infinitely many primes in every arithmetic progression of the form www.math.columbia.edu /~rama/chapters/chap15.html   (194 words)

Dirichlet series analogues of q-series This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The basic hypergeometric series, whilst widely used in mathematics, is itself an analogue for the ordinary hypergeometric series developed by Gauss in the early nineteenth century. We present a variety of results that are demonstrably analogues of many classical formulae from the theories of ordinary and basic hypergeometric series. wwwmaths.anu.edu.au /research.reports/mrr/00.014   (94 words)

Atlas: The universality of Dirichlet series by Antanas Laurincikas In the report the universality of Dirichlet series of certain cusp forms, the joint universality of Lerch zeta-functions and the universality of Dirichlet series attached to finite Abelian groups will be discussed. The universality of given Dirichlet series means that any analytic function can be uniformly approximated on some sets by translations of that Dirichlet series. There exists the Linnik-Ibragimov conjecture that all Dirichlet series are universal. atlas-conferences.com /cgi-bin/abstract/cacf-21   (142 words)

INI : Abstracts : RMAW05 : Multiple Dirichlet series, an historical survey Multiple Dirichlet series (L-functions of several complex variables) are Dirichlet series in one complex variable whose coefficients are again Dirichlet series in other complex variables. It was found recently by Diaconu-Goldfeld-Hoffstein that the moment conjectures of random matrix theory, such as the Keating-Snaith conjecture, would follow if certain multiple Dirichlet series had meromorphic continuation to a a particular tube domain. These series arise naturally in the theory of moments of zeta and L-functions. www.newton.cam.ac.uk /programmes/RMA/Abstracts5/goldfeld.html   (146 words)

On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip -- from Mathematica Information Center On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip We describe computations which show that each of the first 12069 zeros of the Ramanujan tau-Dirichlet series of the form s + it in the region 0 On the Zeros of the Ramanujan tau-Dirichlet Series in the Critical Strip -- from Mathematica Information Center library.wolfram.com /infocenter/Articles/3589   (56 words)

CoLab Document Server - On Dirichlet series for sums of squares Borwein, Jonathan M. and Choi, Stephen (2002) On Dirichlet series for sums of squares. Dirichlet series, sum of squares, closed forms, binary quadratic forms, disjoint discriminants, L-functions In Theorem \ref{thm 2.1} below, we are able to generalize the above result and prove that if $f_i$ and $g_i$ are completely multiplicative, then we have\[\sum_{n=1}^\infty \frac{(f_1\ast g_1)(n)\cdot (f_2\ast g_2)(n)}{n^s}=\frac{L_{f_1f_2}(s)L_{g_1g_2}(s)L_{f_1g_2}(s)L_{g_1f_2}(s)}{L_{f_1f_2g_1g_2}(2s)}\] where $L_f(s):=\sum_{n=1}^\infty f(n)n^{-s}$ is the Dirichlet series corresponding to $f$. eprints.cecm.sfu.ca /archive/00000142   (215 words)

The first zeros on the critical line of some Dirichlet L-series This page presents the results of my initial efforts to compute the first zeros, on the critical line, of some Dirichlet L-series. Tables with approximate values of the first zeros of some primitive Dirichlet L-series, Robertson, Some properties of Dirichlet L-series, Journal of Physics A: Mathematical and General, vol. www.ieeta.pt /%7Etos/zeta.html   (517 words)

Dirichlet Convergence Test As we shall see, this is a generalization of the alternating series. Let s be a series of complex numbers, its partial sums bounded by the radius r. If t is itself a convergent series, such as a geometric series, it is absolutely convergent, its terms being positive. www.mathreference.com /lc-ser,diri.html   (497 words)

DC MetaData for:An algorithm for nonharmonic signal analysis using Dirichlet series on convex polygons This article presents a new algorithm for nonharmonic signal analysis using Dirichlet series $$ f(z) = \sum_{\lambda\in\Lambda} \kappa_{f}(\lambda)\frac{e^{\lambda z}}{L^{\prime}(\lambda)}, \quad z\in D, $$ on a convex polygon $D$ as a generalization of Fourier series. An algorithm for nonharmonic signal analysis using Dirichlet series on convex polygons DC MetaData for:An algorithm for nonharmonic signal analysis using Dirichlet series on convex polygons ibb.gsf.de /preprints/2001/pp01-21.html   (135 words)

The composition operators on the space of Dirichlet series with square summable coefficients., Julia Gordon, Håkan Hedenmalm The composition operators on the space of Dirichlet series with square summable coefficients. The composition operators on the space of Dirichlet series with square summable coefficients., Julia Gordon, Håkan Hedenmalm projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.mmj/1030132413   (38 words)

AIF : Tome 31 fascicle 4 -- 1981 Arithmetic applications of the study of the values at negative integers of Dirichlet series associated with a polynomial in some cases, where these series can be continued to annalif.ujf-grenoble.fr /Vol31/E314_1/E314_1.html   (62 words)

Dirichlet series related D analogues of the q gamma and q beta functions Dirichlet series related D analogues of the q gamma and q beta functions Abstract Recent papers by the author have shown that new Dirichlet summations are obtainable from known basic hypergeometric series identities. A natural extension of these ideas is in the study of D-analogues for the q-gamma and q-beta functions. www.math.anu.edu.au /research.reports/mrr/96.019   (95 words)

Geoff Campbell's Mathematics Homepage which enable me to write down new Dirichlet series summation formulas which are demonstrably analogues for known basic hypergeometric series summations. Professor Askey is an expert on Hypergeometric Series as they apply to the theory of Special Functions. Professor Andrews, of Pennsylvania State University, is a leading figure among Number Theorists and has a wealth of knowledge of q-series and their applications. www.geocities.com /CapeCanaveral/Launchpad/9416   (551 words)

Random Dirichlet functions This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip_{1/2} of functions obeying a Lipschitz condition with exponent 1/2. We show that if one inserts random plus and minus signs before the coefficients of the power series of a function in the Dirichlet space, then the resulting series is almost surely a multiplier of the Dirichlet space. However, unlike the lacunary situation, no corresponding ``almost sure'' Lipschitz result is possible for random series: we exhibit a Dirichlet function with no randomization in Lip_{1/2}. www.mth.msu.edu /~shapiro/Pubvit/Downloads/RanDir/RanDir.html   (140 words)

Powell's Books - Modular Functions & Dirichlet Series 2ND Edition by Tom D Apostol The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. Powell's Books - Modular Functions and Dirichlet Series 2ND Edition by Tom D Apostol Among the major topics covered are Rademacher's convergent series for the partition modular function, Lehner's congruences for the Fourier coefficients of the modular function j, and Hecke's theory of entire forms with multiplicative Fourier coefficients. www.powells.com /biblio?isbn=0387971270   (508 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.