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# Topic: Dirichlet series

###### In the News (Fri 24 May 13)

 Dirichlet character - Wikipedia, the free encyclopedia 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. 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. en.wikipedia.org /wiki/Dirichlet_character   (333 words)

 Series (mathematics) - Wikipedia, the free encyclopedia 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. 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. 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   (Site not responding. Last check: 2007-10-21) Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845. Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties. 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. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dirichlet.html   (2043 words)

 Search Results for Series 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. 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. In a series of articles he generalized results in the classical Fourier theory to more general spaces (Hilbert spaces) by investigating various orthonormal systems of functions, their summability and the representation of functions in such systems by Fourier series and Fourier integrals and more general integral transformations. www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Series&CONTEXT=1   (15235 words)

 Search Results for Dirichlet Many details of the Dirichlet family are given in [6] where it is shown that the Dirichlets came from the neighbourhood of Liege in Belgium and not, as many had claimed, from France. Dirichlet was appointed to the Berlin Academy in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn's two sisters. In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Gottingen on 30 July. www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Dirichlet&CONTEXT=1   (5322 words)

 PlanetMath: Dirichlet series The following are the basic convergence properties of Dirichlet series. There is nothing profound about their proofs, which can be found in [1] and in various other works on complex analysis and analytic number theory. This is version 4 of Dirichlet series, born on 2003-10-09, modified 2004-02-22. planetmath.org /encyclopedia/DirichletSeries.html   (275 words)

 PlanetMath: Dirichlet L-series This is probably the first instance of using complex analysis to prove a purely number theoretic result. 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 This is version 8 of Dirichlet L-series, born on 2003-01-20, modified 2004-07-29. planetmath.org /encyclopedia/DirichletLSeries.html   (169 words)

 L-functions and elliptic curves 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". Dirichlet functions are of great importance in number theory because of the following formal property. 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. www.mbay.net /~cgd/flt/flt06.htm   (2077 words)

 Connexions - Content   (Site not responding. Last check: 2007-10-21) The Fourier Series is the representation of continuous-time, periodic signals in terms of complex exponentials. The Dirichlet conditions suggest that discontinuous signals may have a Fourier Series representation so long as there are a finite number of discontinuities. The Dirichlet conditions are the sufficient conditions to guarantee existence and convergence of the Fourier series or the Fourier transform. cnx.rice.edu /content/similarity?objectId=m12130   (418 words)

 Abstract from Pacific Journal of Mathematics - 208-1-7 - Haakan Hedenmalm and Eero Saksman   (Site not responding. Last check: 2007-10-21) 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 $\sum_{n=0}^\inftya_n^2<+\infty$. These series converge in the half plane $\re s >\frac12$ and define a functions that are locally $L^2$ on the boundary $\re s=\frac12$. 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. nyjm.albany.edu:8000 /PacJ/2003/208-1-7nf.htm   (146 words)

 Meandering Convergence of a Dirichlet Series   (Site not responding. Last check: 2007-10-21) 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. This leads to the interesting "Dirichlet" series pi 1 1 1 1 1 1 1 1 1 1 -- = 1 + - - - + - + - + -- - -- - -- - -- + -- + -- +... For another example of this kind of "logarithmic wave" propogating through the natural numbers, see The Half-Totient Tree. www.mathpages.com /home/kmath483.htm   (256 words)

 Analytic Number Theory   (Site not responding. Last check: 2007-10-21) Dirichlet used arithmetic progressions to show that every arithmetic progression kn+h, where h and k are relatively prime, contains infinitely many primes. This is called a Dirichlet series with coefficients f(n), and the function F(s) is called a generating function of the coefficients. 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)

 dszweb2 The last result shows that every Dirichlet series is essentially a case of a Dirichlet integral. It remains to establish a very important property of functions defined by Dirichlet series, namely that they are holomorphic, i.e. 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). www.maths.lancs.ac.uk /~jameson/dszweb   (1037 words)

 [No title]   (Site not responding. Last check: 2007-10-21) The Dirichlet series are introduced to explore the multiplicative properties of these functions. The study of Dirichlet series leads naturally to a proof of Dirichlet's theorem on the infinitude of primes in arithmetic progressions. 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)

 [No title]   (Site not responding. Last check: 2007-10-21) ABSTRACT (text-based) The purpose of this dissertation is to develop an explicit formula for the Fourier-Jacobi coefficients of certain adelic Eisenstein series on the non-tube domain GU(2,1) by extending a technique used by Siegel, Baily, Tsao, and Karel in the context of rational tube domains. 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. 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. www.cs.brandeis.edu /~tim/Papers/Dissertation/abstract   (255 words)

 Dirichlet characters   (Site not responding. Last check: 2007-10-21) The first zeros on the critical line of some Dirichlet L-series... Johan Bosman, Lenny Taelman On sums of sums of values of Dirichlet characters NA... Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet charac... www.scienceoxygen.com /math/324.html   (140 words)

 The first zeros on the critical line of some Dirichlet L-series   (Site not responding. Last check: 2007-10-21) This page presents the results of my initial efforts to compute the first zeros, on the critical line, of some Dirichlet L-series. Riemann, in a path-breaking paper [2], conjectured that the non-trivial zeros of this function have real part equal to 1/2 (the critical line). It is conceivable, but unlikely, that one or more pairs of very close zeros may have escaped detection. www.ieeta.pt /%7Etos/zeta.html   (517 words)

 Math 506 Extra Credit   (Site not responding. Last check: 2007-10-21) 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. In number theory, the typical example is to construct a Dirichlet series. This is the Dirichlet series for the constant sequence 1. www.math.ksu.edu /math506/m506xc2.html   (391 words)

 INI : Abstracts : RMAW05 : Multiple Dirichlet series, an historical survey   (Site not responding. Last check: 2007-10-21) 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. These series arise naturally in the theory of moments of zeta and L-functions. 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. www.newton.cam.ac.uk /programmes/RMA/Abstracts5/goldfeld.html   (146 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)

 CoLab Document Server - On Dirichlet series for sums of squares   (Site not responding. Last check: 2007-10-21) Borwein, Jonathan M. and Choi, Stephen (2002) On Dirichlet series for sums of squares. One of the applications of Theorem \ref{thm 2.1} is to obtain closed forms, in terms of $\zeta (s)$ and Dirichlet $L$-functions, for the generating functions of $r_N(n), r_N(n)^2, r_{2,P}(n)$ and $r_{2,P}(n)^2$. Dirichlet series, sum of squares, closed forms, binary quadratic forms, disjoint discriminants, L-functions eprints.cecm.sfu.ca /archive/00000142   (215 words)

 Atlas: The universality of Dirichlet series by Antanas Laurincikas   (Site not responding. Last check: 2007-10-21) 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. 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. atlas-conferences.com /cgi-bin/abstract/cacf-21   (142 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 -- 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 library.wolfram.com /infocenter/Articles/3589   (56 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 Andrews, of Pennsylvania State University, is a leading figure among Number Theorists and has a wealth of knowledge of q-series and their applications. Professor Askey is an expert on Hypergeometric Series as they apply to the theory of Special Functions. www.geocities.com /CapeCanaveral/Launchpad/9416   (551 words)

 Dirichlet series inversion and prime number coutnign function... - Physics Help and Math Help - Physics Forums Dirichlet series inversion and prime number coutnign function... Seriously, take the time to learn latex and proper formatting. I think that the most important contribution is to give a formula to invert a dirichlet series by means of the formula www.physicsforums.com /showthread.php?t=38530   (1654 words)

 Amazon.com: Books: Modular Functions and Dirichlet Series in Number Theory (Graduate Texts in Mathematics)   (Site not responding. Last check: 2007-10-21) Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications. 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. In addition to the correction of misprints, minor changes in the exercises and an updated bibliography, this new edition includes an alternative treatment of the transformation formula for the Dedekind eta function, which appears as a five-page supplement to Chapter 3. www.amazon.com /exec/obidos/tg/detail/-/0387971270?v=glance   (761 words)

 Random Dirichlet functions   (Site not responding. Last check: 2007-10-21) 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. 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. 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)

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