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Linnik's theorem - Wikipedia, the free encyclopedia Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It is possible to bound L, and the following table shows the progress that has been made. It is known that L ≤ 2 for almost all integers d ([1] gives this as a consequence of results of Bombieri, Freidlander and Iwaniec). en.wikipedia.org /wiki/Linnik's_theorem   (322 words)

B.L. Julia - Statistical Theory of Numbers In 1989 we are accustomed to the occurrence of theta functions and modular forms in the quantum theory of strings and in the partition functions of statistical mechanics on a two-dimensional torus. Dirichlet L functions and other generalisations of the zeta function are then briefly described. The case of Dirichlet L functions is also interesting because of the absence of pole. www.maths.ex.ac.uk /~mwatkins/zeta/Julia.htm   (1204 words)

[No title] Whereas the study of the quantum mechanical saddle oscillator by means of periodic orbit theory is an interesting investigation, the similarity of Lorentzian smoothing in the density of states, respectively in the density of zeros for $\zeta(z)$, seems to be coincidental. Basic properties of Riemann's zeta-function and Dirichlet's $L$-functions are surveyed; it may be a bit misleading for a non-specialist that the zero-free region for the zeta-function (in Theorem 8.6) is not given in the sharpest known form. The method consists in evaluating Dirichlet series, and several of their derivatives, at a set of regularly spaced values; this is done by using the fast Fourier transform to reduce the problem to the evaluation of rational functions. www.math.niu.edu /~rusin/known-math/99/zeta   (3655 words)

function concept references G Ferraro, Functions, functional relations, and the laws of continuity in Euler, Historia Math. A Kopácková, Phylogenesis of the concept of a function (Czech), in Mathematics throughout the ages II (Czech), (Prometheus, Prague, 2001), 46-80. J Lützen, The development of the concept of function from Euler to Dirichlet. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Printref/Functions.html   (536 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 continuation, Gauss sum, gamma function, functional equation, symmetric, analytic continuation, primitive character, conductor, Riemann zeta function, trivial character, primes, product, 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   (180 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. Since this result demonstrates the functional equation for L(E,s), it verifies the Hasse-Weil conjecture when E is a modular curve (i. gyral.blackshell.com /flt/flt06.htm   (2077 words)

Catalogue of GP/PARI Functions: Arithmetic functions This function also allows vector and matrix arguments, in which case the operation is recursively applied to each component of the vector or matrix. The content of a rational function is the ratio of the contents of the numerator and the denominator. If x is of type integer, rational, polynomial or rational function, the result is a two-column matrix, the first column being the irreducibles dividing x (prime numbers or polynomials), and the second the exponents. pari.math.u-bordeaux.fr /dochtml/html/Arithmetic_functions.html   (5169 words)

Abstract from Pacific Journal of Mathematics - 207-2-8 - E. Kowalski and P. Michel   (Site not responding. Last check: 2007-10-31) For many $L$-functions of arithmetic interest, the values on or close to the edge of the region of absolute convergence are of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of $\zeta(s)$ for $\Reel(s)=1$). Other examples are the Dirichlet $L$-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square $L$-functions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upper-bound, on average, for the number of zeros of the $L$-functions which are very close to $1$. nyjm.albany.edu:8000 /PacJ/2002/207-2-8nf.htm   (153 words)

Publications Córdoba, A.; Fefferman, C.L. ; Seco, L. A number-theoretic estimate for the Thomas-Fermi density. Garunkshtis, R. ; Laurinchikas, A. Steuding, I. An approximate functional equation for the Lerch zeta function.(Russian) Mat. Steuding, J. Dirichlet series associated to periodic arithmetic functions and the zeros of Dirichlet $L$-functions. www.uam.es /gruposinv/ntatuam/publicationsesp.html   (1274 words)

4-dim HyperDiamond Lattice A Riemann zeta function is a function which is analytic in the complex plane, with the possible exception of a simple pole at one, and which is characterized by an Euler product and a functional identity. The Euler zeta function, the Dirichlet zeta functions, and the Hecke zeta functions are examples of Riemann zeta functions. A proof of the Riemann hypothesis for the zeta function of order v and character X for the adelic plane is obtained from the maximal dissipative transformations in the Sonine spaces of order v and character X for the adelic plane when the zeta function is analytic in the complex plane. www.valdostamuseum.org /hamsmith/PrimeFC.html   (5922 words)

PlanetMath: Dirichlet eta function , the Dirichlet eta function is defined as Cross-references: pole, Riemann zeta function, Dirichlet series, theorem, alternating series, converges, series This is version 4 of Dirichlet eta function, born on 2004-08-03, modified 2006-08-20. planetmath.org /encyclopedia/DirichletEtaFunction.html   (79 words)

[No title]   (Site not responding. Last check: 2007-10-31) A generalisation of Artin's conjecture for primitive roots Some results on the distribution of values of additive functions on the set of pairs of positive integers. Some formulas for the Riemann zeta function at odd integer argument resulting from Fourier expansions of the Epstein zeta function journals.impan.gov.pl /cgi-bin/shvold?aa29   (131 words)

Some Constants from Number theory In this section we are interested in the evaluation of the Riemann zeta prime function but this time the primes taken in account are of the form 4k+1 or of the form 4k+3. The proof is the same for the two identities and similar to the proof given in the first section for the Riemann zeta function. The Dirichlet function must here be replaced by two other similar functions. numbers.computation.free.fr /Constants/Miscellaneous/constantsNumTheory.html   (1589 words)

AMS Journals :: Print and Electronic Functional distribution of $L(s, \chi_d)$ with real characters and denseness of quadratic class numbers. A theorem on zeta functions associated with polynomials. Common zeros of theta functions and central Hecke L-values of CM number fields of degree 4. www.mathaware.org /joursearch/servlet/DoSearch?f1=msc&v1=11M06.   (208 words)

Matches for: He wrote the first accessible books on calculus, created the theory of circular functions, and discovered new areas of research such as elliptic integrals, the calculus of variations, graph theory, divergent series, and so on. It took hundreds of years for his successors to develop in full the theories he began, and some of his themes are still at the center of today's mathematics. His pioneering work on elliptic integrals is the precursor of the modern theory of abelian functions and abelian integrals. www.mathaware.org /bookstore?fn=20&arg1=whatsnew&item=EULER   (347 words)

zeta functions and L-functions The analogies between function fields and number fields had been known since Dedekind’s time (at least in characteristic zero), but Artin’s work was perhaps the first to take the base field to have positive characteristic as opposed to subfields of the complex numbers. Artin also (later) developed a quite general theory of L-functions which, once again by purely algebraic means, defined functions akin to the zeta function for general number fields and for function fields. Artin may thus be seen to have been working to ‘geometrize or at least "algebraicize") number theory’ while Weil was trying to ‘arithmetize geometry’, and Weil has remarked on the excitement with which he and his colleagues in those days awaited new numbers of the journals which regularly contained Artin’s work. www.maths.ex.ac.uk /~mwatkins/zeta/directoryofL-functions.htm   (1010 words)

Linear Algebra and its Applications B. Parlett A Recurrence Among the Elements of Functions of Triangular Matrices. Ronald L. Smith The Moore-Penrose inverse of a retrocirculant. 169--174 L. Kelly and L. Watson ${Q}$-matrices and spherical geometry 175--189 F. Giles and W. Pulleyblank Total dual integrality and integer polyhedra. www.math.utah.edu /ftp/pub/tex/bib/toc/linala1970.html   (2396 words)

11M: Zeta and L-functions: analytic theory Estimating the number of primes less than N with zeros of the zeta function. Dirichlet series related to zeta function and arithmetic functions Artin's conjecture on zeta functions of number fields. www.math.niu.edu /~rusin/known-math/index/11MXX.html   (243 words)

Algebra and Number Theory Seminar These are orthonormal bases formed by translates and dilations of a single function; the Haar basis is the prototypical example. Such wavelets are specified by a scaling function, which is a solution of a functional difference equation, called a dilation equation. In certain cases the number of occurrences of a particular generator in an arbitrary word may be a well defined function, and it is then an interesting question to explore the average value. www.math.psu.edu /rvaughan/nthsems03.html   (1787 words)

Mathematics of Computation Part I: Functions whose Early Derivatives are Continuous 101--135 J. Varah Computing Invariant Subspaces of a General Matrix When the Eigensystem Is Poorly Conditioned. Piecewise Continuous Functions and Functions with Poles near the Interval [0, 1]. 905--920 L. Lardy An Extrapolated Gauss--Seidel Iteration for Hessenberg Matrices. www.math.utah.edu /pub/tex/bib/toc/mathcomp1970.html   (6646 words)

Annals of Mathematics, II. Series, Vol. 151, No. 3, pp. 1175-1216, 2000 Another cubic moment considered in this paper is related to Maass cusp forms, and then there is also a contribution coming from the continuous spectrum which turns out to be the sixth moment of the Dirichlet $L$-function $L(1/2+ir, \chi)$ over an interval $[-R, R]$. In this case, cubes of the central values of the $L$-functions attached to Maass cusp forms for the full modular group were averaged over a short spectral interval, and a subconvexity estimate in the spectral aspect was deduced invoking again the non-negativity property of the $L$-values in question. This page was last modified: 22 Jan 2002. www.univie.ac.at /EMIS/journals/Annals/151_3/7.html   (310 words)

Amazon.com: Complex Functions: An Algebraic and Geometric Viewpoint: Books: Gareth A. Jones,David Singerman   (Site not responding. Last check: 2007-10-31) At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. There are several advantages in using the set C of complex numbers as the domain of definition of functions. Indeed, instead of praising this as a pleasantly geometrical part of function theory, it is perhaps even more satisfying to treat it geometrically altogether (cf. www.amazon.com /Complex-Functions-Algebraic-Geometric-Viewpoint/dp/052131366X   (1074 words)

Andrew Ledoan   (Site not responding. Last check: 2007-10-31) Explicit formulas for the triple correlation of zeros of functions in the Selberg class, work in preparation. Distribution of imaginary parts of zeros of Dirichlet $L$-functions, work in preparation. (with E. Alkan and A. Zaharescu) Dirichlet $L$-functions and the index of visible points, to appear in Illinois Journal of Mathematics. www.math.uiuc.edu /~ledoan   (397 words)

Riemann zeta function and Fractal nature of zeroes This page gives a few links to my work on the Riemann zeta function and Dirichlet L-functions (pdf file), and the fractal nature of the distribution of their zeroes. I have studied the zeroes over fifteen orders of magnitude for the Riemann function, and found that the distributions are remarkably similar. The similarity between the fractal structure of the zeroes for the Dirichlet L functions and the Riemann function is also striking. www.geocities.com /oshanker/RiemannZeta/ZetaFunctions.htm   (365 words)

Cwikel   (Site not responding. Last check: 2007-10-31) In 2002, J.B. Conrey and K. Soundararajan showed that there are infinitely many Dirichlet $L$-functions which do not vanish on the critical segment. Let $\mathcal{G}$ be the family of Dirichlet $L$-functions they considered. Following their work, a similar analytic study (especially the non-trivial real zeros but also the size on the critical line) of a family $\mathcal{F}$ of Rankin-Selberg L-functions which has the same symmetry type than $\mathcal{G}$ (namely the symplectic one) was undertaken during my thesis. www.math.princeton.edu /~seminar/2004-05-sem/RicottaAbstract10-25-2004.html   (255 words)

Publications of Richard H. Hudson (USC Math Department) Bays, Carter; Hudson, Richard H., Zeroes of Dirichlet $L$-functions and irregularities in the distribution of primes. Hudson, Richard H.; Markham, Thomas L., Alfred T. Brauer as a mathematician and teacher. Hudson, Richard H., A theorem on totally multiplicative functions. www.math.sc.edu /~hudson/publications_hudson.html   (923 words)

third   (Site not responding. Last check: 2007-10-31) [6] Inversion formula of Dirichlet polynomials and the approximate functional equation of Dirichlet's $L$-functions, Arch. [5] On the approximate functional equation of Dirichlet $L$-functions, Quart. [2] Fourth power mean value of Dirichlet's $L$-functions, International Symposium in Memory of Hua Loo Keng, Vol. www.prime.sdu.edu.cn /third/04wangwei.html   (122 words)

spring.htm The main topic in Math 722 is the basic theory of functions of one complex variable: III) The Riemann ζ function and Dirichlet L functions with application to sharp estimates for the remainder in the prime number theorem and the number of primes in arithmetic progressions. IV) The large sieve and applications to primes in arithmetic progressions and zero density theorems for Dirichlet L functions. www.math.wisc.edu /graduate/spring.htm   (1986 words)

[No title]   (Site not responding. Last check: 2007-10-31) To appear in the Proceedings of the RIMS Symposium on Analytic Number Theory and Surrounding Areas, October 18-22, 2004, RIMS Kokyuroku, with Jianya Liu. Distribution of zeros of Dirichlet L-functions and the least prime in arithmetic progressions The pair correlation of zeros of the Riemann zeta function and distribution of primes www.math.uiowa.edu /~yey/number.html   (164 words)

New Page 1 Abstract: The subject of my talk will be the theorem that in every arithmetic progression {a+nq}, where a and q are positive integers having no common factor, and n ranges over the integers from 1 to infinity, there are infinitely many primes. I will define the Dirichlet characters \xi and Dirichlet L-functions L(xi,s). I will show that the non-vanishing of the Dirichlet L-function associated to particular \xi at s=1 is equivalent to the statement that there are infinitely many primes in {a+nq}. www.cs.bgu.ac.il /~andreym/Seminar/Summer2005.htm   (591 words)

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