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Dirichlet's contributions to mathematical probability theory

Hans Fischer
1994 Historia Mathematica
Only a few short papers on probability and error theory by Peter Gustav Lejeune Dirichlet are printed in his Werke.  ...  However, during his Berlin period, Dirichlet quite frequently gave contexte de ses travaux analytiques.  ...  The integral in (6) Here L, M .... are constants depending on the functions f, and coefficients oe, (v = 1 .. s). A closer specification of these constants cannot be found in the lecture notes.  ...

The functional equation of some Dirichlet series

Bruce C. Berndt
1971 Proceedings of the American Mathematical Society
These Dirichlet series involve primitive characters and can be regarded as generalizations of Dirichlet's Lfunctions or of Epstein's zeta-functions.  ...  The functional equation for two classes of Dirichlet series is established.  ...  If »i = l, g = A = 0, and x( -1) = L we have derived the functional equation of Dirichlet's L-functions.  ...

The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression

2013 Archive for History of Exact Sciences
We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic  ...  In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor.  ...  This corollary will enable us to focus on the residue class of m modulo k in the proof of Dirichlet's theorem. Dirichlet characters and L-series Let k be an integer greater than or equal to 1.  ...

The Functional Equation of Some Dirichlet Series

Bruce C. Berndt
1971 Proceedings of the American Mathematical Society
These Dirichlet series involve primitive characters and can be regarded as generalizations of Dirichlet's Lfunctions or of Epstein's zeta-functions.  ...  The functional equation for two classes of Dirichlet series is established.  ...  If »i = l, g = A = 0, and x( -1) = L we have derived the functional equation of Dirichlet's L-functions.  ...

Formalization of the prime number theorem and Dirichlet's theorem [article]

Mario Carneiro
2016 arXiv   pre-print

The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression [article]

2013 arXiv   pre-print
We survey implicit and explicit uses of Dirichlet characters in presentations of Dirichlet's proof in the nineteenth and early twentieth centuries, with an eye towards understanding some of the pragmatic  ...  In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor.  ...  This corollary will enable us to focus on the residue class of m modulo k in the proof of Dirichlet's theorem. Dirichlet characters and L-series Let k be an integer greater than or equal to 1.  ...

Generalized Functions and Dirichlet's Principle

G. Temple
1956 Proceedings of the Royal Society A
The method is to show that any ' minimizing sequence ' represents a generalized function uf which is proved to be harmonic and to be equivalent to an ordinary numerical function.  ...  This paper gives a simple proof of Dirichlet's principle for any bounded domain.  ...  Generalized functions and Dirichlet's principle 4. The weak form of Green's theorem In particular, we shall carry out our discussion of Dirichlet's problem for any bounded domain £2.  ...

Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes [article]

Franz Lemmermeyer
2011 arXiv   pre-print
Dirichlet abandoned since it is in fact based on Dirichlet's results on the asymptotic behaviour of the divisor function.  ...  " zeta function as a product of L-series.Hilbert.  ...

COMPACTNESS AND DIRICHLET'S PRINCIPLE

Jin Keun Seo, Hamdi Zorgati
2014 Journal of the Korea Society for Industrial and Applied Mathematics
Fortunately, Poincaré and Hilbert defended Riemann's use of the Dirichlet principle and found a balance between rigor and intuition.  ...  There is no theorem without rigor, but we should not be a slave of rigor.  ...  Zorgati to the Department of Computational Science and Engineering, Yonsei University, whose kind hospitality and support have been gratefully acknowledged.  ...

A formalized proof of Dirichlet's theorem on primes in arithmetic progression

John; Intel Corporation Harrison
2009 Journal of Formalized Reasoning
We describe the formalization using the HOL Light theorem prover of Dirichlet's theorem on primes in arithmetic progression.  ...  First we describe a simple and efficient proof of the theorem informally, which iis otherwise difficult to find in one self-contained place at an elementary level.  ...  Acknowledgements The author would like to thank Rob Arthan and the anonymous JFR referees, whose comments have significantly improved the final version of this paper.  ...

ON BOUNDARY VALUE PROBLEMS FOR Φ-LAPLACIAN ON THE SEMI-INFINITE INTERVAL

Arnold Lepin, Leonid Lepin, Nickolay Vasilyev
2017 Mathematical Modelling and Analysis
The Dirichlet problem and the problem with functional boundary condition for ϕ-Laplacian on the semi-infinite interval are studied as well as solutions between the lower and upper functions.  ...  The class BB − (I, R) consists of functions β : I → R satisfying the following conditions: for each t ∈ (a, b] there exists a left derivative β l (t) and a limit lim τ →t− β l (τ ), and moreover β l (t  ...  The set of generalized solutions of the ϕ-Laplacian between a lower and an upper function is compact (in the sense of  ) and has the minimum and maximum generalized solutions, and the Dirichlet problem  ...