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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.  ... 
doi:10.1006/hmat.1994.1007 fatcat:vnkzxhfjbbde5fqn7j2psod4ee

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.  ... 
doi:10.1090/s0002-9939-1971-0276181-1 fatcat:agll3zfgirayne4yx7n4t5b4li

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

Jeremy Avigad, Rebecca Morris
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.  ... 
doi:10.1007/s00407-013-0126-0 fatcat:ecjxix7rirfmhclvwtuxhtymim

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.  ... 
doi:10.2307/2038579 fatcat:wynwi2kvhzejrk4yn3af35rbwy

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

Mario Carneiro
2016 arXiv   pre-print
We present the formalization of Dirichlet's theorem on the infinitude of primes in arithmetic progressions, and Selberg's elementary proof of the prime number theorem, which asserts that the number $\pi  ...  If F is a differentiable function with F ′ = f , and f is a positive decreasing function that converges to zero, then g(x) = n≤x f (n) − F (x) converges to some L and |g(x) − L| ≤ f (x).  ...  Introduction Dirichlet's theorem, or the Dirichlet prime number theorem, states that for any N ∈ N and A ∈ Z such that gcd(A, N ) = 1, there are infinitely many primes in the progression A + kN , or equivalently  ... 
arXiv:1608.02029v1 fatcat:mhl5bz5qyzduxixndyoatryudm

$Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two

F. J. Almgren Jr.
1983 Bulletin of the American Mathematical Society  
Accordingly, we study branched coverings of regions in R m which are graphs of multiple valued functions minimizing the integral of Dirichlet's integrand.  ...  Suppose that A is a bounded open subset of R m with smooth boundary, and let k, /, ra, n, Q be positive integers with k > 3, I < n, and m > 2.  ...  EXISTENCE AND REGULARITY OF DIRICHLET INTEGRAL MINIMIZ-ING Q VALUED FUNCTIONS.  ... 
doi:10.1090/s0273-0979-1983-15106-6 fatcat:rk2whbdzjngadlqo5zaf5w6uy4

Green's Function in Some Contributions of 19th Century Mathematicians

Rossana Tazzioli
2001 Historia Mathematica  
Helmholtz, Riemann, Lipschitz, Carl and Franz Neumann, and Betti deduced functions similar to Green's function in order to solve problems in acoustics, electrodynamics, magnetism, theory of heat, and elasticity  ...  Many years before, one of these methods had been developed by Green in 1828, which consists in finding a function-called a Green's function-satisfying certain conditions and appearing in the analytical  ...  ACKNOWLEDGMENT I thank Craig Fraser for his suggestions and advice, which allowed me to improve the first version of the paper.  ... 
doi:10.1006/hmat.2001.2315 fatcat:aslp2iwun5adbcd4r7cwqzgpmm

Distribution of residues modulo p using the Dirichlet's class number formula [article]

Jaitra Chattopadhyay, Bidisha Roy, Subha Sarkar, R. Thangadurai
2019 arXiv   pre-print
In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number  ...  Kumar Murty for going through the manuscript very carefully and for a suggestion to clear our doubts.  ...  Let L(s, ψ) = ∞ n=1 ψ(n) n s be the Dirichlet L-function associated to ψ.  ... 
arXiv:1810.00227v2 fatcat:kxvzfwxyrraprf2plt2oba2cgi

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

Jeremy Avigad, Rebecca Morris
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.  ... 
arXiv:1209.3657v3 fatcat:bqc7azwfhrb67gwuyejqzfdj7q

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.  ... 
doi:10.1098/rspa.1956.0096 fatcat:nutke5wixnhpbncklj45hqycna

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.  ... 
arXiv:1109.1228v1 fatcat:zrli3tlfzfcihftx5jzkzautfm

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.  ... 
doi:10.12941/jksiam.2014.18.193 fatcat:uznetybzvfeadoizy43nn3lo6a

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.  ... 
doi:10.6092/issn.1972-5787/1558 dblp:journals/jfrea/Harrison09 fatcat:dkpkqvwwjrcw3mef3nprcyxixi

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 [13] ) and has the minimum and maximum generalized solutions, and the Dirichlet problem  ... 
doi:10.3846/13926292.2017.1263245 fatcat:7xpp74siffcxjjbmhilidpl4ne

Mathematics and language [chapter]

Jeremy Avigad
2015 Mathematics, Substance and Surmise  
The research on the history of Dirichlet's theorem described in Section 2 can be found in Jeremy Avigad and Rebecca Morris, "The concept of 'character' in Dirichlets theorem on primes in an arithmetic  ...  Morris and I were curious to understand how the understanding of Dirichlet characters evolved over time, and so we studied a number of subsequent presentations of Dirichlet's theorem and related results  ... 
doi:10.1007/978-3-319-21473-3_12 fatcat:6w4osceosbaibexturn2inqdzy
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