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Dirichlet's contributions to mathematical probability theory

1994
*
Historia Mathematica
*

Only a few short papers on probability

doi:10.1006/hmat.1994.1007
fatcat:vnkzxhfjbbde5fqn7j2psod4ee
*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. ...##
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The functional equation of some Dirichlet series

1971
*
Proceedings of the American Mathematical Society
*

These

doi:10.1090/s0002-9939-1971-0276181-1
fatcat:agll3zfgirayne4yx7n4t5b4li
*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*. ...##
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The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression

2013
*
Archive for History of Exact Sciences
*

We survey implicit

doi:10.1007/s00407-013-0126-0
fatcat:ecjxix7rirfmhclvwtuxhtymim
*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. ...##
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The Functional Equation of Some Dirichlet Series

1971
*
Proceedings of the American Mathematical Society
*

These

doi:10.2307/2038579
fatcat:wynwi2kvhzejrk4yn3af35rbwy
*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*. ...##
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Formalization of the prime number theorem and Dirichlet's theorem
[article]

2016
*
arXiv
*
pre-print

We present the formalization of

arXiv:1608.02029v1
fatcat:mhl5bz5qyzduxixndyoatryudm
*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 $\pi ... If F is a differentiable*function*with F ′ = f ,*and*f is a positive decreasing*function*that converges to zero, then g(x) = n≤x f (n) − F (x) converges to some*L**and*|g(x) −*L*| ≤ f (x). ... Introduction*Dirichlet's**theorem*, or the*Dirichlet*prime number*theorem*, states that for any N ∈ N*and*A ∈ Z such that gcd(A, N ) = 1, there are infinitely many primes in the progression A + kN , or equivalently ...##
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$Q$ valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two

1983
*
Bulletin of the American Mathematical Society
*

Accordingly, we study branched coverings of regions in R m which are graphs of multiple valued

doi:10.1090/s0273-0979-1983-15106-6
fatcat:rk2whbdzjngadlqo5zaf5w6uy4
*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*. ...##
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Green's Function in Some Contributions of 19th Century Mathematicians

2001
*
Historia Mathematica
*

Helmholtz, Riemann, Lipschitz, Carl

doi:10.1006/hmat.2001.2315
fatcat:aslp2iwun5adbcd4r7cwqzgpmm
*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. ...##
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Distribution of residues modulo p using the Dirichlet's class number formula
[article]

2019
*
arXiv
*
pre-print

In this article, we study the number of quadratic residues

arXiv:1810.00227v2
fatcat:kxvzfwxyrraprf2plt2oba2cgi
*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. ...##
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The concept of "character" in Dirichlet's theorem on primes in an arithmetic progression
[article]

2013
*
arXiv
*
pre-print

We survey implicit

arXiv:1209.3657v3
fatcat:bqc7azwfhrb67gwuyejqzfdj7q
*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

1956
*
Proceedings of the Royal Society A
*

The method is to show that any ' minimizing sequence ' represents a generalized

doi:10.1098/rspa.1956.0096
fatcat:nutke5wixnhpbncklj45hqycna
*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. ...##
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Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes
[article]

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. ...

##
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COMPACTNESS AND DIRICHLET'S PRINCIPLE

2014
*
Journal of the Korea Society for Industrial and Applied Mathematics
*

Fortunately, Poincaré

doi:10.12941/jksiam.2014.18.193
fatcat:uznetybzvfeadoizy43nn3lo6a
*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. ...##
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A formalized proof of Dirichlet's theorem on primes in arithmetic progression

2009
*
Journal of Formalized Reasoning
*

We describe the formalization using the HOL Light

doi:10.6092/issn.1972-5787/1558
dblp:journals/jfrea/Harrison09
fatcat:dkpkqvwwjrcw3mef3nprcyxixi
*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. ...##
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ON BOUNDARY VALUE PROBLEMS FOR Φ-LAPLACIAN ON THE SEMI-INFINITE INTERVAL

2017
*
Mathematical Modelling and Analysis
*

The

doi:10.3846/13926292.2017.1263245
fatcat:7xpp74siffcxjjbmhilidpl4ne
*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 ...##
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Mathematics and language
[chapter]

2015
*
Mathematics, Substance and Surmise
*

The research on the history of

doi:10.1007/978-3-319-21473-3_12
fatcat:6w4osceosbaibexturn2inqdzy
*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 ...
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