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Mathematics of Computation

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A probabilistic approach to a differential-difference equation arising in analytic number theory


Author: Jean-Marie-François Chamayou
Journal: Math. Comp. 27 (1973), 197-203
MSC: Primary 65C05; Secondary 10K10
DOI: https://doi.org/10.1090/S0025-5718-1973-0336952-X
MathSciNet review: 0336952
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Abstract: The differential-difference equation

\begin{displaymath}\begin{array}{*{20}{c}} {tv'(t) + v(t - 1) = 0,} \hfill & {t ... ...nstant}},} \hfill & {0 \leqq t \leqq 1,} \hfill \\ \end{array} \end{displaymath}

can be solved by the Monte-Carlo method, for the initial condition $ v(t) = {e^{ - \gamma }},0 \leqq t \leqq 1$, where the $ v(t)$ represent the probability density of a random variable:

$\displaystyle t = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\prod\limits_{j = 1}^i {{x_j},} } $

where the $ {x_j}$ are independent and uniformly distributed on (0, 1).

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0336952-X
Keywords: Differential-difference equation, Monte-Carlo method, stochastic processes, elementary prime number theory, explicit machine computations
Article copyright: © Copyright 1973 American Mathematical Society

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