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The Euler-Maclaurin formula for simple integral polytopes.

Yael Karshon1, Shlomo Sternberg, Jonathan Weitsman

  • 1Department of Mathematics, Hebrew University of Jerusalem, Jerusalem 91904, Israel.

Proceedings of the National Academy of Sciences of the United States of America
|January 8, 2003
PubMed
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We present a new Euler-Maclaurin formula with remainder for summing smooth functions over integral points within lattice polytopes. This formula was derived using elementary mathematical methods.

Area of Science:

  • Mathematics
  • Number Theory
  • Discrete Geometry

Background:

  • The Euler-Maclaurin formula relates sums of functions to their integrals.
  • Lattice polytopes are fundamental objects in discrete geometry and number theory.
  • Calculating sums over integral points in polytopes is a significant problem.

Purpose of the Study:

  • To develop a generalized Euler-Maclaurin formula with a remainder term.
  • To apply this formula to sums over integral points in simple integral lattice polytopes.
  • To provide a new tool for analyzing discrete sums in geometric contexts.

Main Methods:

  • Derivation of the Euler-Maclaurin formula using elementary methods.
  • Focus on functions defined on integral points within a polytope.

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  • Inclusion of a remainder term to quantify approximation error.

    • Main Results:

    • A novel Euler-Maclaurin formula tailored for lattice polytope settings.
    • The formula incorporates a precise remainder term.
    • The proof relies on accessible, elementary techniques, enhancing its applicability.

    Conclusions:

    • The derived formula offers an effective method for approximating discrete sums.
    • Elementary proof methods make the result broadly accessible to mathematicians.
    • This work contributes to the understanding of sums over lattice polytopes.