# Approximating sums by integrals only: multiple sums and sums over lattice polytopes

## TL;DR

This paper extends an integral-only summation approximation method to multiple sums and lattice polytopes, offering faster and more memory-efficient alternatives to traditional derivative-based formulas.

## Contribution

It introduces a multi-index extension of the Alt summation formula, applicable to multi-dimensional sums and sums over lattice polytopes, avoiding high-order derivatives.

## Key findings

- The extended Alt formula effectively approximates multi-dimensional sums.
- It improves computational efficiency over traditional Euler--Maclaurin methods.
- Applications include summing divergent series and lattice point enumeration.

## Abstract

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms of the execution time and memory use. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.