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Averages of multiplicative functions along equidistributed sequences

Stephanie Chan, Peter Koymans, Carlo Pagano, Efthymios Sofos

Abstract

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

Averages of multiplicative functions along equidistributed sequences

Abstract

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.
Paper Structure (12 sections, 13 theorems, 202 equations)

This paper contains 12 sections, 13 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. The upper bound
  3. The lower bound
  4. Preliminary lemmas
  5. The upper bound
  6. Start of the proof
  7. Case (i)
  8. Case (ii)
  9. Case (iii)
  10. Case (iv)
  11. Proof of Theorem \ref{['tMain']}
  12. The lower bound

Key Result

Theorem 1.9

Let $\mathcal{A}$ be an infinite set and for each $T\geqslant 1$ define $\chi_T:\mathcal{A}\to [0,\infty)$ to be any function such that both eq:baz1 and eq:baz2 hold. Take a sequence of strictly positive integers $\mathfrak C=(c_a)_{a \in \mathcal{A}}$. Assume that $\mathfrak{C}$ is equidistributed where $M=M(T)$ is as in Definition def:levdistr. Then for all $T\geqslant 1$ we have where the imp

Theorems & Definitions (34)

  • Definition 1.1: Density functions
  • Definition 1.2: A class of functions
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Definition 1.6: Equidistributed sequences
  • Remark 1.7
  • Example 1.8
  • Theorem 1.9: The upper bound
  • Remark 1.10: Wolke's density function assumption
  • ...and 24 more