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On the asymptotic behavior of the integral T(x)

Mihoub Bouderbala, Meselem Karras

Abstract

In this paper, we estimate the integral T(x) mentioned in the title, where {t} denotes the fractional part of the real number t, and x is any positive real number.

On the asymptotic behavior of the integral T(x)

Abstract

In this paper, we estimate the integral T(x) mentioned in the title, where {t} denotes the fractional part of the real number t, and x is any positive real number.
Paper Structure (10 sections, 6 theorems, 77 equations)

This paper contains 10 sections, 6 theorems, 77 equations.

Table of Contents

  1. Introduction and Results
  2. Proof of Theorem 1
  3. Exploring the behavior of the sum $T_{0}(x)$
  4. Exploring the behavior of the sums $T_{d}(x)$ for $d\ge1$
  5. Obtaining the expression of the sum $T_{d,1}(x)$
  6. Obtaining the expression of the sum $T_{d,2}(x)$
  7. Obtaining the expression of the sum $T_{d,3}(x)$
  8. Final expression and behavior of $T_{d}(x)$
  9. Exploring the behavior of the sum $\sum_{d>D}T_{d}(x)$
  10. Proof of Theorem 1

Key Result

Theorem 1.1

For all $x>0$, we have where $\zeta$ denotes the Riemann zeta function.

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 1 more