Number Theory

    

An Improved Bound for the Counting Function F(m,r)

Authors: Theophilus Agama

Let us define the function F(m, r) as the number of integers n in the interval [2^m, 2^{m+1}) such that ι(n) ≤ floor(m + r), where r = c·m / log m for some constant c satisfying 0 larger than c smaller than log 2. Next, define α = c + log 2 − (1/4)(1 − o(1)). By applying the Chernoff inequality from probability theory and using ideas from De et al. (2025), we obtain the following improved upper bound: F(m, c·m / log m) ≤ exp[ α·m − (1 − ε)·(c·m·log log m) / log m ] for any small ε > 0 as m tends to infinity.

Comments: 6 Pages.

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Submission history

[v1] 2025-06-20 20:12:44

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