Distribuion of Leading Digits of Numbers II

Ohkubo, Yukio; Strauch, Oto

doi:10.2478/udt-2019-0003

Abstract

In this paper, we study the sequence (f (p_n))_n≥1,where p_n is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=log_b x^r and f (x)=log_b(x log x)^r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy $D_{N_{i}}^{*} (f (p_{n}), g)$ D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (N_i)_i_≥1 of the natural numbers. Especially for f (x)= log_b x^r, we obtain the effective results for an upper bound of $D_{N_{i}}^{*} (f (p_{n}) g)$ D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

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Distribuion of Leading Digits of Numbers II

Abstract

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