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\title{On the dicrepancy of the Halton sequences}
\author{Emanouil I. Atanassov}
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\maketitle

In this paper we obtain different estimates for the discrepancy of the
Halton sequences. It is known that for given pairwise
relatively prime integers $p_1,\dots ,p_s$ the discrepancy of the
Halton sequence $\sigma \left( p_1,\dots ,p_s\right) $
satisfies $D_N\left( \sigma \right) <c_s\left( p_1,\dots ,p_s\right) \ln
^sN+O(\ln ^{s-1}N)$. We prove that this estimate holds with
$c_s=\frac 1{s!}\prod\limits_{i=1}^s\frac{p_i-1}{\ln p_i}$,
improving the known result
of Meijer. It is shown that when the integers $p_1,\dots ,p_s$ are the first
$s$ primes then $\lim_{s\rightarrow \infty }c_s=0$. It is also proven that
if the prime numbers $p_1,\dots ,p_s$ satisfy certain condition the same
inequality holds with $c_s=\frac{2^s}{s!}\prod\limits_{i=1}^s\frac{p_i(2+\ln
p_i)}{(p_i-1)\ln p_i}\left( 1+\sum_{i=1}^s\ln p_i\right) $. Then for every
distinct primes $p_1,\dots ,p_s$, a modified Halton sequence $\sigma \left(
p_1,\dots ,p_s\right) $ is constructed, so that the same estimate for its
discrepancy holds unconditionally.
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