Periodica Mathematica Hungarica

, Volume 10, Issue 4, pp 261–271

On the order of the error function of the square-full integers

• D. Suryanarayana
Article

Abstract

LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that
$$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$
where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that$$\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )$$ for every ε>0. In this paper, we prove the following results on the assumption of R.H.:
$$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$
(1)
$$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$
(2)
for any integer ν≥1.

In fact, we prove some general results and deduce the above from them.

On the basis of (1) and (2) above, we conjecture that$$\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-\nulldelimiterspace} {12}} + \varepsilon } )$$ under the assumption of R.H.

AMS (MOS) subject classifications (1970)

Primary 10H15 Secondary 10H25

Key words and phrases

Square-full integers Riemann Zeta function Riemann hypothesis

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