On the seventh power moment of Δ(x)

Let [Formula: see text] be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text] which improves the previous result.

On some upper bounds for the zeta-function and the Dirichlet divisor problem

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

The Dirichlet Divisor Problem of Arithmetic Progressions

We present an elementary method for studying the problem of getting an asymptotic formula that is better than Hooley’s and Heath-Brown’s results for certain cases.

On some mean value results for the zeta-function and a divisor problem

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt<< HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt << T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of the ?*-function comes from the fact that it is the analogue of E(T), the error term in the mean square formula for |?(1/2+it)|2. We also show, if E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt << j,? T7/6+j/4+? (j=1,2,3).