Aleksandar Ivic

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We investigate the pointwise and mean square order of the function Z 2 (s), where Z k (s) = ∞ 1 |ζ(1 2 + ix)| 2k x −s dx, k ∈ N. Three conjectures involving Z 2 (s) and certain exponential sums of Hecke series in short intervals are formulated, which have important consequences in zeta-function theory. A new order result for Z 2 (s) is obtained, and the(More)
For a fixed integer k ≥ 3, and fixed 1 2 < σ < 1 we consider T 1 |ζ(σ + it)| 2k dt = ∞ n=1 d 2 k (n)n −2σ T + R(k, σ; T), where R(k, σ; T) = o(T) (T → ∞) is the error term in the above asymptotic formula. Hitherto the sharpest bounds for R(k, σ; T) are derived in the range min(β k , σ * k) < σ < 1. We also obtain new mean value results for the zeta-function(More)
We obtain, for T ε ≤ U = U(T) ≤ T 1/2−ε , asymptotic formulas for Z 2T T (E(t + U) − E(t)) 2 dt, Z 2T T (∆(t + U) − ∆(t)) 2 dt, where ∆(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square formula for |ζ(1 2 + it)|. Upper bounds of the form O ε (T 1+ε U 2) for the above integrals with biquadrates instead of(More)
Several estimates for the convolution function C[f (x)] := x 1 f (y)f (x/y)(dy/y) and its iterates are obtained when f (x) is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for T 0 |ζ(1/2 + it)| 2k dt (k = 1, 2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and(More)
Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of |ζ(1 2 + it)|. If E * (t) = E(t) − 2π∆ * (t/2π) with ∆ * (x) = −∆(x) + 2∆(2x) − 1 2 ∆(4x), then we obtain the asymptotic formula T 0 (E * (t)) 2 dt = T 4/3 P 3 (log T) + O ε (T 7/6+ε), where P 3 is a polynomial of degree(More)