Aleksandar Ivic

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We investigate the pointwise and mean square order of the function Z2(s), where Zk(s) = ∫∞ 1 |ζ(12 + ix)|2kx−s dx, k ∈ N. Three conjectures involving Z2(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 Z2(s) is obtained, and the function Zk(s)(More)
The aim of this paper is to provide asymptotic formulas for the 2k–th moment of the Riemann zeta-function ζ(s) and some related Dirichlet series in the so-called “critical strip” 12 < σ = Re s < 1. For the zeta-function our results are relevant when k ≥ 3 is a fixed integer, where henceforth s = σ + it will denote a complex variable. Mean values of ζ(s) on(More)
which is valid for any complex s, it follows that ζ(s) has zeros at s = −2,−4, . . . . These zeros are traditionally called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s), of which the smallest ones (in absolute value) are 12 ± 14.134725 . . . i. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical(More)
It is proved that, for T ε 6 G = G(T ) 6 2 √ T , ∫︁ 2T T (︁ I1(t+G,G)− I1(t, G) )︁2 dt = TG 3 ∑︁ j=0 aj log (︁√ T G )︁ +Oε(T 1+εG1/2 + T 1/2+εG2) with some explicitly computable constants aj (a3 > 0) where, for fixed k ∈ N, Ik(t, G) = 1 √ π ∫︁ ∞ −∞ |ζ( 1 2 + it+ iu)| 2ke−(u/G) 2 du. The generalizations to the mean square of I1(t+U,G)−I1(t, G) over [T, T+H](More)
We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε, asymptotic formulas for Z 2T T (E(t+ U)− E(t)) dt, Z 2T T (∆(t+ U)−∆(t)) 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 Oε(T 1+εU2) for the above integrals with biquadrates instead of square are(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)) dt = T P3(logT ) +Oε(T ), where P3 is a polynomial of degree three in log T with(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)