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Aleksandar Ivic

Publications and source records attributed to Aleksandar Ivic.

14 recordsLinked to original sources
arXiv2014

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

Let $Δ(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $Δ^*(x) = -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$. We show that $$ \int_T^{T+H}Δ^*\bigl(\frac{t}{2π}\bigr)|ζ(1/2+it)|^2dt \;\ll\; HT^{1/6}\log^{7/2}T \quad(T^{2/3+\varepsilon} \le H = H(T) \le T), $$ $$ \int_0^TΔ(t)|ζ(1/2+it)|^2dt \;\ll\; T^{9/8}(\log T)^{5/2},$$ and obtain asymptotic formulae for $$ \int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^2 |ζ(1/2+it)|^2dt,\quad \int_0^T{\Bigl(Δ^*\bigl(\frac{t}{2π}\bigr)\Bigr)}^3|ζ(1/2+it)|^2dt. $$ The importance of the $Δ^*$-function comes from the fact that it is the analogue of $E(T)$, the error term in the mean square formula for $|ζ(1/2+it)|^2$. We also show, if $E^*(T) := E(T) - 2πΔ^*(T/(2π))$, $$ \int_0^T E^*(t)E^j(t)|ζ(1/2+it)|^2dt \; \ll_{j,\varepsilon}\; T^{7/6+j/4+\varepsilon}\quad(j= 1,2,3). $$

math.NT↗
arXiv2011

On the general additive divisor problem

We obtain a new upper bound for $\sum_{h\le H}Δ_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $Δ_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $ζ(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.

math.NT↗
arXiv2010

Convolutions and mean square estimates of certain number-theoretic error terms

We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for $f(x)$. Some applications are given, in particular to $|ζ(1/2+ix)|^{2k}$ and the classical Rankin--Selberg problem from analytic number theory.

math.NT↗
arXiv2007

On the divisor function and the Riemann zeta-function in short intervals

We obtain, for $T^ε\le U=U(T)\le T^{1/2-ε}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(Δ(t+U) - Δ(t))^2 dt, $$ where $Δ(x)$ 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 square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|ζ(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed.

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arXiv2006

Some remarks on the moments of $|ζ(1/2+it)|$ in short intervals

Some new results on power moments of the integral $$ J_k(t,G) = {1\over\sqrtπG} \int_{-\infty}^\infty |ζ(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2}du \qquad(t \asymp T, T^ε\le G \ll T, k\in\N) $$ are obtained when $k=1$. These results can be used to derive bounds for moments of $|ζ(1/2+it)|$.

math.NT↗
arXiv2006

On the Riemann zeta-function and the divisor problem III

Let $Δ(x)$ 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 $E^*(t) = E(t) - 2πΔ^*(t/2π)$ with $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$ and we set $\int_0^T E^*(t) dt = 3πT/4 + R(T)$, then we obtain $$ R(T) = O_ε(T^{593/912+ε}), \int_0^TR^4(t) dt \ll_εT^{3+ε}, $$ and $$ \int_0^TR^2(t) dt = T^2P_3(\log T) + O_ε(T^{11/6+ε}), $$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.

math.NT↗
arXiv2006

On some mean square estimates in the Rankin-Selberg problem

An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function $Δ(x;ξ) (0\leξ\le1)$, the error term in the Rankin-Selberg problem weighted by $ξ$-th power of the logarithm. Mean square estimates for $Δ(x;ξ)$ are proved.

math.NT↗
arXiv2005

On the integral of the error term in the Dirichlet divisor problem

Several results are obtained concerning the function $Δ_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding mean square integral. The mean square integral of $Δ_2(x)$ is investigated in detail.

math.NT↗
arXiv2003

A mean value result involving the fourth moment of $|ζ(1/2+it)|$

If $(k,\ell)$ is an exponent pair such that $k+\ell<1$, then we have $$ \int_1^T|ζ(1/2+it)|^4|ζ(σ+it)|^2dt \ll_εT^{1+ε}\quad(σ> \min({5\over6},\max(\ell-k, {5k+\ell\over4k+1})), $$ while if $(k,\ell)$ is an exponent pair such that $3k+\ell<1$, then we have $$ \int_1^T|ζ(1/2+it)|^4|ζ(σ+it)|^4dt \ll_εT^{1+ε}\quad(σ> {11k+\ell+1\over8k+2}). $$

math.NT↗
arXiv2003

On some mean value results involving $|ζ(1/2+it)|$

Several problems involving $E(T)$ and $E_2(T)$, the error terms in the mean square and mean fourth moment formula for $|ζ(1/2+it)}$ are discussed. In particular, it is proved that $$ \int_0^T E(t)E_2(t)dt \ll_ T^{7/4}(\log T)^{7/2}(\log\log T). $$

math.NT↗
arXiv2003

On the moments of Hecke series at central points II

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{κ_j\le T}α_j H_j(1/2) = ({T\overπ})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$ and the Bruggeman--Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^ε)$.

math.NT↗
arXiv2003

Some identities for the Riemann zeta-function

Several identities for the Riemann zeta-function $ζ(s)$ are proved. For example, if $s = σ+ it$ and $σ> 0$, then $$ \int_{-\infty}^\infty |{(1-2^{1-s})ζ(s)\over s}|^2dt = {π\overσ}(1 - 2^{1-2σ})ζ(2σ). $$

math.NT↗