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Aleksandar Ivić

Publications and source records attributed to Aleksandar Ivić.

At least 19 recordsLinked to original sources
arXiv2018

On the distribution of values of the argument of the Riemann zeta-function

Let $S(t) \;:=\; \frac{\displaystyle 1}{\displaystyle π}\arg ζ(\frac{1}{2} + it)$. We prove that, for $T^{\,27/82+\varepsilon} \le H \le T$, we have $$ {\rm mes}\Bigl\{t\in [T, T+H]\;:\; S(t)>0\Bigr\} = \frac{H}{2} + O\left(\frac{H\log_3T}{\varepsilon\sqrt{\log_2T}}\right), $$ where the $O$-constant is absolute. A similar formula holds for the measure of the set with $S(t)<0$, where $\log_kT = \log(\log_{k-1}T)$. This result is derived from an asymptotic formula for the distribution of values of $S(t)$, which is uniform in the relevant parameters, and this is of crucial importance. This in fact depends on the distribution of values of the Dirichlet polynomial which approximates $S(t)$, namely ($p$ denotes primes) $$V_{y}(t)\,=\,\sum\limits_{p\le y}\frac{\sin{(t\log{p})}}{\sqrt{p}}.$$

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arXiv2018

On certain sums over ordinates of zeta-zeros II

Let $γ$ denote the imaginary parts of complex zeros $ρ= β+iγ$ of $ζ(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{γ> 0}γ^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critical line $\int_1^T|G(1/2+it)|^2\,dt$ are revisited. This paper is a continuation of work begun by the second author in 2001.

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arXiv2018

A note on the zeros of the zeta-function of Riemann

It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.

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arXiv2018

On sums of squares of $|ζ(\frac12+iγ)|$ over short intervals

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|ζ(1/2+iγ)|^2$$ and some related integrals is presented, where $γ\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. It is shown unconditionally that the above sum is $\,\ll H\log^2T\log\log T\,$ for $\,T^{2/3}\log^4T \ll H \le T$.

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arXiv2017

On certain integrals involving the Dirichlet divisor problem

We prove that $$ \int_1^XΔ(x)Δ_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^XΔ(x)Δ_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $Δ_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $ζ^k(s)$ ($Δ_2(x) \equiv Δ(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $Δ_k(x)$. We also obtain the analogues of the above bounds when $\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|ζ(1/2+it)|$.

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arXiv2017

On the multiplicites of zeros of $ζ(s)$ and its values over short intervals

We investigate bounds for the multiplicities $m(β+iγ)$, where $β+iγ\,$ ($β\ge \1/2, γ>0)$ denotes complex zeros of $ζ(s)$. It is seen that the problem can be reduced to the estimation of the integrals of the zeta-function over "very short" intervals. A new, explicit bound for $m(β+iγ)$ is also derived, which is relevant when $β$ is close to unity. The related Karatsuba conjectures are also discussed.

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arXiv2016

Moments of Hardy's function over short intervals

Let as usual $Z(t) = ζ(1/2+it)χ^{-1/2}(1/2+it)$ denote Hardy's function, where $ζ(s) = χ(s)ζ(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\log T)^{k^2} \ll_{k,α} \sum_{T<γ\le T+H}\max_{γ\le τ_γ\le γ^+} |ζ(1/2 + iτ_γ)|^{2k} \ll_{k,α} H(\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $γ, γ^+$ denote ordinates of consecutive complex zeros of $ζ(s)$ and $T^α\le H \le T$, where $α$ is a fixed constant such that $0<α\le 1$. This sharpens and generalizes a result of M.B. Milinovich.

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arXiv2016

On the distribution of positive and negative values of Hardy's $Z$-function

We investigate the distribution of positive and negative values of Hardy's function $$ Z(t) := ζ(1/2+it){χ(1/2+it)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s). $$ In particular we prove that $$ μ\bigl(I_{+}(T,T)\bigr) \;\gg T\; \qquad \hbox{and}\qquad μ\bigl(I_{-}(T, T)\bigr) \; \gg \; T, $$ where $μ(\cdot)$ denotes the Lebesgue measure and \begin{align*} { I}_+(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)>0\bigr\}, { I}_-(T,H) &\;=\; \bigl\{T< t\le T+H\,:\, Z(t)<0\bigr\}. \end{align*}

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arXiv2016

Hardy's function $Z(t)$ - results and problems

This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s). $$ In particular, we discuss the first and third moment of $Z(t)$ (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.

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arXiv2015

On a cubic moment of Hardy's function with a shift

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad ζ(s) = χ(s)ζ(1-s) $$ is Hardy's function. The cubic moment of $Z(t)$ is also discussed, and a mean value result is presented which supports the author's conjecture that $$ \int_1^TZ^3(t)\,dt \;=\;O_\varepsilon(T^{3/4+\varepsilon}). $$

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arXiv2015

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

Let $d(n)$ be the number of divisors of $n$, let $$ Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1) $$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$ \int_0^TΔ^k(t)|ζ(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N) $$ are given. This complements the results of the paper Ivić-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.

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arXiv2015

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

Let $d(n)$ be the number of divisors of $n$, let $γ$ denote Euler's constant and $$ Δ(x) := \sum_{n\le x}d(n) - x(\log x + 2γ-1) $$ denote the error term in the classical Dirichlet divisor problem, and let $ζ(s)$ denote the Riemann zeta-function. It is shown that $$ \int_0^TΔ(t)|ζ(1/2+it)|^2\,dt \ll T(\log T)^{4}. $$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $$ \int_1^{T}Δ^{k}(t)|ζ(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-η_k+\varepsilon}), $$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and where $$η_2= 3/20, η_3= η_4=1/10,\ η_5=3/80,\ η_6=35/4742,\ η_7=17/6312,\ η_8=8/9433.$$ The results depend on the power moments of $Δ(t)$ and $E(T)$, the classical error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$.

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arXiv2014

Some Applications of Laplace Transforms in Analytic Number Theory

In this overview paper, presented at the meeting DANS14, Novi Sad, July3-7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of $|ζ(1/2+it)|$, and a discussion of two functional equations connected to a work of Prof. Bogoljub Stanković.

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arXiv2014

Hybrid moments of the Riemann zeta-function

The "hybrid" moments $$ \int_T^{2T}|ζ(1/2+it)|^k{(\int_{t-G}^{t+G}|ζ(1/2+ix)|^\ell dx)}^m dt $$ of the Riemann zeta-function $ζ(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_ε(T^{1+ε}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\ell,m$, and the explicitly determined range of $G = G(T;k,\ell,m)$. The application to a mean square bound for the Mellin transform function of $|ζ(1/2+ix)|^4$ is given.

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arXiv2013

The divisor function and divisor problem

The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their connection with the powers moments of $|ζ(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the lecture given at the conference ERDOS100 in Budapest, July 1-5, 2013.

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arXiv2013

On a hybrid fourth moment involving the Riemann zeta-function

We provide explicit ranges for $σ$ for which the asymptotic formula \begin{equation*} \int_0^T|ζ(1/2+it)|^4|ζ(σ+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(σ)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as $T\rightarrow \infty$, when $1\leq j \leq 6$, where $ζ(s)$ is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An application to a divisor problem is also given

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arXiv2013

On some mean value results for the zeta-function in short intervals

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote 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 $Δ^*(x) := -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}πT + R(T)$, then we obtain a number of results involving the moments of $|ζ(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|ζ(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T) $$ are also treated.

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