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Aleksandar Ivić

Publications and source records attributed to Aleksandar Ivić.

At least 37 records · Page 2Linked to original sources
arXiv2012

On some mean square estimates for the zeta-function in short intervals

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$ and we set $\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$, then we obtain $$ \int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T $$ and $$ HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T, $$ for $T^{2/3+ε}\le H \le T$.

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arXiv2011

On the Rankin-Selberg problem in short intervals

If $$ Δ(x) \;:=\; \sum_{n\leqslant x}c_n - Cx\qquad(C>0) $$ denotes the error term in the classical Rankin-Selberg problem, then we obtain a non-trivial upper bound for the mean square of $Δ(x+U) - Δ(x)$ for a certain range of $U = U(X)$. In particular, under the Lindelöf hypothesis for $ζ(s)$, it is shown that $$ \int_X^{2X} \Bigl(Δ(x+U)-Δ(x)\Bigr)^2\,{\roman d} x \;\ll_ε\; X^{9/7+ε}U^{8/7}, $$ while under the Lindelöf hypothesis for the Rankin-Selberg zeta-function the integral is bounded by $X^{1+ε}U^{4/3}$. An analogous result for the discrete second moment of $Δ(x+U)-Δ(x)$ also holds.

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arXiv2010

On some relations for Mellin transforms of Hardy's function

Some relations involving the Mellin and Laplace transforms of powers of the classical Hardy function $$ Z(t) := ζ(1/2+it)\bigl(χ(1/2+it)\bigr)^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ are obtained. In particular, we discuss some mean square identities and their consequences.

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arXiv2010

On the Mellin transforms of powers of Hardy's function

Various properties of the Mellin transform function $$ {\cal M}_k(s) := \int_1^\infty Z^k(x)x^{-s}dx $$ are investigated, where $$ Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ is Hardy's function and $ζ(s)$ is Riemann's zeta-function. Connections with power moments of $|ζ(1/2+it)|$ are established, and natural boundaries of ${\cal M}_k(s)$ are discussed.

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arXiv2010

On some problems involving Hardy's function

Some problems involving the classical Hardy function $$ Z(t) := ζ(1/2+it)\bigl(χ(1/2+it)\bigr)^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its positive and negative values and the primitive of $Z(t)$. Some analogous problems for the mean square of $|ζ(1/2+it)|$ are also discussed.

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arXiv2010

Higher moments of the error term in the divisor problem

It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}Δ^4_k(x)\d x \ll_εX^ε\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $Δ_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"ı}--type formula for $Δ_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $Δ_2(x)$.

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arXiv2009

On the moments of the Riemann zeta-function in short intervals

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^θ$ with fixed $0<θ\le 1$, $$ \int_T^{T+H}|ζ(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$. The proof is based on the recent method of K. Soundararajan for counting the occurrence of large values of $\log|ζ(1/2+it)|$, who proved that $$ \int_0^{T}|ζ(1/2+it)|^{2k} dt \ll_εT(\log T)^{k^2+ε}. $$

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arXiv2009

On the integral of Hardy's function

If $Z(t) = χ^{-1/2}(1/2+it)ζ(1/2+it)$ denotes Hardy's function, where $ζ(s) = χ(s)ζ(1-s)$ is the functional equation of the Riemann zeta-function, then it is proved that $$ \int_0^T Z(t)\d t = O_\e(T^{1/4+\e}). $$

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arXiv2009

On the mean square of the divisor function in short intervals

We provide upper bounds for the mean square integral $$ \int_X^{2X}(Δ_k(x+h) - Δ_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $Δ_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by $ζ^k(s)$.

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arXiv2009

On the mean square of the Riemann zeta-function in short intervals

It is proved that, for $T^ε\le G = G(T) \le {1\over2}\sqrt{T}$, $$ \int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_ε(T^{1+ε}+ T^{1/2+ε}G^2) $$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G) = {1\over\sqrtπ}\int_{-\infty}^\infty |ζ(1/2+it+iu)|^{2k} {\rm e}^{-(u/G)^2} du. $$ The generalizations to the mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.

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arXiv2007

Sums of the error term function in the mean square for $ζ(s)$

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|ζ(1/2+it)|^2 dt - T\Bigl(\log {T\over2π} + 2γ-1\Bigr)$$ is the error term in the mean square formula for $|ζ(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\rm e}^{2πm}$ and for the partial quotients in its continued fraction expansion.

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arXiv2007

On the fourth moment in the Rankin-Selberg problem

If $$ Δ(x) := \sum_{n\le x}c_n - Cx $$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$ \int_0^X Δ^4(x)\d x \ll_εX^{3+ε},\quad \int_0^X Δ_1^4(x)\d x \ll_εX^{11/2+ε}, $$ where $Δ_1(x) = \int_0^xΔ(u) du$. The latter bound is, up to `$ε$', best possible.

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arXiv2007

On the Riemann zeta-function and the divisor problem IV

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$, then it is proved that $$ \int_0^T|E^*(t)|^3dt \ll_εT^{3/2+ε}, $$ which is (up to `$ε$' best possible) and $ζ(1/2+it) \ll_εt^{ρ/2+ε}$ if $E^*(t) \ll_εt^{ρ+ε}$.

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arXiv2006

On the mean square of the zeta-function and the divisor problem

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 $Δ^*(x) = -Δ(x) + 2Δ(2x) - {1\over2}Δ(4x)$, then we obtain the asymptotic formula $$ \int_0^T (E^*(t))^2 {\rm d} t = T^{4/3}P_3(\log T) + O_ε(T^{7/6+ε}), $$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.

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arXiv2006

The Laplace and Mellin transforms of powers of the Riemann zeta-function

This paper gives a survey of known results concerning the Laplace transform $$ L_k(s) := \int_0^\infty |ζ(1/2+ ix)|^{2k}{\rm e}^{-sx}{\rm d} x \qquad(k \in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) := \int_1^\infty|ζ(1/2+ ix)|^{2k}x^{-s}{\rm d} x\qquad(k\in N), $$ where the integral is absolutely convergent for $\R s \ge c(k) > 1$. Also some new results on these integral transforms of $|ζ(1/2+ ix)|^{2k}$ are given, which have important connections with power moments of the Riemann zeta-function $ζ(s)$.

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