arXiv subjects
Publications and source records attributed to Aleksandar Ivić.
The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |ζ({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = σ+ it$) is used to obtain estimates for $$ \sum_{r=1}^R\int_{t_r-G}^{t_r+G}|ζ(1/2+it)|^{2k}{\rm d} t\quad(T < t_1 < >... < t_R < 2T), $$ where $t_{r+1} - t_r \ge G (r =1,..., R-1), T^ε\le G \le T^{1-ε$. These results can be used to derive bounds for the moments of $}|ζ(1/2+it)|$.
Upper bound estimates for the exponential sum $$ \sum_{K<κ_j\le K'<2K} α_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over κ_j})) \qquad(T^ε\le K \le T^{1/2-ε}) $$ are considered, where $α_j = |ρ_j(1)|^2(\coshπκ_j)^{-1}$, and $ρ_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_j = κ_j^2 + {1\over4}$ to which the Hecke series $H_j(s)$ is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with $H_j(\hf)$ or $H_j^2(\hf)$ replacing $H_j^3(1/2)$ are also considered. The above sum is conjectured to be $\ll_εK^{3/2+ε}$, which is proved to be true in the mean square sense.
First part of this paper was published in CEJM (2)(4) (2004), 1-15. It is proved now that $$ \int_0^T|E^*(t)|^5{\rm d}t \ll_εT^{2+ε}. $$ Here $$ E^*(t) = E(t) - 2πΔ^*(t/2π), Δ^*(x) = - Δ(x) +2Δ(2x) - {1\over2}Δ(4x), $$ where $E(t)$ is the error term in the mean square formula for $|ζ(1/2+it)|$ and $Δ(x)$ is the error term in the Dirichlet divisor problem. It is also shown how bounds for moments of $|E^*(t)|$ lead to bounds for moments of $|ζ(1/2+it)|$.
The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |ζ(1/2+ix)|^{2k}x^{-s}{\rm d} x (k = 1,2,...)$ is investigated. Analytic continuation and mean square estimates of ${\cal Z}_k(s) $ are discussed, as well as connections with power moments of $|ζ(1/2+ix)|$, with the special emphasis on the cases $k = 1,2$.
Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $ζ(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $σ= 1$. A zero-density counting function involving multiplicities is also discussed.
Let ${\cal Z}_1(s) = \int_1^\infty |ζ({1\over2}+ix)|^2x^{-s}{\rm d}x (σ= \Re s > 1)$. A result concerning analytic continuation of ${\cal Z}_1(s)$ to $\bf C$ is proved, and also a result relating the order of ${\cal Z}_1(σ+ it) (1/2 \le σ\le 1, t\ge t_0)$ to the order of ${\cal Z}_1({1\over2}+it)$.
Let $Δ(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X Δ^3(x){\rm d}x = BX^{7/4} + O_ε(X^{β+ε}) \qquad(B > 0) $$ and $$ \int_1^X Δ^4(x){\rm d}x = CX^2 + O_ε(X^{γ+ε}) \qquad(C > 0) $$ with $β= 7/5, γ= 23/12$. This improves on the values $β= 47/28, γ= 45/23$, due to K.-M. Tsang. A result on the integrals of $Δ^3(x)$ and $Δ^4(x)$ in short intervals is also proved.
The Laplace transform of $|ζ(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the non-Euclidean Laplacian.
In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|ζ(\s+it)|^4g(t){\rm d}t\qquad(\hf < σ< 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function $$\eqalign{E_2(T,σ) &: =\int_0^T|ζ(σ+it)|^4{rm d}t - {ζ^4(2σ)\overζ(4σ)}T -{T\over3-4σ}{({T\over2π} )}^{2-4σ}{ζ^4(2-2σ)\overζ(4-4σ)}\cr& - T^{2-2σ}(a_0(σ) + a_1(σ)\log T + a_2(σ)\log^2T),\cr} $$ the error term in the asymptotic formula for the fourth moment of $|ζ(σ+it)|$.
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 $$ \int_0^T(E^*(t))^4 dt \ll_εT^{16/19+varepsilon}$$, which is the first non-trivial bound for higher moments of $E^*(t)$. The method of proof also provides an upper bound for sums of fourth powers of mean square integrals of $|ζ(1/2 + it)|$ over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for $|ζ(1/2 + it)|$. Among the chief ingredients in the proof is a recent result of Robert--Sargos on the distribution of four square roots of integers, plus an approach of M. Jutila that involves the use of Airy integrals to deal with the ensuing exponential sums.
Mean square estimates for $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $|ζ(1/2+ix)|^4$.
Power moments of $$ J_k(t,G) = {1\over\sqrtπG} \int_{-\infty}^\infty |ζ(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2} du \qquad(t \asymp T, T^ε\le G \ll T),$$ where $k$ is a natural number, are investigated. The results that are obtained are used to show how bounds for $\int_0^T|ζ(1/2+it)|^{2k} dt$ may be obtained.
An asymptotic formula for the number of $n \le x$ such that $n$ does not divide $P(n)!$ is given, where P(n) is the largest prime factor of $n$.
This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.
Let $γ$ denote imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. Certain sums over the $γ$'s are evaluated, by using the function $G(s) = \sum_{γ>0}γ^{-s}$ and other techniques. Some integrals involving the function $S(T) = (1/π)\argζ(1/2+iT)$ are also considered.
If $P(x)$ is the error term in the circle problem, then it is proved that $$\int_0^\infty P^2(x)e^{-x/T}dx = {1\over4}({T\overπ})^{3/2} \sum_{n=1}^\infty r^2(n)n^{-3/2} - T + O_ε(T^{2/3+ε}), $$ improving the author's earlier exponent 5/6. The new bound is obtained by using results of F. Chamizo on the correlated sum $\sum_{n\le x}r(n)r(n+h)$, where $r(n)$ is the number of representations of $n$ as a sum of two squares.
We discuss some problems in number theory posed by Djuro Kurepa (1907-1993), including his classical left factorial hypothesis that an odd prime $p$ does not divide $0! + 1! + ... + (p-1)!$.
An overview of results and problems concerning the asymptotic formula for $\int_0^T|ζ(1/2+it)|^4dt$ is given, together with a discussion of modern methods from spectral theory used in recent work on this subject.