arXiv subjects
Publications and source records attributed to Aleksandar Ivić.
Small values of $|ζ(1/2+it)|$ are investigated, using the value distribution results of A. Selberg. This gives an asymptotic formula for $μ(\{0 < t \le T : |ζ(1/2+it)| \le c\})$. Some related problems involving values of $|ζ(1/2+it)|$ and gaps between the consecutive ordinates of zeros of $ζ(s)$ are also discussed.
A discussion involving the evaluation of the sum $\sum_{0<γ\le T} |ζ(1/2+iγ)|^2$ is presented, where $γ$ denotes imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. Three theorems involving certain integrals related to this sum are proved, and the sum is unconditionally shown to be $\ll T\log^2T\log\log T$.
Sums of squares of $|ζ(1/2+it)|$ over short intervals are investigated. Known upper bounds for the fourth and twelfth moment of $|ζ(1/2+it)|$ are derived. A discussion concerning other possibilities for the estimation of higher power moments of $|ζ(1/2+it)|$ is given.
A sharp asymptotic formula for the sum of reciprocals of $π(n)$ is derived, where $π(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivić and L. Panaitopol.
Let P(n) denote the largest prime factor of $n \ge 2, P(1) = 1$, and let $$ β(n) = \sum_{p|n}p, \Beta(n) = \sum_{p^α||n}αp, \Beta_1(n) = \sum_{\p^α||n}p^α$$ denote "large" additive functions. A survey of results on the functions $P(n), β(n), \Beta(n)$ and $\Beta_1(n)$ is presented, as well as some new results and open problems.
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.
For a fixed integer $k\ge 3$ and fixed $1/2 < σ> 1$ we consider $$ \int_1^T |ζ(σ+ it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2σ}T + R(k,σ;T), $$ where $R(k,σ;T) = o(T) (T\to\infty)$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,σ;T)$ are given for certain ranges of $σ$. We also obtain new mean value results for the zeta-functions of holomorphic cusp forms and the Rankin-Selberg series.
We prove $\sum_{K-G}\le κ_j \le K+G}α_j H_j^3(1/2) \ll_εGK^{1+ε}$ for $K^ε\le G \le K$, where $α_j = |ρ_j(1)|^2(\cosh πκ_j)^{-1}$, and $ρ_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $λ_j = κ_j^2 + 1/4$ to which the Hecke series $H_j(s)$ is attached. This result yields the new bound $H_j(1/2) \ll_εκ_j^{1/3+ε}$.
An overview of results and problems concerning the asymptotic behaviour for summatory functions of a certain class of additive functions is given. The class of functions in question involves Karamata's regular variation. Some new Abelian and Tauberian results for additive functions of the form $F(n) = \sum_{p^α||n}αh(p)$ are given.
Estimates for $Z_2(s) = \int_1^|infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, both pointwise and in mean square. It is shown how these estimates can be used to bound $E_2(T)$, the error term in the asymptotic formula for $\int_0^T |ζ(1/2+it)|^4dt$.
Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|ζ(1/2+it)|$ on the critical line, and zeros of a class of convolution functions. The first two topics are classical, and the remaining ones are connected with the author's research.