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Aleksander Simonič

Publications and source records attributed to Aleksander Simonič.

16 recordsLinked to original sources
arXiv2025

An explicit form of Ingham's zero density estimate

Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq T$. We provide an explicit version of this result with the exponent $(7-5\sigma)/(2-\sigma)$ of the logarithmic factor. In addition, we also provide an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line.

math.NT↗
arXiv2024

Conditional estimates for $L$-functions in the Selberg class II

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the region $0\leq\sigma-1/2\ll 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish both uniform bounds and completely explicit estimates by also assuming the strong $\lambda$-conjecture. In addition to providing estimates for a large set of functions, our results improve the best known estimates for specific functions in the Selberg class including the lower bounds for the Riemann zeta function close to the critical line.

math.NT↗
arXiv2024

Conditional estimates for $L$-functions in the Selberg class

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+δ\leqσ<1$, fixed $δ\in(0,1/2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish uniform bounds for $|3/4-σ|\leq 1/4-1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$, $|1-σ|\leq 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$ and $σ=1$, and completely explicit estimates by assuming also the strong $λ$-conjecture.

math.NT↗
arXiv2023

Conditional estimates for the logarithmic derivative of Dirichlet $L$-functions

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L'/L\right)\left(σ,χ\right)$ of Dirichlet $L$-functions, where $χ$ is a primitive character modulo $q\geq 10^{30}$ and $1/2+1/\log{\log{q}}\leqσ\leq 1-1/\log\log q$. In addition, for $σ=1$ we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.

math.NT↗
arXiv2022

Atkinson's formula for the mean square of $ζ(s)$ with an explicit error term

We provide an explicit $O\left(\log^2{T}\right)$-term of the celebrated Atkinson's formula for the error term $E(T)$ of the second power moment of the Riemann zeta-function on the critical line. As an application, we obtain an explicit version of well-known estimate $E(T)\ll_{\varepsilon} T^{\frac{1}{3}+\varepsilon}$.

math.NT↗
arXiv2022

Estimates for $L$-functions in the critical strip under GRH with effective applications

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood's well known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application we provide conditional and effective estimate for the Mertens function.

math.NT↗
arXiv2020

Quasimodularity of the $k$th Residual Cranks

We establish quasimodularity for a family of residual crank generating functions defined on overpartitions. We also show that the second moments of these $k$th residual cranks admit a combinatoric interpretation as weighted overpartition counts.

math.NT↗
arXiv2020

A note on a straight gravity tunnel through a rotating body

It is well-known that the straight gravity tunnel between any two different positions on a non-rotating Earth, which has uniform density, is traversable, i.e., an object initially at rest will reach its destination through the gravity tunnel in both directions. Moreover, the time taken to fall is always constant. These facts are no longer true if rotation is allowed. The aim of this note is to derive the necessary and sufficient condition for traversability of straight gravity tunnels through a rotating physical body with spherically symmetric gravitational field. Fall-through times are expressed in a closed form for linear and constant gravitational fields. In conclusion, these models are compared to numerically obtained data using the internal structure of the Earth.

physics.pop-ph↗
arXiv2019

Explicit zero density estimate for the Riemann zeta-function near the critical line

In 1946, A. Selberg proved $N(σ,T) \ll T^{1-\frac{1}{4} \left(σ-\frac{1}{2}\right)} \log{T}$ where $N(σ,T)$ is the number of nontrivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}>σ$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this estimate, together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta-function on the critical line.

math.NT↗
arXiv2019

Platonic configurations of points and lines

We present some methods for constructing connected spatial geometric configurations $(p_{q}, n_{k})$ of points and lines, preserved by the same rotations (and reflections) of Euclidean space $E^{3}$ as the chosen Platonic solid. In this paper we are primarily interested in balanced configurations $(n_{3}), (n_{4})$ and $(n_{5})$, but also in unbalanced configurations $(p_{3},n_{4}), (p_{3}, n_{5})$ and $(p_{4}, n_{5})$.

math.CO↗
arXiv2019

On Littlewood's proof of the prime number theorem

In this note we examine Littlewood's proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the non-vanishing of Riemann's zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

math.NT↗
arXiv2017

Lehmer pairs and derivatives of Hardy's $Z$-function

Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $ζ$-function to be a Lehmer pair in terms of derivatives of Hardy's $Z$-function. Furthermore, we connect Newman's conjecture with stationary points of the $Z$-function, and present some numerical results.

math.NT↗
arXiv2016

Elementary approach to the Hartogs extension theorem

In this paper we present a proof of Hartogs' extension theorem, following T. Sobieszek's paper from 2003. Hartogs' theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is "a several complex variables theorem" in nature because its conclusion is false in the complex plane. Sobieszek's proof is quite remarkable because he uses, stated in his paper without proofs, only higher-dimensional identity principle for holomorphic functions and Cauchy's integral formula for compact sets. We proved this two theorems here, making this exposition self-contained. The only background required is an undergraduate course in real and complex analysis and in point-set topology.

math.CV↗
arXiv2015

The Ahlfors lemma and Picard's theorems

The article introduces Ahlfors' generalization of the Schwarz lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings, including the celebrated Picard's theorems. The article concludes with a brief insight into the theory of Kobayashi hyperbolic complex manifolds.

math.CV↗