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Łukasz Pańkowski

Publications and source records attributed to Łukasz Pańkowski.

15 recordsLinked to original sources
arXiv2026

On the sign changes of $ψ(x)-x$

We improve the lower bound for $V(T)$, the number of sign changes of the error term $ψ(x)-x$ in the Prime Number Theorem in the interval $[1,T]$ for large $T$. We show that \[ \liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{γ_{0}}π+\frac{1}{60} \] where $γ_{0}=14.13\ldots$ is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated $k$-function, over $4\cdot10^{21}$ times better than previously known estimates of this type.

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arXiv2025

Universality of the zeta function in short intervals

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for $H=(\log T)^B$ with an explicitly given $B>0$. Unconditionally, we show that for the same $H$ the set of real numbers $τ\in[T,T+H]$ such that $ζ(s+iτ)$ approximates an arbitrary given analytic function has a positive upper density.

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arXiv2023

Notes on Universality in Short Intervals and Exponential Shifts

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin\v cikas in a problem session of a recent workshop on universality.

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arXiv2020

Joint value-distribution of shifts of the Riemann zeta-function

We prove that any non-zero complex values $z_1,\ldots,z_n$ can be approximated by the following integral shifts of the Riemann zeta-function $ζ(s+id_1τ),\ldots,ζ(s+id_nτ)$ for infinitely many $τ$, provided $d_1,\ldots,d_n\in\mathbb{N}$ and $s$ is a fixed complex number lying in the right open half of the critical strip.

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arXiv2020

Joint Extreme values of $L$-functions

We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class which have polynomial Euler product and satisfy Selberg's orthonormality condition. We show that on every vertical line $s=σ+it$ with $σ\in(1/2,1)$, these $L$-functions simultaneously take large values of size $\exp\left(c\frac{(\log t)^{1-σ}}{\log\log t}\right)$ inside a small neighborhood.

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arXiv2019

Zeros of $L(s)+L(2s)+\cdots+L(Ns)$ in the region of absolute convergence

In this paper we show that for every Dirichlet $L$-function $L(s,χ)$ and every $N\geq 2$ the Dirichlet series $L(s,χ)+L(2s,χ)+\cdots+L(Ns,χ)$ have infinitely many zeros for $σ>1$. Moreover we show that for many general $L$-functions with an Euler product the same holds if $N$ is sufficiently large, or if $N=2$. On the other hand we show with an example the the method doesn't work in general for $N=3$.

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arXiv2018

Note on the number of divisors of reducible quadratic polynomials

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum $\sum_{n \leq x} d( n ) d (n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.

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arXiv2016

Joint universality for dependent $L$-functions

We prove that, for arbitrary Dirichlet $L$-functions $L(s;χ_1),\ldots,L(s;χ_n)$ (including the case when $χ_j$ is equivalent to $χ_l$ for $j\ne k$), suitable shifts of type $L(s+iα_jt^{a_j}\log^{b_j}t;χ_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.

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arXiv2016

Large values of L-functions from the Selberg class

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with Jörn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the "resonance method", which was introduced by Soundararajan, while the second proof uses methods from Diophantine approximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above.

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arXiv2015

Joint universality for Lerch zeta-functions

For $0<α, λ\leq 1$, the Lerch zeta-function is defined by $L(s;α, λ)$$:= \sum_{n=0}^\infty e^{2πiλn} (n+α)^{-s}$, where $σ>1$. In this paper, we prove joint universality for Lerch zeta-functions with distinct $λ_1,\ldots,λ_m$ and transcendental $α$.

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arXiv2015

Value distribution for the derivatives of the logarithm of $L$-functions from the Selberg class in the half-plane of absolute convergence

In the present paper, we show that, for every $δ>0$, the function $(\log {\mathcal{L}}(s))^{(m)}$, where $m\in {\mathbb{N}} \cup \{ 0\}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty a(n) n^{-s}$ is an element of the Selberg class ${\mathcal{S}}$, takes any value infinitely often in any strip $1<\Re(s) <1+δ$, provided $\sum_{p\leq x} |a (p)|^2 \sim κπ(x)$ for some $κ>0$. In particular, ${\mathcal{L}} (s)$ takes any non-zero value infinitely often in the strip $1<\Re(s)<1+δ$, and the first derivative of ${\mathcal{L}} (s)$ vanishes infinitely often.

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arXiv2015

Joint universality and generalized strong recurrence with rational parameter

We prove that, for every rational $d\ne 0,\pm 1$ and every compact set $K\subset\{s\in\mathbb{C}:1/2<\Re(s)<1\}$ with connected complement, any analytic non-vanishing functions $f_1,f_2$ on $K$ can be approximated, uniformly on $K$, by the shifts $ζ(s+iτ)$ and $ζ(s+idτ)$, respectively. As a consequence we deduce that the set of $τ$ satisfying $|ζ(s+iτ)-ζ(s+idτ)|<\varepsilon$ uniformly on $K$ has a positive lower density for every $d\ne 0$.

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arXiv2013

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned above despite of the existence of the zeros off the critical line. Next we prove that the Barnes multiple zeta-functions associated to rational or transcendental parameters have infinitely many zeros off the critical line. By using this fact, we show that the Shintani multiple zeta-functions have infinitely many complex zeros under some conditions. As corollaries, we show that the Mordell multiple zeta-functions, the Euler-Zagier-Hurwitz type of multiple zeta-functions and the Witten multiple zeta-functions have infinitely many complex zeros off the critical line.

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arXiv2012

Corrigendum for "The generalized strong recurrence for non-zero rational parameters" Archiv der Mathematik 95 (2010), 549-555

In the present paper, we prove that self-approximation of $\log ζ(s)$ with $d=0$ is equivalent to the Riemann Hypothesis. Next, we show self-approximation of $\log ζ(s)$ with respect to all nonzero real numbers $d$. Moreover, we partially filled a gap existing in "The strong recurrence for non-zero rational parameters" and prove self-approximation of $ζ(s)$ for $0 \ne d=a/b$ with $|a-b|\ne 1$ and $\gcd(a,b)=1$.

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