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Johan Andersson

Publications and source records attributed to Johan Andersson.

At least 19 recordsLinked to original sources
arXiv2025

On Simulation-Guided LLM-based Code Generation for Safe Autonomous Driving Software

Automated Driving System (ADS) is a safety-critical software system responsible for the interpretation of the vehicle's environment and making decisions accordingly. The unbounded complexity of the driving context, including unforeseeable events, necessitate continuous improvement, often achieved through iterative DevOps processes. However, DevOps processes are themselves complex, making these improvements both time- and resource-intensive. Automation in code generation for ADS using Large Language Models (LLM) is one potential approach to address this challenge. Nevertheless, the development of ADS requires rigorous processes to verify, validate, assess, and qualify the code before it can be deployed in the vehicle and used. In this study, we developed and evaluated a prototype for automatic code generation and assessment using a designed pipeline of a LLM-based agent, simulation model, and rule-based feedback generator in an industrial setup. The LLM-generated code is evaluated automatically in a simulation model against multiple critical traffic scenarios, and an assessment report is provided as feedback to the LLM for modification or bug fixing. We report about the experimental results of the prototype employing Codellama:34b, DeepSeek (r1:32b and Coder:33b), CodeGemma:7b, Mistral:7b, and GPT4 for Adaptive Cruise Control (ACC) and Unsupervised Collision Avoidance by Evasive Manoeuvre (CAEM). We finally assessed the tool with 11 experts at two Original Equipment Manufacturers (OEMs) by conducting an interview study.

cs.SE↗
arXiv2024

Mittag-Leffler type theorems for Helson zeta-functions

Let $f$ be a zero-free analytic function on $\Re(s) \geq 1$. We prove that there exists an entire zero-free function $g$ and a Helson zeta-function $\zeta_\chi(s)=\sum_{n=1}^\infty \chi(n) n^{-s}$, where $\chi(n)$ is a completely multiplicative unimodular function such that $f(s)=g(s) \zeta_\chi(s)$ for $\Re(s)>1$. By the Mittag-Leffler theorem this implies that a Helson zeta-function may have meromorphic continuation from $\Re(s)>1$ to the complex plane with a prescribed set of zeros and poles in the half plane $\Re(s)<1$. This improves on results of Seip and Bochkov-Romanov who proved the same result in the strip $21/40<\Re(s)<1$ and conditional on the Riemann hypothesis in the strip $1/2< \Re(s)<1$. Our results also gives information on maximum domains of meromorphicity and analyticity of Helson zeta-functions and show that any open connected set $U$ that includes the half plane $\Re(s) >1$, may be a maximum domain of meromorphicity or of analyticity for a Helson zeta-function. This extends results of Bhowmik and Schlage-Puchta to Dirichlet series with Euler products.

math.NT↗
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 Laurin\v{c}ikas 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.

math.NT↗
arXiv2023

Discrete universality, continuous universality and hybrid universality are equivalent

Recently Sourmelidis proved that the discrete universality theorem is equivalent to the continuous universality theorem for zeta-functions. He treats both the zero-free universality theorem and the strong universality theorem. Unfortunately in the zero-free case his result is conditional on a Riemann hypothesis, and in the strong universality case he only proves the implication in one direction. We prove this equivalence unconditionally, and also prove an equivalence with hybrid universality. While the main application of our result is on Dirichlet series and zeta-functions, our proof is general and does not use the fact that the functions in question can be represented by Dirichlet series.

math.NT↗
arXiv2021

Polynomial approximation avoiding values in sets II

We prove some results on when functions on compact sets $K \subset \mathbb C$ can be approximated by polynomials avoiding values in given sets. We also prove some higher dimensional analogues. In particular we prove that a continuous function from a compact set $K \subset \mathbb R^n$ without interior points to $\mathbb R^n$ can be uniformly approximated by a polynomial mapping avoiding values in any given countable set $A \subset \mathbb R^n$, giving a real $n$-dimensional analogue of a recent version of Lavrentiev's theorem of Andersson and Rousu. We also prove the same result for infinite dimensional Banach spaces.

math.CA↗
arXiv2021

On the growth of the $L^p$ norm of the Riemann zeta-function on the line Re$(s)=1$

We prove that if $δ>0$ and $p$ is real then $$ \sup_T \int_T^{T+δ} |ζ(1+it)|^p dt <\infty,$$ if and only if $-1 1) $$ which with the exception of an additional $\log \log \log T$ factor in the second estimate coincides with conditional (under the Riemann hypothesis) order estimates. We also prove weaker unconditional order estimates.

math.NT↗
arXiv2020

Joint universality on the half plane of absolute convergence

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint universality results for Dirichlet $L$-functions. In contrast to classical universality, we do not need that the Dirichlet series in question have an analytic continuation beyond their region of absolute convergence. Also we may allow weaker orthogonality conditions for pairs of Dirichlet series than in the previous joint universality results of Lee-Nakamura-Pańkowski. We take care to avoid using the Ramanujan conjecture in our proof and hence as a consequence of our universality theorem, we obtain stronger results on zeros of linear combinations of $L$-functions in the half plane of absolute convergence than previous results of Booker-Thorne and Righetti. For example as a consequence of our main universality result we have that certain linear combinations of Hecke $L$-series coming from Maass wave forms have infinitely many zeros in any strip $1<$Re$(s)<1+δ$.

math.NT↗
arXiv2020

Universality of the Hurwitz zeta-function on the half plane of absolute convergence

Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<α<1, α\neq \frac 1 2$ then $f(s)$ may be uniformly approximated arbitrarily closely by $ζ(1+iT+iδs,α)$ on $K$ for some $T,δ>0$, where $ζ(s,α)$ denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.

math.NT↗
arXiv2020

On the Balasubramanian-Ramachandra method close to Re(s)=1

We study the problem on how to get good lower estimates for the integral $$ \int_T^{T+H} |ζ(σ+it)| dt, $$ when $H \ll 1$ is small and $σ$ is close to $1$, as well as related integrals for other Dirichlet series, by using ideas related to the Balasubramanian-Ramachandra method. We use kernel-functions constructed by the Paley-Wiener theorem as well as the kernel function of Ramachandra. We also notice that the Fourier transform of Ramachandra's Kernel-function is in fact a $K$-Bessel function. This simplifies some aspects of Balasubramanian-Ramachandra method since it allows use of the theory of Bessel-functions.

math.NT↗
arXiv2019

Polynomial approximation avoiding values in countable sets

We generalize a version of Lavrentév's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial without zeros on the set K, so that the polynomial can avoid values from any given countable set. We also prove a corresponding version of Mergelyan's theorem when the interior of K is a finite union of Jordan domains, pairwise separated by a positive distance.

math.CV↗
arXiv2019

On the universality of the Epstein zeta function

We study universality properties of the Epstein zeta function $E_n(L,s)$ for lattices $L$ of large dimension $n$ and suitable regions of complex numbers $s$. Our main result is that, as $n\to\infty$, $E_n(L,s)$ is universal in the right half of the critical strip as $L$ varies over all $n$-dimensional lattices $L$. The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same approach we also prove that, as $n\to\infty$, $E_n(L_1,s)-E_n(L_2,s)$ is universal in the full half-plane to the right of the critical line as $(L_1,L_2)$ varies over all pairs of $n$-dimensional lattices. Finally, we prove a more classical universality result for $E_n(L,s)$ in the $s$-variable valid for almost all lattices $L$ of dimension $n$. As part of the proof we obtain a strong bound of $E_n(L,s)$ on the critical line that is subconvex for $n\geq 5$ and almost all $n$-dimensional lattices $L$.

math.NT↗
arXiv2018

Voronin Universality in several complex variables

We prove the Voronin universality theorem for the multiple Hurwitz zeta-function with rational or transcendental parameters in $\mathbb{C}^n$ answering a question of Matsumoto. In particular this implies that the Euler-Zagier multiple zeta-function is universal in several complex variables and gives the first example of a Dirichlet series that is universal in more than one variable.

math.NT↗
arXiv2016

Non universality on the critical line

We prove that the Riemann zeta-function is not universal on the critical line by using the fact that the Hardy Z-function is real, and some elementary considerations. This is a related to a recent result of Garunkstis and Steuding. We also prove conditional and partial results for non universality on the lines Re(s)=σfor 0<σ<1/2 and together with our recent result for non universality on the line Re(s)=1 it will mostly answer the question of on what lines the zeta-function is universal.

math.NT↗
arXiv2016

On questions of Cassels and Drungilas-Dubickas

We answer a question of Drungilas-Dubickas in the affirmative under the assumption of standard conjectures on smooth numbers in polynomial sequences. This gives evidence against the "Dubickas Conjecture", which Kačinskaitė and Laurinčikas proved implies universality results for the Hurwitz zeta-function with certain algebraic irrational parameters. Under these standard conjectures we also prove some results that confirms observations of Worley relating to a problem of Cassels on the multiplicative dependence of algebraic numbers shifted by integers.

math.NT↗
arXiv2013

Bounded prime gaps in short intervals

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof of Theorem 1, but we also explain how a sharper variant of Zhang's Theorem 2 would imply the same result for shorter intervals.

math.NT↗
arXiv2012

Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions

We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose interior is a Jordan domain, with nonvanishing polynomials. This result was proved earlier by the author in the case of a compact set K without interior points, and independently by Gauthier for this case and the case of strictly starlike compact sets. We apply this result on the Voronin universality theorem for compact sets K of this type, where the usual condition that the function is nonvanishing on the boundary can be removed. We conjecture that this version of Mergelyan's theorem might be true for a general set K with connected complement and show that this conjecture is equivalent to a corresponding conjecture on Voronin Universality.

math.CV↗
arXiv2012

On generalized Hardy classes of Dirichlet series

We generalize the Hardy class H^2 of Dirichlet series studied by Hedenmalm, Lindqvist, Olofsson, Olsen, Saksman, Seip and others to consider more general Dirichlet series. We prove some results on this class, such as estimates for its logarithmic L^1-norm in short intervals. We relate this to, and use these results to make a recent nonvanishing result of Dirichlet series of ours more explicit. In particular we give an application on the Hurwitz zeta-function.

math.CV↗