arXiv subjects

Ralph Furmaniak

Publications and source records attributed to Ralph Furmaniak.

3 recordsLinked to original sources

arXiv2026

Bohr's Last Problem Under the Entirety Hypothesis: A Survey with Initial Reductions

Bohr's last problem (1952) asks whether every ordinary Dirichlet series with nonzero Lindel\"of order function $\mu$ has $\mu'(\omega_\mu{-}0)\le-1$; a negative answer would imply Lindel\"of for $\zeta$. Kahane (1989) refuted this with half-plane counterexamples. We study the refinement for series with entire continuation of order $\le 1$: the Analytic Lindel\"of Hypothesis that $\mu$ is piecewise linear with integer slopes. Deforming the Mellin integral to the strip boundary reduces $\mu_L$ to a residue sum over singularities of the generating function on $|x|=1$, giving $\mu_L(\sigma)=\max(0,\tfrac12-\sigma+\rho)$. For classical $L$-functions this sum is the functional-equation dual, and bounding it is Lindel\"of; for self-similar or random singularities it is a Rajchman Fourier transform. We show Kahane's half-plane examples fail entirety, his entire random examples have integer slopes a.s., and Lerch-Lindel\"of implies ALH. Our central construction is the Cantor Dirichlet series $L(s)=\sum\hat\nu(n)n^{-s}$, with $\nu$ the ternary Cantor measure. Its Kaczorowski--Perelli twist spectrum is empty; we prove $\mu_L(\tfrac12)\le\tfrac18$ unconditionally via a Montgomery--Vaughan argument on the product variable $(m_1+\alpha)(m_2+\alpha)$, where a Vieta identity guarantees distinct frequencies. A Cantor-weighted Hurwitz second-moment conjecture would give $\mu_L(\tfrac12)=0$.

math.NT↗

arXiv2026

An Explicit Entire Function of Order One with All Zeros on a Line and Bounded in a Half-Plane

We construct a single explicit entire function $\Xi_c(s)$ of order 1, with all zeros provably on $Re(s) = 1/2$, satisfying a functional equation $\Xi_c(s) = \Xi_c(1-s)$, whose normalized form $Z_c(s) = \Xi_c(s)/[\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)]$ is uniformly bounded for $Re(s) > 1 + \delta$ yet satisfies $\sup_t|Z_c(1+it)| = +\infty$. The function thus satisfies an analogue of the Riemann Hypothesis together with the sharp bounded/unbounded transition at $\sigma = 1$ characteristic of $\zeta$. The transition is controlled by a Dirichlet series $D(s) = \sum e^{-ik\theta} p_k^{-s}$ whose absolute convergence for $\sigma > 1$ and divergence at $\sigma = 1$ drive the dichotomy. The key technical input is a dyadic large-sieve estimate establishing the linearization condition that connects the Hadamard product to $D$. The construction and proofs were developed in collaboration with Claude (Anthropic); see Acknowledgments.

math.NT↗

arXiv2018

On the Complexity of the Weighted Fused Lasso

The solution path of the 1D fused lasso for an $n$-dimensional input is piecewise linear with $\mathcal{O}(n)$ segments (Hoefling et al. 2010 and Tibshirani et al 2011). However, existing proofs of this bound do not hold for the weighted fused lasso. At the same time, results for the generalized lasso, of which the weighted fused lasso is a special case, allow $Ω(3^n)$ segments (Mairal et al. 2012). In this paper, we prove that the number of segments in the solution path of the weighted fused lasso is $\mathcal{O}(n^2)$, and that, for some instances, it is $Ω(n^2)$. We also give a new, very simple, proof of the $\mathcal{O}(n)$ bound for the fused lasso.

cs.LG↗