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Detecting zeros of Dirichlet $L$-functions via the Riemann zeta-function
Masatoshi Suzuki
https://arxiv.org/abs/2508.17701 https://arxiv.org/pdf/2508.17701…

Detecting zeros of Dirichlet $L$-functions via the Riemann zeta-function
Assuming the Riemann hypothesis, we show that a certain vertical distribution of the nontrivial zeros of the Riemann zeta-function is equivalent to the generalized Riemann hypothesis for Dirichlet $L$-functions. Furthermore, under both the Riemann hypothesis and the generalized Riemann hypothesis, we show that the nontrivial zeros of Dirichlet $L$-functions can be detected through those of the Riemann zeta-function.

From Partial Exchangeability to Predictive Probability: A Bayesian Perspective on Classification
Marcio Alves Diniz
https://arxiv.org/abs/2508.16716 https://

From Partial Exchangeability to Predictive Probability: A Bayesian Perspective on Classification
We propose a novel Bayesian nonparametric classification model that combines a Gaussian process prior for the latent function with a Dirichlet process prior for the link function, extending the interpretative framework of de Finetti representation theorem and the construction of random distribution functions made by Ferguson (1973). This approach allows for flexible uncertainty modeling in both the latent score and the mapping to probabilities. We demonstrate the method performance using simula…

Dirichlet $L$-functions on the Critical Line
Sami Vihko
https://arxiv.org/abs/2506.16115 https://arxiv.org/pdf/2506.16115

Dirichlet $L$-functions on the Critical Line
In this paper we prove that the Dirichlet $L$-functions $L(1/2+ix,χ_q)$, where $χ_q$ is uniformly random Dirichlet character modulo $q$, converges to a random Schwartz distribution $ζ_{\mathrm{rand}}$, which is related to (complex) Gaussian multiplicative chaos. This is the same limiting object that appeared in [34], where the authors proved that the random shifts of the Riemann zeta function on the critical line $ζ(1/2+ix+iωT)$, where $ω\sim \mathrm{Unif} ([0,1])$ and $x\in \mathbb{R}$, …

A Nonlinear Nonlocal Problem for the Caputo Fractional Subdiffusion Equation
Ravshan Ashurov, Rajapboy Saparboyev, Navbahor Nuraliyeva
https://arxiv.org/abs/2506.19516

A Nonlinear Nonlocal Problem for the Caputo Fractional Subdiffusion Equation
In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of order \(0 < α< 1\), along with homogeneous Dirichlet boundary conditions. The nonlocal initial condition is of the form \( u(x,0) = g(x, u(x,T)) \), where \(g\) is a nonlinear function satisfying a Lipschitz condition. The main challenge arises from the implici…

Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval
Anton Alexa
https://arxiv.org/abs/2506.18914 https://

Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval
We define a second-order differential operator $\hat{C}$ on the Hilbert space $L^2([-v_c, v_c])$, constructed from a smooth deformation function $C(v)$. The operator is considered on the Sobolev domain $H^2([-v_c, v_c]) \cap H^1_0([-v_c, v_c])$ with Dirichlet boundary conditions. We prove that $\hat{C}$ is essentially self-adjoint by verifying its symmetry and computing von Neumann deficiency indices, which vanish. All steps are carried out explicitly. This result ensures the mathematical consi…

Ces\`{a}ro-type operators acting on Dirichlet spaces
Petros Galanopoulos, Daniel Girela
https://arxiv.org/abs/2507.13502 https://arxi…

Cesàro-type operators acting on Dirichlet spaces
If $(η)=\{ η_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Cesàro-type operator $\mathcal C_{(η)}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb D$ as follows: If $f$ is an analytic function in $\mathbb D$, $f(z)=\sum_{n=0}^\infty a_nz^n$ ($z\in \mathbb D$), then $\mathcal C_{(η)}(f)$ is formally defined by $$\mathcal C_{(η)}(f)(z)=\mathcal C_{\{η_n\}}(f)(z)=\sum_{n=0}^\infty η_n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{…

BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians
Qingsong Gu, Hua Qiu
https://arxiv.org/abs/2507.14858 https://

BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians
Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $Ω\subset K$ whose boundary $\partial Ω$ is a graph-directed self-similar set, we prove that the eigenvalue counting function $ρ^Ω(x)$ of the Laplacian with Dirichlet or Neumann boundary conditions (Neumann only for connected $Ω$) has an explicit second term as $x\to +\infty$, beyond the dominant Weyl term. If $\partialΩ$ has a strong iterated structure, …

Large solutions for subordinate spectral Laplacian
Ivan Bio\v{c}i\'c, Vanja Wagner
https://arxiv.org/abs/2506.16780 https://arxiv…

Large solutions for subordinate spectral Laplacian
We find a large solution to a semilinear Dirichlet problem in a bounded $C^{1,1}$ domain for a non-local operator $ϕ(-Δ\vert_{D})$, an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained, and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the semi…

Neumann scalar determinants on constant curvature disks
Soumyadeep Chaudhuri
https://arxiv.org/abs/2507.05159 https://arxiv.org/pdf/2…

Neumann scalar determinants on constant curvature disks
Working in the $ζ$-function regularisation scheme, we find certain infinite series representations of the logarithms of massive scalar determinants, $\det(Δ+m^{2})$ for arbitrary $m^2$, on finite round disks of constant curvature ($R=\frac{2η}{L^2}, η=0,\pm1$) with Neumann boundary conditions. The derivation of these representations relies on a relation between the Neumann determinants on the disks and the corresponding Dirichlet determinants via the determinants of the Dirichlet-to-Neumann…

On slice regular fractal-fractional Dirichlet type spaces
Jos\'e Oscar Gonz\'alez-Cervantes, Carlos Alejandro Moreno-Mu\~noz, Juan Bory-Reyes
https://arxiv.org/abs/2506.14172

On slice regular fractal-fractional Dirichlet type spaces
In this paper, we study some families of right modules of quaternionic slice regular functions induced by a generalized fractal-fractional derivative with respect to a truncated quaternionic exponential function on slices. Important Banach spaces of slice regular functions, namely the Bergman and the Dirichlet modules, are two important elements of one of our families.

Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric L\'evy noise
Juan J. Jim\'enez
https://arxiv.org/abs/2507.12656 https…

Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise
In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{L}u=\dotξ$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric Lévy noise $\dotξ$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fracti…

Robin Green Function Estimates and a Model of Mammalian Lungs
Guy David, Stefano Decio, Max Engelstein, Marcel Filoche, Svitlana Mayboroda, Marco Michetti
https://arxiv.org/abs/2507.13168

Robin Green Function Estimates and a Model of Mammalian Lungs
The present paper establishes delicate properties of the Green function with Robin boundary conditions, in particular, elucidating the nature of the passage between the Dirichlet-like and Neumann-like behavior. This yields sharp quantifiable bounds on the corresponding harmonic measure and proves the phase transition in the behavior of the total flow earlier conjectured in physics literature in concert with the efficacy of mammalian lungs.

On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points
Fei Liu, Yinghan Zhang
https://arxiv.org/abs/2506.22883

On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points
In this paper, we investigate the problem of the existence of the bounded harmonic functions on a simply connected Riemannian manifold $\widetilde{M}$ without conjugate points, which can be compactified via the ideal boundary $\widetilde{M}(\infty)$. Let $\widetilde{M}$ be a uniform visibility manifold which satisfy the Axiom $2$, or a rank $1$ manifold without focal points, suppose that $Γ$ is a cocompact discrete subgroup of $Iso(\widetilde{M})$, we show that for a given continuous function …

Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
Thomas Wright
https://arxiv.org/abs/2507.10780 https://arxiv.org/…

Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We that under the assumption of an exceptional character $χ$ with $L(1,χ)=o\left((\log D)^{-5}\right)$, for any $q

Explicit Formulas For Generalized Polylogarithmic Integrals, Euler Sums, And BBP-Type Series
Ali Olaikhan
https://arxiv.org/abs/2507.04205 https://

Explicit Formulas For Generalized Polylogarithmic Integrals, Euler Sums, And BBP-Type Series
This paper explores closed-form expressions for some polylogarithm integrals with integrands containing five parameters. These closed form expressions are given in terms of the Lerch transcendent function, which reduces, in some cases, to well-known special functions, such as the Riemann zeta, Dirichlet eta, Dirichlet beta, the Hurwitz zeta, and the polygamma functions. Related Euler sums and BBP-type series will also be discussed.

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function
Piotr Gwiazda, Jakub Wo\'znicki, Aneta Wr\'oblewska-Kami\'nska, Aleksandra Zimmermann
https://arxiv.org/abs/2506.13305

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function
We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the Itô's formula in Orlicz spaces. Thi…

Computation of the Totient Summatory Function
Lucas Augustus Brown
https://arxiv.org/abs/2506.07386 https://arxiv.org/pdf/2506.07386

Computation of the Totient Summatory Function
An algorithm is devised for computing $Φ(n) = ϕ(1) + ϕ(2) + \cdots + ϕ(n)$ in time $\widetildeΘ(n^{2/3})$ and space $\widetildeΘ(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet hyperbola method and the Mertens function. The algorithm is then used to compute $Φ(10^{19}) = 30396355092701331435065976498046398788$.

P\'{o}lya's conjecture for Dirichlet eigenvalues of annuli
Nikolay Filonov, Michael Levitin, Iosif Polterovich, David A. Sher
https://arxiv.org/abs/2505.21737

Pólya's conjecture for Dirichlet eigenvalues of annuli
We prove Pólya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques, and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon Pólya's original estima…

An Addendum to Plouffe's Ramanujan Identities
Segun Olofin Akerele
https://arxiv.org/abs/2507.00040 https://arxiv.org/pdf/2507.00…

An Addendum to Plouffe's Ramanujan Identities
We introduce a new class of polylogarithm sums closely related to a family studied by L. Vepštas in 2010. These generalized sums depend on two free parameters and yield closed-form expressions involving the Dirichlet eta function. Additionally, we present an alternative proof for a hyperbolic sum originally discussed by Ramanujan.

On the motion of a point charge in a plate capacitor considering influence effects
Heinz-J\"urgen Schmidt, Thomas Br\"ocker
https://arxiv.org/abs/2506.06132

On the motion of a point charge in a plate capacitor considering influence effects
A point charge between the plates of a capacitor generates an influence charge distribution on the plates that modify the electric field acting upon the point charge. This effect is described by the well-known Dirichlet Green's function for the two parallel conducting plate problem for which we derive an infinite mirror charge series representation. At the line perpendicular to the plates and passing through the point charge this Green's function and hence the total force can be explicitly eval…

Evaluating $\zeta(s)$ At Odd Positive Integers Using Automatic Dirichlet Series
L\'aszl\'o T\'oth
https://arxiv.org/abs/2508.04151 https://arxi…

Evaluating $ζ(s)$ At Odd Positive Integers Using Automatic Dirichlet Series
In this paper, we use the Thue-Morse sequence and the paperfolding sequence to build a Dirichlet series that evaluates to a linear combination of the Riemann zeta function at odd positive integers and odd powers of $π$. In doing so, we also provide an alternative proof of a 2015 result by Allouche and Sondow.

Bayesian Conformal Prediction via the Bayesian Bootstrap
Graham Gibson
https://arxiv.org/abs/2508.01418 https://arxiv.org/pdf/2508.01418

Bayesian Conformal Prediction via the Bayesian Bootstrap
Reliable uncertainty quantification remains a central challenge in predictive modeling. While Bayesian methods are theoretically appealing, their predictive intervals can exhibit poor frequentist calibration, particularly with small sample sizes or model misspecification. We introduce a practical and broadly applicable Bayesian conformal approach based on the influence-function Bayesian bootstrap (BB) with data-driven tuning of the Dirichlet concentration parameter, α. By efficiently approxima…

Twisted second moment of primitive cubic L-functions
Ziwei Hong, Zhiyong Zheng
https://arxiv.org/abs/2506.14656 https://arxiv.org/pdf…

Twisted second moment of primitive cubic L-functions
We investigate the mean value of the twisted second moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ cubic\\ genus(χ)=g}}χ(h_1)\barχ(h_2)|L_q(\frac{1}{2}, χ)|^2, \end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive cubic character $χ$. Employing a double Dirichlet series approach, we establish an error term of size

Sharp inequalities and asymptotics for polyharmonic eigenvalues
Davide Buoso, Pedro Freitas
https://arxiv.org/abs/2506.12791 https://…

Sharp inequalities and asymptotics for polyharmonic eigenvalues
We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in $\mathbb R^{d}$. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the correspo…

Tatuzawa's theorem for Rankin-Selberg $L$-functions
Gergely Harcos, Jesse Thorner
https://arxiv.org/abs/2508.10844 https://arxiv.org/pdf/2508.10844

Tatuzawa's theorem for Rankin-Selberg $L$-functions
Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a pos…

Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient
Phuong Le
https://arxiv.org/abs/2508.08859 https://arxiv.org/…

Monotonicity in half-spaces for singular quasilinear elliptic problems involving the gradient
We study positive solutions to the problem $-Δ_p u + \vartheta |\nabla u|^q = \frac{1}{u^γ} + f(u)$ in $\mathbb{R}^N_+$ with the zero Dirichlet boundary condition, where $p>1$, $γ>0$, $0

Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
Mr Nathan Creighton
https://arxiv.org/abs/2507.08640

Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
We consider the large deviations at the order of the variance for the central value of a family of $L$-functions among the members with bounded discriminant. When there is an upper bound on an integer moment of the central value twisted by a short Dirichlet polynomial, we can establish upper bounds on the density of members exhibiting a large central value. We adapt the techniques from Arguin and Bailey for large deviations of the Riemann zeta function to prove results on the degree two family …

Boundary estimates for singular elliptic problems involving a gradient term
Phuong Le
https://arxiv.org/abs/2508.07360 https://arxiv.org/pdf/2508.07360

Boundary estimates for singular elliptic problems involving a gradient term
We study the behavior of weak solutions to the singular quasilinear elliptic problem $-Δ_p u + \vartheta |\nabla u|^q = \frac{1}{u^γ} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $γ>0$, $0

This https://arxiv.org/abs/2503.17228 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Mean value of cubic $L$-funcitons with fixed genus
We investigate the mean value of the first moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ cubic\\ genus(χ)=g}}L_q(\frac{1}{2}, χ), \end{equation*} where $L_q(s,χ)$ denotes the $L$-function associated with primitive cubic character $χ$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{7}{8}+\varepsilon)g}$.

A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application
Qingfeng Sun, Hui Wang, Yanxue Yu
https://arxiv.org/abs/2505.23573

A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application
Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $χ_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $χ$ be a primitive Dirichlet character modulus $q$ with $χ\neqχ_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s, f \otimes χ)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t, f…