arXiv subjects
Publications and source records attributed to Ali Saraeb.
We prove a converse theorem for functional equations of Dirichlet $L$-functions. Under mild assumptions, we prove that these functional equations for $L$-series of the form $\sum_{n\ge 1} f(n) n^{-s}$ force the coefficient function $f$ to be a primitive Dirichlet character. Consequently, these functional equations force the existence of an Euler product.
We develop a resurgent approach to the problem of unique continuation of mock theta functions across their natural boundary. The starting point is the representation of the associated Mordell-Appell integrals as Laplace transforms of resurgent functions, which serve as the primary analytic objects. By rotating the Laplace contour by $\pi$, i.e. onto the Stokes line, one obtains, in all known cases, the mock-modular relations between the Mordell-Appell integrals and the corresponding unary series in $\hat q=e^{-\pi i \tau}$ and $\hat q_1=e^{-\pi i (-1/\tau)}$. We then prove that these relations admit a unique solution on the $q$-side, expressed in terms of $q=e^{\pi i \tau}$ and $q_1=e^{\pi i (-1/\tau)}$, with coefficients determined by the corresponding Mordell-Appell integrals. This yields a canonical continuation across the natural boundary, given by a resurgent extension of the classical principle of permanence of relations, and singles out a distinguished family of mock theta functions in each group. We present a complete analysis for the order 3 and 5 cases (mf3 and mf5). The method extends naturally to higher orders; a general theory will appear in a separate paper.
Let $\mathbb{S}^1$ denote the unit circle. We introduce and develop the analytic and bias theory of unimodular fake M\"obius functions, i.e. multiplicative functions $\mathfrak{f}:\mathbb{N} \to \mathbb{S}^1 \cup \{0\}$ whose prime-power values are prescribed by a fixed sequence $\{\varepsilon_k\}_{k\ge1}$ via the rule $\mathfrak{f}(p^k)=\varepsilon_k$ for every prime $p$ and every $k\ge1$. A key feature of these functions is that their Dirichlet series admit a factorization into complex powers of the Riemann zeta function. Our main analytic result is an explicit formula for the smoothed summatory function $\sum_{n\ge1}\mathfrak{f}(n)e^{-n/x}$, consisting of a leading main term together with a sequence of lower-order terms. The formula may be viewed as an extension of the Selberg-Delange method and is expected to be of independent interest. As an application, we introduce a notion of bias at a natural scale and obtain an explicit criterion distinguishing persistent bias, apparent bias, and no bias for unimodular fake M\"obius functions.
This paper presents two concrete applications of Artificial Intelligence to algorithmic and analytic number theory. Recent benchmarks of large language models have mainly focused on general mathematics problems and the currently infeasible objective of automated theorem proving. In the first part of this paper, we relax our ambition and focus on a more specialized domain: we evaluate the performance of the state-of-the-art open-source large language model Qwen2.5-Math-7B-Instruct on algorithmic and computational tasks in algorithmic number theory. On a benchmark of thirty algorithmic problems and thirty computational questions taken from classical number-theoretic textbooks and Math StackExchange, the model achieves at least 0.95 accuracy (relative to the true answer) on every problem or question when given an optimal non-spoiling hint. The second part of the paper empirically verifies a folklore conjecture in analytic number theory stating that the modulus \(q\) of a Dirichlet character \(\chi\) is uniquely determined by the initial nontrivial zeros \(\{\rho_1,\dots,\rho_k\}\) (for some \(k\in\mathbb{N}\)) of the corresponding Dirichlet \(L\)-function \(L(s,\chi)\). We train a LightGBM multiclass classifier to predict the conductor \(q\) for 214 randomly chosen Dirichlet \(L\)-functions from a vector of statistical features of their initial zeros (moments, finite-difference statistics, FFT magnitudes, etc.). The model empirically verifies the conjecture for small \(q\), achieving at least 93.9\% test accuracy when sufficient statistical properties of the zeros are incorporated. For the second part of the paper, code and dataset are available.
We propose a numerical method for approximating and discovering zeros of the Dirichlet L-function L(s, chi) corresponding to real Dirichlet characters chi.
This paper presents an algorithmic method for generating random orthogonal matrices \(A\) that satisfy the property \(A^t S A = S\), where \(S\) is a fixed real invertible symmetric or skew-symmetric matrix. This method is significant as it generalizes the procedures for generating orthogonal matrices that fix a general fixed symmetric or skew-symmetric bilinear form. These include orthogonal matrices that fall to groups such as the symplectic group, Lorentz group, Poincar\'e group, and more generally the indefinite orthogonal group, to name a few. These classes of matrices play crucial roles in diverse fields such as theoretical physics, where they are used to describe symmetries and conservation laws, as well as in computational geometry, numerical analysis, and number theory, where they are integral to the study of quadratic forms and modular forms. The implementation of our algorithms can be accomplished using standard linear algebra libraries.
In this paper, we generalize Rademacher's proof of the transformation law of the eta function to the Jacobi theta function using Residue calculus.
In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of $\theta_1(z, \tau)$.
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.
We present a new proof of the transformation law of $\vartheta_1$ under the action of the generator of the full modular group $\Gamma$ using Siegel's method.