arXiv subjects

Alexandru Pascadi

Publications and source records attributed to Alexandru Pascadi.

12 recordsLinked to original sources

arXiv2026

Bilinear forms with Kloosterman sums via quadratic characters

We prove new bounds for bilinear forms with Kloosterman sums, valid for all moduli c. In the critical range where the summation length is the square root of the modulus, the saving over the trivial bound is c^{-1/32}, improving on all previous approaches even for prime moduli. This is based on a new connection to quadratic character sums. Applications to moments of twisted L-functions and to the large sieve for exceptional Maass forms are given.

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arXiv2025

Non-abelian amplification and bilinear forms with Kloosterman sums

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums of length $\sqrt{c}$, our method produces a non-trivial bound for all moduli except near-primes, saving $c^{-1/12}$ for products of two primes of the same size. Combining this with previous results for prime moduli, we achieve savings beyond the P\'olya-Vinogradov range for all moduli. We give applications to moments of twisted cuspidal $L$-functions, and to large sieve inequalities for exceptional cusp forms with composite levels.

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arXiv2025

Large sieves for $\mathrm{GL}_n$ and applications

Let $\mathfrak{F}_n$ be the set of unitary cuspidal automorphic representations of $\mathrm{GL}_n$ over a number field $F$, and let $S\subseteq\mathfrak{F}_n$ be an arbitrary finite subset. Given $π_0\in\mathfrak{F}_{n_0}$, we establish large sieve inequalities for the families $\{L(s,π)\colon π\in S\}$ and $\{L(s,π\timesπ_0)\colon π\in S\}$ that, unlike previous results, are independent of progress towards the generalized Ramanujan conjecture, and simultaneously handle the Dirichlet coefficients of $L$, $L^{-1}$, and $\log L$. We also give the first such result that improves upon the trivial bound for short sums. We present several applications, including: (1) the strongest bound for $\sum_{π\in S}|L(\frac{1}{2},π)|^2$ that holds for arbitrary $S$, (2) significant improvements to zero density estimates for families of automorphic and Rankin--Selberg $L$-functions, counting violations to the generalized Riemann hypothesis near $\mathrm{Re}(s)=1$, (3) the removal of all unproven hypotheses in the conditional log-free zero density estimate for families of Rankin--Selberg $L$-functions proved by Brumley, Thorner, and Zaman, and (4) an improvement of the density theorem for non-archimedean Langlands parameters due to Lichtman and Pascadi, counting violations to the generalized Ramanujan conjecture.

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arXiv2025

On the exponents of distribution of primes and smooth numbers

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve weights). This completely eliminates the dependency on Selberg's eigenvalue conjecture in previous works of Lichtman and the author, which built in turn on results of Maynard and Drappeau. We rely on recent large sieve inequalities for exceptional Maass forms of the author for additively-structured sequences, and on a related result of Watt for multiplicatively-structured sequences. As applications, we prove refined upper bounds for the counts of twin primes and consecutive smooth numbers up to $x$.

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arXiv2025

Smooth numbers in arithmetic progressions to large moduli

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66/107-o(1)}$. This overcomes a longstanding barrier of $x^{3/5-o(1)}$ present in previous works of Bombieri-Friedlander-Iwaniec, Fouvry-Tenenbaum, Drappeau, and Maynard. We build on Drappeau's variation of the dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers-Iwaniec type estimates for sums of Kloosterman sums.

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arXiv2024

Density theorems for $\text{GL}_n$ via Rankin-Selberg $L$-functions

We obtain density theorems for cuspidal automorphic representations of $\text{GL}_n$ over $\mathbb{Q}$ which fail the generalized Ramanujan conjecture at some place. We depart from previous approaches based on Kuznetsov-type trace formulae, and instead rely on $L$-function techniques. This improves recent results of Blomer near the threshold of the pointwise bounds.

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arXiv2024

Several new product identities in relation to two-variable Rogers-Ramanujan type sums and mock theta functions

Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's septuple identity. We view these series expansions as representations in canonical bases of certain vector spaces of quasiperiodic meromorphic functions (related to sections of line and vector bundles), and find new identities for two nonuple products, an undecuple product, and several two-variable Rogers-Ramanujan type sums. Our main theorem explains a correspondence between the septuple product identity and the two original Rogers-Ramanujan identities, involving two-variable analogues of fifth-order mock theta functions. We also prove a similar correspondence between an octuple product identity of Ewell and two simpler variations of the Rogers-Ramanujan identities, which is related to third-order mock theta functions, and conjecture other occurrences of this phenomenon. As applications, we specialize our results to obtain identities for quotients of generalized Dedekind eta functions and mock theta functions.

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arXiv2024

Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$

We prove new large sieve inequalities for the Fourier coefficients $\rho_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski's previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions.

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arXiv2023

The sparse regularity method with Schatten norms and entropy

We introduce a regularity method for sparse graphs, with new regularity and counting lemmas which use the Schatten-von-Neumann norms to measure uniformity. This leads to $k$-cycle removal lemmas in subgraphs of mildly-pseudorandom graphs, and also in graphs lacking a quasi-smooth family of bipartite subgraphs, extending results of Conlon, Fox, Sudakov and Zhao. We give some applications in additive combinatorics: one about translation-invariant linear equations in subsets of mildly-pseudorandom sets, one about such equations in generalized Sidon sets, and one about polygonal patterns in subsets of $\mathbf{Z}^2$ with few parallelograms (giving a two-dimensional analogue for a result of Prendiville). Separately, our regularity lemma implies a dense graph removal lemma with mild constant dependencies, in graphs whose spectral $L^{2-\varepsilon}$ norms are small.

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arXiv2019

Patterns of primes in the Sato-Tate conjecture

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $μ_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval $I\subseteq [-1, 1]$, let $p_{I,n}$ denote the $n$th prime such that $a_E(p)/(2\sqrt{p})\in I$. We show $\liminf_{n\to\infty}(p_{I,n+m}-p_{I,n}) < \infty$ for all $m\ge 1$ for "most" intervals, and in particular, for all $I$ with $μ_{ST}(I)\ge 0.36$. Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.

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arXiv2018

A Graphical Approach to Finding the Frobenius Number, Genus and Hilbert Series of a Numerical Semigroup

This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have multiple inputs and outputs. If such a construction is possible, then determining whether the studied semigroup is symmetric, or finding explicit forms of its Apéry set and Hilbert series, is reduced to straightforward computations assisted by a MAPLE program we made available on arXiv. This approach applies to many of the cases considered in literature, including semigroups generated by arithmetic and geometric sequences, compound sequences, progressions of the form $a^n, a^n + a, \ldots, a^n + a^{n-1}$, triangular and tetrahedral numbers, certain Fibonacci triplets, etc. After explaining the general approach in more detail, the paper studies the types of edges that can be used as building blocks of a reduction graph, as well as a series of operations that serve to modify or combine valid reduction graphs. In the end of the paper, we use these techniques to solve the Frobenius problem for 7 new classes of numerical semigroups.

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arXiv2018

Computer-Assisted Proofs of Congruences for Multipartitions and Divisor Function Convolutions, based on Methods of Differential Algebra

This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in $\mathbb{Z}/q\mathbb{Z}$, implemented in a couple of MAPLE programs available as ancillary files on arXiv. The first results of the paper are Ramanujan-type congruences of the form $p^{*k}(qn+r) \equiv_q 0$ and $σ^{*k}(qn+r) \equiv_q 0$, where $p(n)$ and $σ(n)$ are the partition and divisor functions, $q > 3$ is prime, and $^{*k}$ denotes $k^{th}$-order self-convolution. We prove all the valid congruences of this form for $q \in \{5, 7, 11\}$, including the three Ramanujan congruences, and a nontrivial one for $q = 17$. All such multipartition congruences have already been settled in principle up to a numerical verification due to D. Eichhorn and K. Ono via modular forms, but our proofs are purely algebraic. On the other hand, the majority of the divisor function congruences are new results. We then proceed to search for more general congruences modulo small primes, concerning linear combinations of $σ^{*k}(pn+r)$ for different values of $k$, as well as weighted convolutions of $p(n)$ and $σ(n)$ with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N. C. Bonciocat.

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