## 10 Citations

A note on pseudo-polynomials divisible only by a sparse set of primes

- Mathematics
- 2020

For certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a…

Derangements and the $p$-adic incomplete gamma function.

- Mathematics
- 2020

We introduce a new $p$-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath…

On the Joint Distribution of the Roots of Pairs of Polynomial Congruences

- Mathematics
- 2020

Let f(x) be a primitive irreducible polynomial with integer coefficients of degree greater than one. In 1964, Hooley showed that the sequence of normalized roots u/n, where f(u) = 0(n), ordered in…

Exponential sums, twisted multiplicativity and moments

- Mathematics
- 2021

We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of…

Fine-scale distribution of roots of quadratic congruences

- Mathematics
- 2021

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence μ ≡ D mod m, with D > 0 square-free and D 6≡ 1 mod 4. This is achieved by translating the…

Sectorial equidistribution of the roots of $x^2 + 1$ modulo primes

- Mathematics
- 2021

The equation x+1 = 0 mod p has solutions whenever p = 2 or 4n+1. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots…

Structural results on lifting, orthogonality and finiteness of idempotents

- MathematicsRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
- 2021

In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely…

On primary pseudo-polynomials (Around Ruzsa's Conjecture)

- Mathematics
- 2020

Every polynomial P (X) ∈ Z[X] satisfies the congruences P (n + m) ≡ P (n) mod m for all integers n,m ≥ 0. An integer valued sequence (an)n≥0 is called a pseudo-polynomial when it satisfies these…

On pseudo-polynomials divisible only by a sparse set of primes and $\a$-primary pseudo-polynomials

- Mathematics
- 2020

We explore two questions about pseudo-polynomials, which are functions f : N → Z such that k divides f(n+ k)− f(n) for all n, k. First, for certain arbitrarily sparse sets R, we construct…

Parameterizing roots of polynomial congruences

- Mathematics
- 2020

We use the arithmetic of ideals in orders to parameterize the roots $\mu \pmod m$ of the polynomial congruence $F(\mu) \equiv 0 \pmod m$, $F(X) \in \mathbb{Z}[X]$ monic, irreducible and degree $d$.…

## References

SHOWING 1-10 OF 27 REFERENCES

Exponential sums with reducible polynomials

- Mathematicsdiscrete Analysis
- 2019

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we…

A note on pseudo-polynomials divisible only by a sparse set of primes

- Mathematics
- 2020

For certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a…

On pseudo-polynomials

- Mathematics, Philosophy
- 1971

In a recent paper [1] I made the following Definition. The function f: ℤ + ∪ {0} → ℤ is a pseudo-polynomial if for all integers n ≥ 0, k ≥ 1.

The smallest root of a polynomial congruence

- Mathematics
- 2020

Fix f(t) ∈ Z[t] having degree at least 2 and no multiple roots. We prove that as k ranges over those integers for which the congruence f(t) ≡ 0 (mod k) is solvable, the least nonnegative solution is…

ERDŐS–TURÁN WITH A MOVING TARGET, EQUIDISTRIBUTION OF ROOTS OF REDUCIBLE QUADRATICS, AND DIOPHANTINE QUADRUPLES

- Mathematics
- 2011

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of…

Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

- Mathematics, Computer ScienceANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2020

The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input"Sheaves, except for parameters in a high-codimension subset.

Algebraic twists of modular forms and Hecke orbits

- Mathematics
- 2012

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a…

Sommes de modules de sommes d'exponentielles

- Mathematics
- 2003

Keywords: Kloosterman sums ; absolute value ; mean value ; exponential sums Reference TAN-ARTICLE-2003-002doi:10.2140/pjm.2003.209.261 Record created on 2008-11-14, modified on 2017-05-12