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arXiv:2402.03999 (Published 2024-02-06)
An average version of Cilleruelo's conjecture for families of $S_n$-polynomials over a number field
Categories: math.NTFor $ f\in\mathbb{Z}[X] $ an irreducible polynomial of degree $ n $, the Cilleruelo's conjecture states that$$\log(\mbox{lcm}(f(1),\dots,f(M)))\sim(n-1)M\log M$$as $ M\rightarrow+\infty $, where $ \mbox{lcm}(f(1),\dots,f(M)) $ is the least common multiple of $f(1),\dots,f(M)$. It's well-known for $ n=1 $ as a consequence of Dirichlet's Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for $S_n$-polynomials with integral coefficients over a fixed extension $K/\mathbb{Q}$ by considering the least common multiple of ideals of $\mathcal{O}_K$.
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arXiv:1808.05416 (Published 2018-08-16)
Estimates of Fourier coefficients of cusp forms associated to cofinite Fuchsian subgroups
Categories: math.NTIn this article, we prove an average version of the Ramanujan-Petersson conjecture for cusp forms associated to arbitrary cofinite Fuchsian subgroups.
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arXiv:1601.05324 (Published 2016-01-20)
On General Prime Number Theorems with Remainder
Comments: 15 pagesCategories: math.NTWe show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all} n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all} n \in \mathbb{N},$$ where $N$ and $\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299--307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Ces\`aro sense.