Featured Researches

Number Theory

(Logarithmic) densities for automatic sequences along primes and squares

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares ( n 2 ) n≥0 and primes ( p n ) n≥1 exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lemańczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function.

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Number Theory

A Chabauty-Coleman bound for surfaces

Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve C of genus g?? defined over a number field F , with Jacobian of rank at most g?? . Namely, in the case F=Q , if p>2g is a prime of good reduction, then the number of rational points of C is at most the number of F p -points plus a contribution coming from the canonical class of C . We prove a result analogous to Coleman's bound in the case of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman's explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a p -adic analytic subgroup with a subvariety of A by means of overdetermined systems of differential equations in positive characteristic.

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Number Theory

A Density Theorem for Sp(4)

Strong bounds are obtained for the number of automorphic forms for the group ? 0 (q)?�Sp(4,Z) violating the Ramanujan conjecture at any given unramified place, which go beyond Sarnak's density hypothesis. The proof is based on a relative trace formula of Kuznetsov type, and best-possible bounds for certain Kloosterman sums for Sp(4) .

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Number Theory

A Dynamical Analogue of Sen's Theorem

We study the higher ramification structure of dynamical branch extensions, and propose a connection between the natural dynamical filtration and the filtration arising from the higher ramification groups: each member of the former should, after a linear change of index, coincide with a member of the latter. This is an analogue of Sen's theorem on ramification in p -adic Lie extensions. By explicitly calculating the Hasse-Herbrand functions of such branch extensions, we are able to show that this description is accurate for some families of polynomials, in particular post-critically bounded polynomials of p -power degree. We apply our results to give a partial answer to a question of Berger (in arXiv:1411.7064) and a partial answer to a question about wild ramification in arboreal extensions of number fields (raised in both arXiv:math/0408170 and arXiv:1511.00194).

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Number Theory

A Family of Supercongruences Involving Multiple Harmonic Sums

In recent years, the congruence ??i+j+k=p i,j,k>0 1 ijk ?��?2 B p?? (modp), first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers p r with the indexes summing up to m p r where m is coprime to p , where all the indexes are also coprime to p .

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Number Theory

A Hardy-Ramanujan type inequality for shifted primes and sifted sets

We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.

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Number Theory

A Linear Division-Based Recursion with Number Theoretic Applications

A simple remark on infinite series is presented. This applies to a particular recursion scenario, which in turn has applications related to a classical theorem on Euler's phi-function and to recent work by Ron Brown on natural density of square-free numbers.

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Number Theory

A Look at Chowla's Problem

In this paper we look at the history behind Chowla's problem on the solutions to L(1,f)=0 for periodic f . We focus on the results given by Baker, Birch and Wirsing on the topic. We briefly discuss recent results due to Chatterjee, Murty and Pathak which give a full solution when combined with the work of Baker, Birch and Wirsing.

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Number Theory

A Note on Je{?}manowicz' Conjecture for Non-primitive Pythagorean Triples

Let (a,b,c) be a primitive Pythagorean triple parameterized as a= u 2 ??v 2 , b=2uv, c= u 2 + v 2 ,\ where u>v>0 are co-prime and not of the same parity. In 1956, L. Je{?}manowicz conjectured that for any positive integer n , the Diophantine equation (an ) x +(bn ) y =(cn ) z has only the positive integer solution (x,y,z)=(2,2,2) . In this connection we call a positive integer solution (x,y,z)??2,2,2) with n>1 exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case v=2, u is an odd prime. As an application we show the truth of the Je{?}manowicz conjecture for all prime values u<100 .

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Number Theory

A Theorem of Congruent Primes

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter than existing proofs, but the method may be useful in other contexts. The proof of Theorem 1 tracks the set of solutions and this set branches as a binary tree. Conditions set to the theorem restricts the branches so that only one branch is left. Following this branch gives either a solution or a contradiction. In Theorem 1 it leads to a contradiction. The interest is in the proof method, which maybe can be generalized to non-primes.

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