Content uploaded by Ram Murty

Author content

All content in this area was uploaded by Ram Murty on Aug 16, 2018

Content may be subject to copyright.

Content uploaded by Ram Murty

Author content

All content in this area was uploaded by Ram Murty on Aug 16, 2018

Content may be subject to copyright.

... An unsolved Artin's conjecture asserts that δ p = p − 1 for infinitely many primes p if a is not a square. In [19], Erdős and Murty obtained a nontrivial lower bound on δ p , which implies ...

... By Theorem 1, such r/p belongs to V a, f , which implies the density of V a, f in [0, 1]. For infinitely many prime numbers p, the exponent 0.0896 can be improved by combining [19] with a subsequent result of Baker and Harman [26] which yields the exponent 0.677 for p in (5). By a result of Heath-Brown (Corolary 2 of [27]), there are at most three primes a for which Artin's conjecture fails to hold. ...

... Finally, assume that A ≥ 2 and d ≥ 1 are two fixed integers. Then, by the abovementioned result [19], for almost all primes p the order of A modulo p is at least p 1/2 . Thus, the order of the multiplicative group generated by A modulo p d , where p is any of those almost all primes, is at least p 1/2 as well. ...

In this paper, we explicitly describe all the elements of the sequence of fractional parts {af(n)/n}, n=1,2,3,…, where f(x)∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2 is an integer. We also show that each value w={af(n)/n}, where n≥nf and nf is the least positive integer such that f(n)≥n/2 for every n≥nf, is attained by infinitely many terms of this sequence. These results combined with some earlier estimates on the gaps between two elements of a subgroup of the multiplicative group Zm* of the residue ring Zm imply that this sequence is everywhere dense in [0,1]. In the case when f(x)=x this was first established by Cilleruelo et al. by a different method. More generally, we show that the sequence {af(n)/nd}, n=1,2,3,…, is everywhere dense in [0,1] if f∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2, d≥1 are integers such that d has no prime divisors other than those of a. In particular, this implies that for any integers a≥2 and b≥1 the sequence of fractional parts {an/nb}, n=1,2,3,…, is everywhere dense in [0,1].

... In [19], it was shown that if ord(A, N) was somewhat larger than N 1/2 (and N satisfies a further genericity condition), then all eigenfunctions in H N are uniformly distributed. Note that the condition holds for almost all primes [7]. Separately, it was shown that ord(A, N) is sufficiently large for almost all integers N. ...

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm Sp}(2g,\mathbb Z)$, which we take to be ergodic.Under some natural assumptions, we show that there is a density one sequence of integers $N$ so that as $N$ tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant $N$ are uniformly distributed. For the two-dimensional case ($g=1$), this was proved by P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.

... While it seems likely that such variants of Lemma 3·1 hold (one can present a similar heuristic as for Artin's conjecture), our understanding is very limited. Unconditionally, we only know that given an integer a with |a| > 1, the order of a modulo p is almost always > p 1/2 [3]. Under GRH one has Lemma 3·1, and an involved variant of Hooley's classical (conditional) solution of Artin's conjecture yields an analogue of Lemma 3·1 in the case where P is of degree 2 and remains irreducible modulo p, as shown by Roskam [20]. ...

Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree d whose Galois group is $S_d$ . Let $(a_n)$ be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence $(a_n)$ is positive.

... belongs to the ε-fraction of bad multipliers in Theorem 3 and Corollary 4) for all primes p. This happens in general whenever the residue a ∈ F p has a representative in Z of size < exp H (μ) (or more generally if a is the reduction modulo a prime ideal of norm p of a fixed algebraic number with Mahler measure < exp H (μ)). On the other hand, it is well-known and easy to check that apart from a density zero family of primes, any given a ∈ Z \ {−1, 0, 1} has multiplicative order at least c √ p/ log p, and it is also known modulo GRH that given any function ε( p) > 0 tending to 0 when p → +∞, a has multiplicative order even at least ε( p) p for a density one set of primes, see [8]. ...

We study the Markov chain $x_{n+1}=ax_n+b_n$ on a finite field ${\mathbb {F}}_p$, where $a \in {\mathbb {F}}_p^{\times }$ is fixed and $b_n$ are independent and identically distributed random variables in ${\mathbb {F}}_p$. Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of $a \in {\mathbb {F}}_p^\times$. We also obtain weaker, but unconditional, upper bounds for the mixing time.

... In this article, we are concerned about the case when {x n } is a linear recurrence sequence in F p . For a simple sequence 2 n (mod p), it follows combining a result of Erdős and Murty [6] and a result of Glibichuk [13] that for almost all primes p, every residue class modulo p can be represented in the following form 2 n 1 + · · · + 2 n 8 (mod p), for certain positive integers n 1 , · · · , n 8 . ...

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski’s bound for exponential sums attached to certain linear recurrence sequences over finite fields.

Mersenne primes and Fermat primes may be thought of as primes of the form $\Phi_m(2)$, where $\Phi_m(x)$ is the $m$th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.

For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.

Since Hooley’s seminal 1967 resolution of Artin’s primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH).

Given $(a_1, \dots , a_n, t) \in \mathbb {Z}_{\ge 0}^{n + 1}$, the Subset Sum problem ($\mathsf {SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum _{i \in S} a_i = t$. Bellman (1957) gave a pseudopolynomial time dynamic programming algorithm which solves the Subset Sum in O(nt) time and O(t) space.In this work, we present search algorithms for variants of the Subset Sum problem. Our algorithms are parameterized by k, which is a given upper bound on the number of realisable sets (i.e. number of solutions, summing exactly t). We show that $\mathsf {SSUM}$ with a unique solution is already $\mathsf {NP}$-hard, under randomized reduction. This makes the regime of parametrized algorithms, in terms of k, very interesting.Subsequently, we present an $\tilde{O}(k\cdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a ${\mathsf {poly}}(knt)$-time and $O(\log (knt))$-space deterministic algorithm that finds all the realisable sets for a subset sum instance. Our algorithms use analytic and number-theoretic techniques.
KeywordsSubset sumPower seriesIsolation lemmaHamming weightInterpolationLogspaceNewton’s identities

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such that $n^2=\sum_{i=1}^m c_i g^{k_i}$ is $o\left(\left(\log M \right)^{m-1/2}\right)$. We also obtain a similar bound when dealing with more general inequalities $\left|Q(n)-\sum_{i=1}^m c_i\lambda^{k_i}\right|\le B,$ where $Q\in {\mathbb C}[X]$ and also $\lambda\in {\mathbb C}$ (while $B$ is a real number).

ResearchGate has not been able to resolve any references for this publication.