Igor Shparlinski
We consider a generalisation of the classical Lehmer problem about the distribution of modular inverses in arithmetic progression, introduced by E. Alkan, F. Stan and A. Zaharescu. Using bounds of sums of multiplicative characters instead of traditionally applied to this kind of problem Kloosterman sums, we improve their results in several directions.
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We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang.
New evidence for Rémond's generalisation of Lehmer's conjecture
本文将Pottmeyer结果推广至几乎分裂半阿贝尔簇，证明商去饱和闭包后有理点群为自由群，并复现Rémond的Kummer理论证明。
Lehmer's totient problem asks if there exists a composite number $d$ such that its totient divide $d-1$. In this article we generalize the Lehmer's totient problem in algebraic number fields. We introduce the notion of a Lehmer number. Lehmer numbers are defined to be the natural numbers which obey the Lehmer's problem in the ring of algebraic integers of a number field.
Toward a generalization of Lehmer's problem to adelic curves
本文探讨了Lehmer问题在adelic曲线上的推广，研究其在有限生成字段中的应用。
We find the minimal non-trivial integer variable group determinant for any dihedral group of order less than $3.79\times 10^{47}$. We think of this as the Lind-Lehmer problem for the dihedral group. We give a complete description of the determinants for some dihedral groups including $D_{2p}$ and $D_{4p}$.
We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its obstruction index over $\Q^{ab}$, the maximal abelian extension of $\Q$. We prove up to $\eps$ the sharpest conjecture that can be formulated.
The Euler's totient function $ φ(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ φ(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.
We draw connections between the various conjectures which are included in G. Rémond's generalized Lehmer problems. Specifically, we show that the degree one form of his conjecture for the multiplicative group is, in a sense, almost as strong as the strong form. Our insights into the conjectures apply a basic result on what may be called Diophantine approximation in the metric induced by the canonical height. We also present a previously unpublished partial result of Rémond on his conjectures.
We introduce vector space norms associated to the Mahler measure by using the L^p norm versions of the Weil height recently introduced by Allcock and Vaaler. In order to do this, we determine orthogonal decompositions of the space of algebraic numbers modulo torsion by Galois field and degree. We formulate L^p Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p = 1 case and the Schinzel-Zassenhaus conjecture in the p = infinity case.
A composite number $n$ is called a Lehmer number when $ϕ(n) | n - 1$, where $ϕ$ is the Euler totient function. Lehmer's totient problem asks if there exist any composite numbers $n$ such that $ϕ(n)| n-1$? No such numbers are known. In this paper we establish an Euler Totient Inequality and relate some new parameters to Lehmer numbers. As an application of what we have done, we show that for all prime numbers $p$ and for all odd square-free numbers $n$, at most one of $n$ or $pn$ is a Lehmer number. Finally we suggest some open problems for the future investigations on the Lehmer numbers.
Lehmer's totient problem consists of determining the set of positive integers $n$ such that $φ(n)|n-1$ where $φ$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a composite number such that $φ(n)|(n-1)^k$. The relation between $k$-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also $k$-Lehmer numbers.
We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number τ_0 = 1.17628... .
Sequences related to Lehmer's problem
本文研究Mahler测度定义下与Lehmer问题相关的整数序列性质，探讨其潜在解法。
On existence of generalised Lehmer numbers modulo primes
本文推广Lehmer数为模奇素数$q$的$q$-元组，计数并给出带显式误差项的渐近估计，利用特征和与Deligne界。
A problem of D. H. Lehmer in short intervals. I
研究与模q乘法逆元奇偶性不同的整数个数问题，利用算术指数对和素变量不完全克洛斯特曼和估计，在特定条件下得到短区间内此类整数个数渐近公式并证明素变量版本，改进了前人结果。
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $φ(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $φ(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each…
A new characterization of prime numbers and solution to Lehmer's conjecture on Euler's totient function
利用初等对称多项式与数论结果解决 Lehmer 猜想，并由此得到素数的新刻画。
We formulate Lehmer's Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components then its Lehmer constant vanishes.
Waring's problem involving D.H. Lehmer numbers
本文证明每个充分大的整数，除非模16余15或16，可表示为14个D.H. Lehmer数的四次方之和；且每个充分大的整数可表示为16个D.H. Lehmer数的四次方之和。
A problem of D. H. Lehmer in short intervals. II
研究与模\(q\)乘法逆元奇偶性不同的整数个数问题，通过对含克洛斯特曼分数的双线性形式估计，得到小于\(q^{1/2}\)的此类整数个数渐近公式及误差项二阶矩估计。
We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algeb…
On Lehmer's problem and related problems
本文研究了Lehmer问题及相关问题，证明了合数N满足特定等式时M的上界，并对unitary totient function、Dedekind function及unitary divisors之和给出了类似结论。
We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.
We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.
Motivated by a famous question of Lehmer about the Mahler measure we study and solve its analytic analogue.
We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an absolute power of a ``log'' (independant of the dimension of $A$). We thus extend the results of F. Amoroso and S. David on the same problem on a multiplicative group $\mathbb{G}_m^n$. When $A$ is an elliptic curve and $V=\bar{P}$ is the set of conjugates of a non torsion $\bar{k}$-point, we reobtain the result of M. Laurent on the elliptic Lehmer's problem.
Lehmer's totient problem asks whether there exists any composite number $n$ such that $φ(n) \, \mid \, (n-1)$, where $φ$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n > 10^{30}$. In this paper, we develop a new approach to the problem via some recent results in group theory related to a function $ψ$ (the sum of order of elements of a group) and show that if $k φ(n) = n-1$ for some integer $k$, then $k$ must be $\geq 3$, and actually, if $5, 7, 11, 13 \not | n$, $k \geq 4$. This implies that any counterexample must be such that $n > 10^{8171}$ …
On Pillai's Problem involving Lucas sequences of the second kind
本文研究第二类Lucas序列与完全幂之差的Pillai型丢番图方程，证明在至少存在三个解时，解的大小和特征多项式系数均有上界。
The trigonometric polynomial on sums of two squares, an additive problem and generalisation
本文研究两互素平方和奇数集合上的三角多项式，证明其在有理点附近的强估计，并结合受转移原理启发的圆法解决三类正密度子集元素相加的三表示问题。
Generalisation of Baker's Forcing Method to Arbitrary Prime and NP-hardness of Several $p$-adic Optimisations
研究将贝克强制方法推广到任意素数的p进数优化，证明了p进数线性回归、2进数动态神经网络及相关p进数优化问题部分推广的NP难。
On a Generalisation of a Function of Ron Graham's
本文证明了 Kagey 和 Rajesh 关于满足 $g(n)=2n$ 的 $n$ 的特征猜想，并对 Ron Graham 函数 $g(n)$ 到 $m$ 次幂的推广给出了若干新结果。
A Totient Function Associated with Variants of Groups
本文引入一种新的陶辛函数，结合欧拉函数及其1869年Schemmel扩展，探讨其评估方法及数论问题，面向广泛受众。
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