# A proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

@article{VergerGaugry2017APO, title={A proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus}, author={Jean-Louis Verger-Gaugry}, journal={arXiv: Number Theory}, year={2017} }

The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f α (z) associated with the dynamical zeta functions ζ α (z) of the Renyi–Parry arithmetical dynamical systems, for α an algebraic integer α of house α greater than 1, (ii) the discovery of lenticuli of poles of ζ α (z) which uniformly equidistribute at the limit on a limit " lenticular " arc of the unit circle, when α tends to 1 + , giving rise to a continuous lenticular…

## 3 Citations

### A remark on a recent proof of Lehmer's Conjecture [arXiv:1709.03771]

- Mathematics
- 2018

Over one year ago, a very long preprint posted on arXiv [arXiv:1709.03771] and HAL announced a proof of Lehmer's Conjecture (and of other related results). Unfortunately, as was remarked by several…

### On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials

- MathematicsArnold Mathematical Journal
- 2018

The class $${\mathcal {B}}$$B of lacunary polynomials $$f\,(x)\ :=\ -1\ +\ x\ +\ x^{n}\ +\ x^{m_{1}}\ +\ x^{m_{2}}\ +\ \cdots \ +\ x^{m_{s}}$$f(x):=-1+x+xn+xm1+xm2+⋯+xms, where $$s\ \geqslant \…

### Errata and Addenda to Mathematical Constants

- Mathematics
- 2016

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always…

## References

SHOWING 1-10 OF 787 REFERENCES

### On a conjecture of A. Schinzel and H. Zassenhaus

- Mathematics
- 1993

Suppose that α is not a root of unity. In 1933 Lehmer [6] posed the following question: is it true that there exists an absolute constant δ > 0 such that M(α) > 1 + δ? In 1965 Schinzel and Zassenhaus…

### Problème de Lehmer en caractéristique finie

- Mathematics
- 1995

Let IFq (T) be the rational function field over the finite field IFq of characteristic p > 0. Let C (x) be the canonical height associated with the Carlitz module C. We prove an analog of the real…

### Classical and Elliptic Polylogarithms and Special Values of L-Series

- Mathematics
- 2000

The Dirichlet class number formula expresses the residue at s = 1 of the Dedekind zeta function ζ F(s) of an arbitrary algebraic number field F as the product of a simple factor (involving the class…

### ON A PROBLEM OF SCHINZEL

- Mathematics
- 1956

Using elementary methods, A. Schinzel and W. Sierpinski obtained some results concerning the distribution of the Euler function (n) and the function σ(n) which represents the sum of divisors of n.…

### An extension of Pólya’s theorem on power series with integer coefficients

- Mathematics
- 1968

This was proved first by Polya [4] for the case in which G is simply connected. A general proof was given later by Polya [5, p. 703]. (It is understood that f(z) is single valued in G.) A more…

### On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

- Mathematics
- 2016

Abstract Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers (θn−1)n≥2 $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1.…

### On the spectrum of the Zhang-Zagier height

- Mathematics, Computer ScienceMath. Comput.
- 2001

An algorithm able to find some algebraic numbers of small height is described, and this search up to degree 64 suggests that the spectrum of n(α) may have a limit point less than 1.292.

### On the dichotomy of Perron numbers and beta-conjugates

- Mathematics
- 2008

Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function $${f_{\beta}(z) =-1 + \sum_{i \geq 1} t_i z^i}$$ associated with…

### On the Rationality of the Zeta Function of an Algebraic Variety

- Mathematics
- 1960

Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its…

### On the distribution of the roots of a polynomial with integral coefficients

- Mathematics
- 1951

The following question has been raised by D. H. Lehmer' in connection with prime number problems: "If e is a positive quantity, to find a polynomial of the form f(z) =z"+alzr-+ +a, where the a's are…