EUDML

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients $λ_{n}$ $(n = 1, 2, ...)$ . We define similar coefficients $λ_{n} (π)$ associated to principal automorphic $L$ -functions $L (s, π)$ over $G L (N)$ . We relate these cofficients to values of Weil’s quadratic functional associated to the representation $π$ on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for $L (s, π)$ . Assuming the...

Haar Type Orthonomal Wavelet Bases in R2.

Jeffrey C. Lagarias; Yang Wang — 1995

The journal of Fourier analysis and applications [[Elektronische Ressource]]

Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors.

Jeffrey C. Lagarias; Yang Wang — 2000

The journal of Fourier analysis and applications [[Elektronische Ressource]]

Simultaneously good bases of a lattice and its reciprocal lattice.

Johan Hastad; Jeffrey C. Lagarias — 1990

Mathematische Annalen

Correction to : “Linear fractional transformations of continued fractions with bounded partial quotients”

Jeffrey C. Lagarias; Jeffrey O. Shallit — 2003

Journal de théorie des nombres de Bordeaux

We fill a gap in the proof of a theorem of our paper cited in the title.

3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell; Jeffrey C. Lagarias — 2015

Acta Arithmetica

The 3x+k function $T_{k} (n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_{k} (\cdot)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_{k} (\cdot)$ . We consider the generating functions $f_{k, m} (z) = \sum_{n > 0, n \to m} z^{n}$ , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k, m} (z)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...

Bounds for the 3x+1 problem using difference inequalities

Ilia Krasikov; Jeffrey C. Lagarias — 2003

Acta Arithmetica

On a two-variable zeta function for number fields

Jeffrey C. Lagarias; Eric Rains — 2003

Annales de l’institut Fourier

This paper studies a two-variable zeta function $Z_{K} (w, s)$ attached to an algebraic number field $K$ , introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When $w = 1$ this function becomes the completed Dedekind zeta function ${\hat{ζ}}_{K} (s)$ of the field $K$ . The function is a meromorphic function of two complex variables with polar divisor $s (w - s)$ , and it satisfies the functional equation $Z_{K} (w, s) = Z_{K} (w, w - s)$ . We consider the special case $K = ℚ$ , where for $w = 1$ this function...

Smooth solutions to the $a b c$ equation: the $x y z$ Conjecture

Jeffrey C. Lagarias; Kannan Soundararajan — 2011

Journal de Théorie des Nombres de Bordeaux

This paper studies integer solutions to the $a b c$ equation $A + B + C = 0$ in which none of $A, B, C$ have a large prime factor. We set $H (A, B, C) = max (| A |, | B |, | C |)$ , and consider primitive solutions ( $\gcd (A, B, C) = 1$ ) having no prime factor larger than ${(log H (A, B, C))}^{κ}$ , for a given finite $κ$ . We show that the $a b c$ Conjecture implies that for any fixed $κ < 1$ the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed $κ > 8$ the $a b c$ equation has infinitely many primitive solutions....