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Li coefficients for automorphic L -functions

Jeffrey C. Lagarias — 2007

Annales de l’institut Fourier

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...

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 ( · ) 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 ( · ) . We consider the generating functions f k , m ( z ) = ∑ n > 0 , n → 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...

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 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....

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