On a positivity property of the Riemann ξ-function
Zero spacing distributions for differenced L-functions
Correction to: "On a positivity property of the Riemann ξ-function" (Acta Arith. 89 (1999), 217-234)
Li coefficients for automorphic -functions
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients . We define similar coefficients associated to principal automorphic -functions over . We relate these cofficients to values of Weil’s quadratic functional associated to the representation 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 . Assuming the...
Haar Type Orthonomal Wavelet Bases in R2.
Orthogonality Criteria for Compactly Supported Refinable Functions and Refinable Function Vectors.
Simultaneously good bases of a lattice and its reciprocal lattice.
Correction to : “Linear fractional transformations of continued fractions with bounded partial quotients”
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
The 3x+k function sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of . We consider the generating functions , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions 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
On a two-variable zeta function for number fields
This paper studies a two-variable zeta function attached to an algebraic number field , 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 this function becomes the completed Dedekind zeta function of the field . The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the special case , where for this function...
Smooth solutions to the equation: the Conjecture
This paper studies integer solutions to the equation in which none of have a large prime factor. We set , and consider primitive solutions () having no prime factor larger than , for a given finite . We show that the Conjecture implies that for any fixed 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 the equation has infinitely many primitive solutions....
On the range of fractional parts {ξ(p/q)ⁿ}
Product-free sets with high density
On the normality of arithmetical constants.
The distribution of trees.
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