Sets with prescribed lower and upper weighted densities.
Seven octonary quadratic forms
Several generating functions for second-order recurrence sequences.
Several inequalities about the number of positive divisors of a natural number .
Several-variable -adic families of Siegel-Hilbert cusp eigensystems and their Galois representations
S-expansions in dimension two
The technique of singularization was developped by C. Kraaikamp during the nineties, in connection with his work on dynamical systems related to continued fraction algorithms and their diophantine approximation properties. We generalize this technique from one into two dimensions. We apply the method to the the two dimensional Brun’s algorithm. We discuss, how this technique, and related ones, can be used to transfer certain metrical and diophantine properties from one algorithm to the others. In...
S-extremal strongly modular lattices
S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.
S-ganze Punkte auf elliptischen Kurven.
SGn of Orders and Group-Rings.
Sharp bounds for the intersection of nodal lines with certain curves
Let be a hyperbolic surface and let be a Laplacian eigenfunction having eigenvalue with . Let be the set of nodal lines of . For a fixed analytic curve of finite length, we study the number of intersections between and in terms of . When is compact and a geodesic circle, or when has finite volume and is a closed horocycle, we prove that is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between and is . This bound is sharp.
Sharp bounds for the number of solutions to simultaneous Pellian equations
Sharp bounds on the mathematical constant e.
Sharp elementary estimates for the sequence of primes
Sharp large deviations for gaussian quadratic forms with applications
Sharp large deviations for Gaussian quadratic forms with applications
Under regularity assumptions, we establish a sharp large deviation principle for Hermitian quadratic forms of stationary Gaussian processes. Our result is similar to the well-known Bahadur-Rao theorem [2] on the sample mean. We also provide several examples of application such as the sharp large deviation properties of the Neyman-Pearson likelihood ratio test, of the sum of squares, of the Yule-Walker estimator of the parameter of a stable autoregressive Gaussian process, and finally of the empirical...
Sharper ABC-based bounds for congruent polynomials
Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial . Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials of degree at most cannot all be congruent modulo . This paper presents two improvements...
Sheaves of bounded p-adic logarithmic differential forms
Shifted B-numbers as a set of uniqueness for additive and multiplicative functions
Shifted primes without large prime factors
Shifted values of the largest prime factor function and its average value in short intervals
We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence is uniformly distributed modulo 1.