New asymptotic formulas for the distribution of left ideals of orders.
New formulations of the class number one problem
New inequalities involving the zeta function.
New integral representations for the square of the Riemann zeta-function
Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such factors occur...
New series involving the zeta function.
New versions of the Nyman-Beurling criterion for the Riemann hypothesis.
Non annulation des fonctions des formes modulaires de Hilbert au point central
La conjecture de Birch et Swinnerton-Dyer donne des estimations fines sur le rang de certaines variétés abéliennes définies sur . Dans le cas des jacobiennes des courbes modulaires, ce problème est équivalent à l’estimation de l’ordre d’annulation en des fonctions des formes modulaires, et a été traité inconditionnellement par Kowalski, Michel et VanderKam. L’objet de ce travail est d’étendre cette approche dans le cas d’un corps totalement réel arbitraire, ce qui nécessite l’utilisation de...
Non-abelian -adic -functions and Eisenstein series of unitary groups – The CM method
In this work we prove various cases of the so-called “torsion congruences” between abelian -adic -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of variables and we obtain more explicit results in the...
Nonlinear exponential twists of the Liouville function
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum The main tool we use is Vaughan’s identity for λ(n).
Non-vanishing of class group -functions at the central point
Let be an imaginary quadratic field, and denote by its class number. It is shown that there is an absolute constant such that for sufficiently large at least of the distinct -functions do not vanish at the central point .
Non-vanishing of -th derivatives of twisted elliptic -functions in the critical point
Let be a modular elliptic curve over
Nonvanishing of quadratic Dirichlet -functions at .
Note on a hypothesis implying the non-vanishing of Dirichlet L-series L(s,χ) for s > 0 and real characters χ
We prove that if χ is a real non-principal Dirichlet character for which L(1,χ) ≤ 1- log2, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ) > 0 for s > 0.
Note on a paper by H. L. Montgomery - II
Note on a Paper by h. l. Montgomery ``omega Theorems for the Riemann Zeta-function''
Note on Dirichlet's L-functions
Note on some greatest common divisor matrices
Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.
Note sur la série
Note sur la série
Notes on the Riemann zeta-function, II