Low lying zeros of families of -functions
Lower bounds for power moments of L-functions
Lower bounds for the conductor of L-functions
Lower order terms for the one-level densities of symmetric power L-functions in the level aspect
Lower order terms in the 1-level density for families of holomorphic cuspidal newforms
Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series
We obtain formulas for computing mean values of Dirichlet polynomials that have more terms than the length of the integration range. These formulas allow one to compute the contribution of off-diagonal terms provided one knows the correlation functions for the coefficients of the Dirichlet polynomials. A smooth weight is used to control error terms, and this weight can in typical applications be removed from the final result. Similar results are obtained for the tails of Dirichlet series. Four examples...
Mean-periodicity and zeta functions
This paper establishes new bridges between zeta functions in number theory and modern harmonic analysis, namely between the class of complex functions, which contains the zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of an arithmetic scheme with its expected analytic shape is shown...
Mittelwerte von Hecke Polynomen.
Mod structure of alternating and non-alternating multiple harmonic sums
The well-known Wolstenholme’s Theorem says that for every prime the -st partial sum of the harmonic series is congruent to modulo . If one replaces the harmonic series by for even, then the modulus has to be changed from to just . One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction...
Modular case of Levinson's theorem
We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-functions, an explicit positive proportion of zeros which lie on the critical line.
Motivic Artin -functions and a motivic Tamagawa number. (Fonctions d'Artin et nombre de Tamagawa motiviques.)
Multiple Dirichlet series over rational function fields
Multiplicative functions dictated by Artin symbols
Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/ℚ, we construct a natural class of completely multiplicative functions whose values are dictated by Artin symbols, and we show that...
Multizetas, perinomal numbers, arithmetical dimorphy, and ARI/GARI
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...
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
Note sur la série
Notes on the Riemann zeta-function, II