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Minoration au point des fonctions L attachées à des caractères de Dirichlet

Pierre Barrucand, Stéphane Louboutin (1993)

Colloquium Mathematicae

Il est connu (voir [1], [3]) que lorsque χ varie parmi les caractères de Dirichlet non quadratiques, nous avons | L ( 1 , X ) | - 1 = O ( L o g ( f χ ) ) . Nous montrons ici qu’en se restreignant aux caractères d’ordre impair donné, nous avons | L ( 1 , X ) | - 1 = o ( L o g ( f χ ) ) . Il serait évidemment bien plus satisfaisant de parvenir à prouver un tel résultat sans restreindre χ à varier parmi des caractères d’ordre fixé. Pour les caractères d’ordre pair, nous ne pouvons établir un tel résultat qu’en nous restreignant aux caractères pour lesquels les conducteurs de χ 2 restent...

Mod p structure of alternating and non-alternating multiple harmonic sums

Jianqiang Zhao (2011)

Journal de Théorie des Nombres de Bordeaux

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the ( p - 1 ) -st partial sum of the harmonic series is congruent to 0 modulo p 2 . If one replaces the harmonic series by ∑ k ≥ 1 1 / n k for k even, then the modulus has to be changed from p 2 to just p . 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

Damien Bernard (2015)

Acta Arithmetica

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.

Multiple zeta values and periods of moduli spaces 𝔐 ¯ 0 , n

Francis C. S. Brown (2009)

Annales scientifiques de l'École Normale Supérieure

We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces 𝔐 0 , n of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on 𝔐 0 , n and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes’ formula iteratively to reduce each period integral to multiple zeta values. We also give a geometric interpretation of the double shuffle...

Multiplicative functions dictated by Artin symbols

Robert J. Lemke Oliver (2013)

Acta Arithmetica

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 K of completely multiplicative functions whose values are dictated by Artin symbols, and we show that...

Currently displaying 441 – 460 of 1029