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New integral representations for the square of the Riemann zeta-function

Andreas Guthmann (1997)

Acta Arithmetica

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...

Non annulation des fonctions L des formes modulaires de Hilbert au point central

Denis Trotabas (2011)

Annales de l’institut Fourier

La conjecture de Birch et Swinnerton-Dyer donne des estimations fines sur le rang de certaines variétés abéliennes définies sur Q . Dans le cas des jacobiennes des courbes modulaires, ce problème est équivalent à l’estimation de l’ordre d’annulation en 1 / 2 des fonctions L 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 p -adic L -functions and Eisenstein series of unitary groups – The CM method

Thanasis Bouganis (2014)

Annales de l’institut Fourier

In this work we prove various cases of the so-called “torsion congruences” between abelian p -adic L -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 n variables and we obtain more explicit results in the...

Nonlinear exponential twists of the Liouville function

Qingfeng Sun (2011)

Open Mathematics

Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum X n 2 X λ ( n ) e 2 π i α n , 0 α The main tool we use is Vaughan’s identity for λ(n).

Non-vanishing of class group L -functions at the central point

Valentin Blomer (2004)

Annales de l’institut Fourier

Let K = ( - D ) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c > 0 such that for sufficiently large D at least c · h p D ( 1 - p - 1 ) of the h distinct L -functions L K ( s , χ ) do not vanish at the central point s = 1 / 2 .

Note on some greatest common divisor matrices

Peter Lindqvist, Kristian Seip (1998)

Acta Arithmetica

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.

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