1,721,002 research outputs found
On the reciprocity law for the twisted second moment of dirichlet L-functions
No full textWe investigate the reciprocity law, studied by Conrey and Young, for the second moment of Dirichlet L-functions twisted by χ(a) modulo a prime q. We show that the error term in this reciprocity law can be extended to a continuous function of a/q with respect to the real topology. Furthermore, we extend this reciprocity result, proving an exact formula also involving shifted moments. We also give an expression for the twisted second moment involving the coefficients of the continued fraction expansion of a/q, and, consequently, we improve upon a classical result of Selberg on the second moment of Dirichlet L-functions with two twists. Finally, we obtain a formula connecting the shifted second moment of the Dirichlet L-functions with the Estermann function. In particular cases, this result can be used to obtain some simple explicit exact formulae for the moments
A congruence sum and rational approximations
No full textWe give a reciprocity formula for a two-variables sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is approximable from below and show that the reciprocity formula is a simple consequence of this fact
The first moment of twisted Hecke -functions with unbounded shifts
No full textWe compute the first moment of twisted Hecke L-functions of weight 2 and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift
THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS
No full text We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0. </jats:p
A weighted one-level density of the non-trivial zeros of the Riemann zeta-function
No full textWe compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by |ζ(12+it)|2k for k=1 and, for test functions with Fourier support in (-12,12), for k=2. As a consequence, for k=1,2, we deduce under the Riemann hypothesis that T(logT)[email protected]@4abf1d3bkjavax.xml.bind.JAXBElement@5508e36djavax.xml.bind.JAXBElement@2bd394ebjavax.xml.bind.JAXBElement@52d251f1javax.xml.bind.JAXBElement@6036c855 non-trivial zeros of ζ, of imaginary parts up to T, are such that ζ attains a value of size (logT)k+o(1) at a point which is within O(1/logT) from the zero
The θ = ∞ conjecture implies the Riemann hypothesis
No full textWe show that the θ = ∞ conjecture implies the Riemann hypothesis
ON THE DISTRIBUTION OF A COTANGENT SUM
No full textAbstract. Maier and Rassias computed the moments and proved a distribution re-sult for the cotangent sum c0(a/q): = − m<
A Note on the Dimension of the Largest Simple Hecke Submodule
No full textFor k≥2 even, let dk,N denote the dimension of the largest simple Hecke submodule of Sk(Γ0(N);Q)new. We show, using a simple analytic method, that dk,N≫kloglogN/log(2p) with p, the smallest prime co-prime to N. Previously, bounds of this quality were only known for N in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning Sk(Γ0(N),χ), with k≥2 an integer and χ an arbitrary nebentypus
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