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1.
Each simple zero of the Riemann zeta function on the critical line with is a center for the flow of the Riemann xi function with an associated period . It is shown that, as , Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture for some exponent , we obtain the upper bound . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert-Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.
2.
Djurdje Cvijovic Jacek Klinowski 《Proceedings of the American Mathematical Society》1997,125(9):2543-2550
It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.
3.
H. Mishou 《Lithuanian Mathematical Journal》2007,47(1):32-47
In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached
with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007. 相似文献
4.
Sami Omar. 《Mathematics of Computation》2001,70(236):1607-1616
Using Weil's explicit formula, we propose a method to compute low zeros of the Dedekind zeta function. As an application of this method, we compute the first zero of the Dedekind zeta function associated to totally complex fields of degree less than or equal to 30 having the smallest known discriminant.
5.
《Discrete Mathematics》2023,346(1):113138
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from vertex-transitive graphs. 相似文献
6.
Let ζ be the Riemann zeta function and δ(x)=1/(2x-1). For all x>0 we have
(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),