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Cotangent sums related to the Riemann Hypothesis for various shifts of the argument
Helmut Maier, Michael Th. Rassias
One of the approaches to the Riemann Hypothesis is the Nyman-Beurling criterion. Cotangent sums play a significant role. Here we investigate the values of these cotangent sums for…
Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis
Helmut Maier, Michael Th. Rassias
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of t…
The error for the second moment of cotangent sums related to the Riemann Hypothesis
Helmut Maier, Michael Th. Rassias
In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error ter…
Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis
Helmut Maier, Michael Th. Rassias
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the Möbius function. The estimate is remarkably sharp in comparison to…
A criterion related to the Riemann Hypothesis
Helmut Maier, Michael Th. Rassias
A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2π}\int_{-\infty}^\infty\left|1-ζA_N\left…
From a cotangent sum to a generalized totient function
Michael Th. Rassias
In this paper we investigate a certain category of cotangent sums and more specifically the sum $$\sum_{m=1}^{b-1}\cot\left(\frac{πm}{b}\right)\sin^{3}\left(2πm\frac{a}{b}\right)\:…
Asymptotics for moments of certain cotangent sums for arbitrary exponents
Helmut Maier, Michael Th. Rassias
In this paper we extend a result on the asymptotics of moments of certain cotangent sums associated to the Estermann and Riemann zeta functions established in a previous paper for…
Asymptotics for moments of certain cotangent sums
Helmut Maier, Michael Th. Rassias
In this paper we improve a result on the order of magnitude of certain cotangent sums associated to the Estermann and the Riemann zeta functions.
A Half-Discrete Hardy-Hilbert-Type Inequality with a Best Possible Constant Factor Related to the Hurwitz Zeta Function
Michael Th. Rassias, Bicheng Yan
Using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a…
A cotangent sum related to zeros of the Estermann zeta function
Michael Th. Rassias
We consider a cotangent sum related to Estermann's Zeta function. We provide an elementary and self-contained improvement of the error term in an asymptotic formula proved by V. I.…
A Hilbert-Type Integral Inequality in the Whole Plane Related to the Hypergeometric Function and the Beta Function
Michael Th. Rassias, Bicheng Yang
A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and th…
A More Accurate Half-Discrete Hardy-Hilbert-Type Inequality with the Best Possible Constant Factor Related to the Extended Riemann-Zeta Function
Michael Th. Rassias, Bicheng Yang
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hype…