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# New inequalities involving Riemann zeta function

Goal: New inequalities involving Riemann zeta function

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## Project log

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In this paper, we give asymptotic formulas that combine the Euler-Riemann zeta function and the Chebyshev-Stirling numbers of the first kind. These results allow us to prove an asymptotic formula related to the $n$th complete homogeneous symmetric function, which was recently conjectured by the second author: $$h_{n}\left(1,\left( \frac{k}{k+1}\right)^2 ,\left( \frac{k}{k+2} \right)^2 ,\ldots \right) \sim \binom{2k}{k}\quad\text{as}\quad n\to\infty.$$A direct proof of this asymptotic formula, due to Gerg\H{o} Nemes, is provided in the appendix.
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In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments ? (2n) and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions [18]. An interesting asymptotic formula related to the nth complete homogeneous symmetric function is conjectured in this context. hn (1, ( k k+1 )2 , ( k k+2 )2 , . . . ) - ( 2k k ) , n→8.
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We find accurate approximations for certain finite differences of the Euler zeta function, $\zeta(x)$.
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In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions. An interesting asymptotic formula related to the n-th complete homogeneous symmetric function is conjectured in this context.
added a project goal
New inequalities involving Riemann zeta function