The theorem proved here extends the author's previous work on Chebyshev series [4] by showing that if f(x) is a member of the class of so-called "Stieltjes functions" whose asymptotic power series 2 anx" about x = 0 is such that ttlogjaj _ hm —;-= r, «log n then the coefficients of the series of shifted Chebyshev polynomials on x e [0, a],2b„Tf(x/a), satisfy the inequality 2 . m log | (log |M) ...

There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that hypothesis is true if only inequality \(\zeta(2) \cdot \prod_{q\leq q_{n}} (1+\frac{1}{q}) > e^{\gamma} \log \theta(q_{n})\) holds for all prime numbers \(q_{n}> 3\), where \(\theta(x)\) Chebyshev function, \(\gamma \approx 0.57721\) Euler-Mascheroni constant, \(\zeta(x)\) zeta f...