An Elementary Proof to eγ log log n > Ψ(n)/n for any n > 30

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

AN EXTENSION OF A RESULT OF ERDŐS AND ZAREMBA

Erdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$ , γ being Euler’s constant, where $\Phi(n)=\sum_{d|n} \frac{\log d}{d}$ .We extend this result to the function $\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\gamma$ . The proof requires a new approach. As an application, we prove that for any $\eta>1$ , any finite sequence of reals $\{c_k, k\in K\}$ , $\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu)$ , where C(η) depends on η only. This improves a recent result obtained by the author.

On Some Series Dependent on a Parameter Which Generalize Euler’s Constant

Series which depend on a parameter and generalize the constant discovered by Euler are introduced and studied. Convergence results are established. An infinite series expansion is obtained from these generalized formulas which can be used to evaluate the generalized constant. Euler’s constant can be obtained as a special case. Some asymptotic results are formulated and limits of some closely related sequences are given at the end.

Generalized Lambert series and arithmetic nature of odd zeta values

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.