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In order to study the analytic properties of the Goldbach generating function we consider a smooth version, similar to the Chebyshev function for the Prime Number Theorem. In this paper we obtain explicit numerical estimates for the average order of its summatory function both in the classical case and in arithmetic progressions. These support the existing asymptotic results, under the (Generalised) Riemann Hypothesis, involving error terms.

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\mathcal{L}(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\mathcal{L}(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\mathcal{L}(s)$, we establish both uniform bounds, and completely explicit estimates by assuming also the strong $\lambda$-conjecture.

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+\delta\leq\sigma<1$, fixed $\delta\in(0,1/2)$ and for functions in the Selberg class except for the identity function. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish uniform bounds for $|3/4-\sigma|\leq 1/4-1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$, $|1-\sigma|\leq 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$ and $\sigma=1$, and completely explicit estimates by assuming also the strong $\lambda$-conjecture.

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the reduced residue system modulo $m$, then under certain explicit extra conditions we must have $\lambda_0+\lambda_1F_p(\alpha_1)+\ldots+\lambda_kF_p(\alpha_k) \neq 0$ for at least one such prime. We also prove an explicit $p$-adic lower bound for the previous linear form. Secondly, we consider the case where we take primes in arithmetic progressions from more than $k\varphi(m)/(k+1)$ residue classes. Then there is an infinite collection of intervals each containing at least one prime which is in those arithmetic progressions and for which we have $\lambda_0+\lambda_1F_p(\alpha_1)+\ldots+\lambda_kF_p(\alpha_k) \neq 0$. We also derive an explicit $p$-adic lower bound for the previous linear form.

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Pad\'e approximations of the second kind., Comment: 27 pages

We prove an explicit error term for the $\psi(x,\chi)$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.

$\tau$-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove that certain values of the $\tau$-Li coefficients lead to existence or non-existence of certain zeros. The first main result gives explicit numbers $N_1$ and $N_2$ such that if all real parts of the $\tau$-Li coefficients are non-negative for all indices between $N_1$ and $N_2$, then the function has non zeros outside a certain region. According to the second result, if some of the real parts of the $\tau$-Li coefficients are negative for some index $n$ between numbers $n_1$ and $n_2$, then there is at least one zero outside a certain region.

In this paper we prove explicit upper and lower bounds for the error term in the Riemann-von Mangoldt type formula for the number of zeros inside the critical strip. Furthermore, we also give examples of the bounds.