Results in this dissertation are divided into four groups and are mainly effective estimates for the Riemann zeta-function
$$ \begin{align*} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}, \quad {\mathrm{Re}\,} s> 1, \end{align*} $$
and associated functions in the Selberg class under the assumption of the (generalised) Riemann hypothesis ((G)RH), that is,
$\zeta (s) \ne 0$
for
${\mathrm {Re}\,}{s}> 1/2$
. The zero-free regions for
$\zeta (s)$
are connected with the distribution of prime numbers, for example, the celebrated prime number theorem
$$ \begin{align*} \sum_{n\le x} \Lambda(n) \sim x \end{align*} $$
is equivalent to the statement that
$\zeta (1+it) \ne 0$
(see [Reference Simonič13]), and that RH is equivalent to
$$ \begin{align*} \sum_{n\le x} \Lambda(n) = x+O(\sqrt x \log^2 x). \end{align*} $$
Here,
$\Lambda (n)$
is the von Mangoldt function, equal to
$\log p$
for
$n = p^k, k \in \mathbb {N}$
, and 0 otherwise. The fundamental connection between
$\zeta (s)$
and
$\Lambda (n)$
is
$$ \begin{align} - \frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}, \quad {\mathrm{Re}\,} s>1, \end{align} $$
which is an immediate consequence of the Euler product for
$\zeta (s)$
. Selberg [Reference Selberg12] found a remarkably simple but powerful method to replace the right-hand side of (1) by the corresponding Dirichlet sum, together with the other terms which emerge from the singularities of
$(\zeta '/\zeta )(s)$
. His equation, known as the Selberg moment formula, and its variants are primary for establishing conditional estimates for
$\log |\zeta (s)|$
and
$|(\zeta '/\zeta )(s)|$
. Although not considered in the thesis, explicit versions of these bounds prove to be useful in connection with the error term in the prime number theorem [Reference Cully-Hugill and Johnston4], as well as with the distribution of prime numbers in intervals [Reference Cully-Hugill and Dudek3] and in arithmetic progressions [Reference Lee8]. A very brief description of the four groups that constitute the thesis is given below.
The first group of results consists of explicit and RH estimates for the moduli of
$S(t)$
, that is, the argument of
$\zeta (1/2 + it)$
, its antiderivative
$$ \begin{align*} S_1(t) = \int_0^t S(u)\,du, \quad t\ge 0, \end{align*} $$
and
$\zeta (1/2 + it)$
. More precisely, explicit estimates of the bounds
$$ \begin{align*} S(t) \ll\frac{\log t}{\log\log t},\quad S_1(t) \ll \frac{\log t}{(\log\log t)^2},\quad \bigg|\zeta\bigg(\frac{1}{2}+it\bigg)\bigg| \le \exp\bigg(O(1)\frac {\log t}{\log\log t}\bigg) \end{align*} $$
are provided. We follow techniques outlined by Selberg [Reference Selberg12] and Fujii [Reference Fujii5, Reference Fujii6], and in the last case, also by Soundararajan [Reference Soundararajan17]. As a corollary, we establish explicit and conditional upper bounds on gaps between consecutive zeros, for example,
$$ \begin{align*} \gamma' - \gamma \le \frac{12.05}{\log\log \gamma} \end{align*} $$
for
$\gamma ' \ge \gamma \ge 10^{2465}$
, where
$\gamma $
and
$\gamma '$
are the ordinates of two consecutive nontrivial zeros. Results from this chapter are published in [Reference Simonič14].
The second group of results consists of explicit and RH estimates for
$\log 1/|\zeta (s)|$
and
$\log |\zeta (s)|$
to the right of the critical line by following techniques developed by Titchmarsh [Reference Titchmarsh18] while using also results on
$S(t)$
and
$S_1(t)$
from the first group. Moreover, we use these bounds to obtain effective and conditional estimates for
$$ \begin{align*} M(x) = \sum_{n\le x} \mu(n) \quad\mbox{and}\quad Q_k(x) = \sum_{n\le x}\sum_{d^k|n} \mu(d), \end{align*} $$
where
$\mu (n)$
is the Möbius function and
$Q_k(x)$
counts the number of k-free numbers not exceeding
$x \ge 1$
. Note that the prime number theorem is equivalent to
$M(x) = o(x)$
and that RH is equivalent to
$M(x) \ll _\varepsilon x^{1/2 +\varepsilon }$
. Our estimates are of the same strength as those of Titchmarsh [Reference Titchmarsh18], and Montgomery and Vaughan [Reference Montgomery, Vaughan, Halberstam and Hooley10], that is,
$$ \begin{align*} M(x) \ll \sqrt{x}\exp{\bigg(\frac{O(1)\log{x}}{\log{\log{x}}}\bigg)}, \quad Q_k(x) = \frac{x}{\zeta(k)} + O_{k,\varepsilon}(x^{{1}/{(k+1)}+\varepsilon}), \end{align*} $$
respectively. Results from this chapter are published in [Reference Simonič15]. It should be noted that better (nonexplicit) results on various bounds on
$\zeta (s)$
exist (see [Reference Carneiro and Chandee1]). Improvements and generalisations of some of these results are currently in progress, and should yield also an improvement over our explicit estimate for
$\zeta (1/2 + it)$
from the first group.
The third group of results consists of GRH estimates for
$|\!\log \mathcal {L}(s)|$
and
$|(\mathcal {L}' /\mathcal {L}) (s)|$
for functions in the Selberg class with a polynomial Euler product, where
${\sigma \ge 1/2 + 1/\!\log \log (c_{\mathcal {L}}|t|)}$
and
$|t|$
is sufficiently large. The shape of these bounds are as in Littlewood [Reference Littlewood9] namely
$$ \begin{align*} \log{\zeta(s)} \ll_{\varepsilon,\sigma_0} (\log{t})^{2(1-\sigma)+\varepsilon} \end{align*} $$
for
$\varepsilon>0$
,
$1/2<\sigma_0\leq\sigma\leq 1$
and
$t$
large, and are thus not the sharpest known. However, GRH can be replaced with a weaker assumption of having no zeros
$\rho = \beta + i\gamma $
with
$\beta> 1/2$
and
$t - \gamma \ll \log \log |t|$
. We provide effective estimates for
$\zeta (s)$
, Dirichlet L-functions with primitive characters and Dedekind zeta-functions, together with an improvement over a particular estimate for
$M(x)$
from the second group of results, for example, for
$x \ge 1$
,
$$ \begin{align*} |M(x)|\le 555.71x^{0.99} + 1.94\cdot 10^{14}x^{0.98} \end{align*} $$
under RH. We also discuss a connection between a particular estimate on the 1-line and several classes of functions. Results from this chapter are published in [Reference Simonič16]. We should mention that our bounds on
$\mathcal {L}(s)$
have been already improved in [Reference Palojärvi and Simonič11], see also the next paragraph.
The fourth group of results consists of effective and GRH estimates for
$|(L'/L)(\sigma ,\chi )|$
for Dirichlet L-functions with primitive characters
$\chi $
modulo q, where
$$ \begin{align*} \frac{1}{2} + \frac{1}{\log\log q} \le \sigma \le 1-\frac{1}{\log\log q} \end{align*} $$
and also
$\sigma = 1$
, by combining methods from Selberg [Reference Selberg12] and from the theory of band-limited functions applied to the Guinand–Weil exact formula. One of the results is that under GRH,
$$ \begin{align*}\bigg|\frac{L'}{L}(1,\chi)\bigg| \le 2\log\log q - 0.4989 + 5.91\frac{(\log\log q)^2}{\log q}\end{align*} $$
for
$q \ge 10^{30}$
, which improves [Reference Ihara, Murty and Shimura7, Corollary 3.3.2]. We provide a similar conditional estimate also for
$|(\zeta '/\zeta ) (1 + it)|$
. Results in this group were obtained in collaboration with A. Chirre and M. V. Hagen, and are published in [Reference Chirre, Hagen and Simonič2]. These techniques are used in [Reference Palojärvi and Simonič11] to obtain GRH estimates for
$|\!\log \mathcal {L}(s)|$
and
$|(\mathcal {L}'/\mathcal {L})(s)|$
for functions in the Selberg class with a polynomial Euler product, where
$$ \begin{align*} \frac{1}{2} + \frac{1}{\log\log q_{\mathcal{L}}|t|^{d_{\mathcal{L}}}} \le \sigma \le 1- \frac{1}{\log\log q_{\mathcal{L}}|t|^{d_{\mathcal{L}}}} \end{align*} $$
and
$|t|$
is sufficiently large. Here,
$q_{\mathcal {L}}$
and
$d_{\mathcal {L}}$
are the conductor and the degree of
$\mathcal {L}$
, respectively. This improves several known results. Moreover, our results are fully explicit under the additional assumption of the strong
$\lambda $
-conjecture.
