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NEW EFFECTIVE RESULTS IN THE THEORY OF THE RIEMANN ZETA-FUNCTION

Published online by Cambridge University Press:  12 December 2023

ALEKSANDER SIMONIČ*
Affiliation:
School of Science, University of New South Wales (Canberra), Campbell, ACT 2612, Australia
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Abstract

Type
PhD Abstract
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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

(1) $$ \begin{align} - \frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}, \quad {\mathrm{Re}\,} s>1, \end{align} $$

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.

Footnotes

Thesis submitted to the University of New South Wales in August 2022; degree approved on 10 November 2022; supervisor Timothy Trudgian.

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