6 Growth exponents for the Riemann zeta function

Definition 6.1 Growth rate of zeta

For any fixed \(\sigma \in \mathbf{R}\), let \(\mu (\sigma )\) denote the least possible (fixed) exponent for which one has the bound

\[ |\zeta (\sigma +it)| \ll |t|^{\mu (\sigma )+o(1)} \]

for all unbounded \(t\).

One can check that for each \(\sigma \), the set of possible candidates for \(\mu (\sigma )\) is closed (by underspill), non-empty, and bounded from below, so that \(\mu (\sigma )\) is well-defined as a real number. An equivalent definition without asymptotic notation, is that \(\mu (\sigma )\) is the least real number such that for every \(\varepsilon {\gt}0\) there exists \(C{\gt}0\) such that

\[ |\zeta (\sigma +it)| \leq C |t|^{\mu (\sigma )+\varepsilon } \]

for all \(t\) with \(|t| \geq C\); equivalently, one has

\[ \mu (\sigma ) = \limsup _{|t| \to \infty } \frac{\log |\zeta (\sigma +it)|}{\log |t|}. \]

Implemented at bound_mu.py as:
Bound_mu

Lemma 6.2 Trivial bound

One has \(\mu (\sigma )=0\) for all \(\sigma \geq 1\).

Implemented at bound_mu.py as:
apply_trivial_mu_bound()

Proof ▶

Immediate from the absolute convergence of the Dirichlet series for both \(\zeta (s)\) and \(1/\zeta (s)\); see e.g., [ 144 , Theorem 1.9 ] .

Lemma 6.3 Convexity

\(\mu \) is convex.

Implemented at bound_mu.py as:
bound_mu_convexity()

Proof ▶

Immediate from the Phragmén–Lindelöf principle; see e.g., [ 144 , § A.8 ] .

Lemma 6.4 Functional equation

One has \(\mu (1-\sigma ) = \mu (\sigma ) + \sigma - 1/2\) for all \(0 \leq \sigma \leq 1/2\).

Implemented at bound_mu.py as:
apply_functional_equation()

Proof ▶

Immediate from the functional equation for \(\zeta \) and asymptotics of the Gamma function; see e.g., [ 144 , (1.23), (1.25) ] .

Lemma 6.5 Left of critical strip

One has \(\mu (\sigma )=1/2-\sigma \) for \(\sigma \leq 0\).

Implemented at bound_mu.py as:
apply_trivial_mu_bound()

Proof ▶

Immediate from Lemmas 6.2, 6.4.

Lemma 6.6 Convexity bounds

One has \(\max (0, 1/2-\sigma ) \leq \mu (\sigma ) \leq (1-\sigma )/2\) for \(0 \leq \sigma \leq 1\).

Implemented at bound_mu.py as:
apply_trivial_mu_bound()

Proof ▶

Immediate from Lemma 6.2, Lemma 6.5, and Lemma 6.6.

6.1 Connection with exponent pairs and dual exponent pairs

Lemma 6.7 Connection with dual exponent pairs

For any \(1/2 \leq \sigma \leq 1\), one has

\[ \mu (\sigma ) \leq \sup _{0 \leq \alpha \leq 1/2} \beta (\alpha ) - \alpha \sigma . \]

Proof ▶

Let \(t\) be unbounded. From the Riemann–Siegel formula (see [ 144 , Theorem 4.1 ] ) one has

\[ \zeta (\sigma +it) \ll \left|\sum _{n \leq \sqrt{t/2\pi }} \frac{1}{n^{\sigma +it}}\right| + |t|^{1/2-\sigma } \left|\sum _{n \leq \sqrt{t/2\pi }} \frac{1}{n^{1-\sigma -it}}\right| + O(1). \]

From dyadic decomposition and Definition 4.2 (and Lemma 2.1) one has for any fixed \(\varepsilon {\gt}0\) that

\[ \sum _{t^\varepsilon \leq n \leq \sqrt{t/2\pi }} \frac{1}{n^{\sigma +it}} \ll |t|^{\sup _{\varepsilon \leq \alpha \leq 1/2} \beta (\alpha ) - \alpha \sigma + o(1)}, \]

while from the triangle inequality one has the crude bound

\[ \sum _{n {\lt} t^\varepsilon } \frac{1}{n^{\sigma +it}} \ll |t|^\varepsilon . \]

Combining the bounds and using underspill, we conclude that

\[ \sum _{n \leq \sqrt{t/2\pi }} \frac{1}{n^{\sigma +it}} \ll |t|^{\sup _{0 \leq \alpha \leq 1/2} \beta (\alpha ) - \alpha \sigma + o(1)}. \]

A similar argument gives

\[ \sum _{n \leq \sqrt{t/2\pi }} \frac{1}{n^{1-\sigma -it}} \ll |t|^{\sup _{0 \leq \alpha \leq 1/2} \beta (\alpha ) - \alpha (1-\sigma ) + o(1)} \]

Since \(\sigma \geq 1/2\) and \(\alpha \leq 1/2\), one has \((1/2-\sigma ) - \alpha (1-\sigma ) \leq -\alpha \sigma \), and hence

\[ \zeta (\sigma +it) \ll |t|^{\sup _{0 \leq \alpha \leq 1/2} \beta (\alpha ) - \alpha \sigma + o(1)} \]

giving the claim.

We remark that this inequality is morally an equality (indeed, it would be if one would restrict the model phases in Definition 4.2 to purely the logarithmic phase \(u \mapsto \log u\)).

The following form of Lemma 6.7 is convenient for applications:

Corollary 6.8 Exponent pairs and \(\mu \)

If \((k,\ell )\) is an exponent pair, then

\[ \mu (\ell -k) \leq k. \]

Implemented at bound_mu.py as:
exponent_pair_to_mu_bound(exp_pair)

Proof ▶

Immediate from Lemma 6.7 and Lemma 5.3. See also [ 144 , (7.57) ] .

Heuristic 6.9 Lindelöf hypothesis

One has \(\mu (1/2)=0\).

Implemented at bound_mu.py as:
bound_mu_Lindelof()

Lemma 6.10

The exponent pair conjecture implies the Lindelöf hypothesis.

Proof ▶

Immediate from Corollary 6.8.

Proposition 6.11 Conjectured value of \(\mu \)

We have the lower bound

\begin{equation} \label{muh} \mu (\sigma ) \geq \max \left(0, \frac{1}{2}-\sigma \right) \end{equation}

for all \(\sigma \in \mathbf{R}\), and equality holds everywhere in 1 if and only if the Lindelöf hypothesis holds.

We remark that this proposition explains why there are no further lower bounds on \(\mu \) in the literature beyond 1; all the remaining known results revolve around upper bounds.

Proof ▶

Clearly equality in 1 implies the Lindelöf hypothesis, while from the trivial bounds in Propositions 6.2, 6.5 and convexity (Lemma 6.6) one we see that the Lindelöf hypothesis implies the upper bound

\[ \mu (\sigma ) \leq \max \left(0, \frac{1}{2}-\sigma \right) \]

for all \(\sigma \). So it suffices to establish the lower bound unconditionally. By the functional equation (Proposition 6.4) it suffices to do this for \(\sigma \geq 1/2\); in fact by convexity it suffices to establish the claim when \(1/2 {\lt} \sigma {\lt} 1\). In this regime, the \(L^2\) mean value theorem (see [ 144 , Theorem 1.11 ] ) gives

\[ \int _0^T |\zeta (\sigma +it)|^2\ dt \asymp T \]

for large \(T\), giving the claim.

6.2 Known bounds on \(\mu \)

Theorem 6.12 Historical bounds

The upper bounds on \(\mu (\sigma )\) given by Table 6.1 are known.

TODO: supplement as many of these citations as possible with derivations from other exponents and relations in the database

Table 6.1 Historical bounds on \(\mu (\sigma )\) for \(1/2 \le \sigma \le 1\), and the exponent pair generating them (if applicable).

Reference	Results	Exponent pair
Hardy–Littlewood (1923) [ 97 ]	\(\mu (1/2) \le 1/6\)	(1/6, 2/3)
Walfisz (1924) [ 287 ]	\(\mu (1/2) \le 193/988\)
Titchmarsh (1932) [ 273 ]	\(\mu (1/2) \leq 27/164\)
Phillips (1933) [ 232 ]	\(\mu (1/2) \leq 229/1392\)
Titchmarsh (1942) [ 276 ]	\(\mu (1/2) \leq 19/116\)
Min (1949) [ 216 ]	\(\mu (1/2) \leq 15/92\)
Haneke (1962) [ 92 ]	\(\mu (1/2) \leq 6/37\)
Kolesnik (1973) [ 170 ]	\(\mu (1/2) \leq 173/1067\)
Kolesnik (1982) [ 172 ]	\(\mu (1/2) \leq 35/216\)
Kolesnik (1985) [ 173 ]	\(\mu (1/2) \leq 139/858\)
Bombieri–Iwaniec (1985) [ 18 ]	\(\mu (1/2) \leq 9/56\)	\((9/56, 1/2+9/56)\)
Watt (1989) [ 293 ]	\(\mu (1/2) \leq 89/560\)	\((89/560, 1/2+89/560)\)
Huxley–Kolesnik (1991) [ 132 ]	\(\mu (1/2) \leq 17/108\)	\((17/108, 1/2+17/108)\)
Huxley (1993) [ 128 ]	\(\mu (1/2) \leq 89/570\)	\((89/570, 1/2+89/570)\)
Huxley (1996) [ 129 ]	\(\mu (1934/3655) \leq 6299/43860\)
Sargos (2003) [ 259 ]	\(\mu (49/51) \leq 1/204\), \(\mu (361/370) \leq 1/370\)
Huxley (2005) [ 131 ]	\(\mu (1/2) \leq 32/205\)	\((32/205, 1/2+32/205)\)
Lelechenko (2014) [ 177 ]	\(\mu (3/5) \leq 1409/12170\), \(\mu (4/5) \leq 3/71\)
Bourgain (2017) [ 23 ]	\(\mu (1/2) \leq 13/84\)	\((13/84, 1/2+13/84)\)
Heath-Brown (2017) [ 113 ]	\(\mu (\sigma ) \le \frac{8}{63}\sqrt{15}(1 - \sigma )^{3/2}\) for \(1/2 \le \sigma \le 1\)
Heath-Brown (2020) [ 58 ]	\(\mu (11/15) \leq 1/15\)

Recorded in literature.py as:
add_literature_bounds_mu()

Some additional bounds are recorded in [ 279 ] by combining various exponential sum estimates.

Theorem 6.13

[ 279 , Theorems 2.4-2.6 ] We have

\[ \mu (\sigma ) \le \begin{cases} (31 - 36\sigma )/84 , & \frac{1}{2} \leq \sigma {\lt} \frac{88225}{153852} = 0.5734\ldots , \\ (220633 - 251324\sigma )/620612 , & \frac{88225}{153852} \leq \sigma {\lt} \frac{521}{796} = 0.6545\ldots , \\ (1333 - 1508\sigma )/3825 , & \frac{521}{796} \leq \sigma {\lt} \frac{53141}{76066} = 0.6986\ldots , \\ (405 - 454\sigma )/1202 , & \frac{53141}{76066} \leq \sigma {\lt} \frac{3620}{5119} = 0.7071\ldots , \\ (49318855 - 52938216\sigma )/170145110 , & \frac{3620}{5119} \leq \sigma {\lt} \frac{52209}{69128} = 0.7552\ldots , \\ (471957 - 502648\sigma )/1682490 , & \frac{52209}{69128} \leq \sigma {\lt} \frac{1389}{1736} = 0.8001\ldots , \\ (2841 - 3016\sigma )/10316 , & \frac{1389}{1736} \leq \sigma {\lt} \frac{134765}{163248} = 0.8255\ldots , \\ (859 - 908\sigma )/3214 , & \frac{134765}{163248} \leq \sigma {\lt} \frac{18193}{21906} = 0.8305\ldots , \\ 5(8707 - 9067\sigma )/180277 , & \frac{18193}{21906} \leq \sigma {\lt} \frac{249}{280} = 0.8892\ldots , \\ (29 - 30\sigma )/130 , & \frac{249}{280} \leq \sigma \leq \frac{9}{10}.\\ \end{cases} \]

Furthermore, for \(1/2 \le \sigma \le 1\), we have

\[ \mu (\sigma ) \le \frac{2}{13}\sqrt{10}(1 - \sigma )^{3/2} = 0.4865\ldots (1 - \sigma )^{3/2}, \]

and

\[ \mu (\sigma ) \le \frac{2}{3^{3/2}}(1 - \sigma )^{3/2} + \frac{103}{300}(1 - \sigma )^{2},\qquad \frac{117955}{118272} \le \sigma \le 1. \]

Recorded in literature.py as:
add_literature_bounds_mu()

Additionally, the series of exponent pairs in Theorem 5.17 imply the following bounds on \(\mu (\sigma )\) close to \(\sigma = 1\).

Theorem 6.14 Heath-Brown [ 113 ] \(\mu \) bounds

For any integer \(k \ge 3\), one has

\[ \mu \left(1 - \frac{3 k^2 - 3 k + 2}{k(k - 1)^2(k + 2)}\right)\le \frac{2}{(k - 1)^2(k + 2)}. \]

Proof ▶

Follows from substituting Theorem 5.17 into 6.8.

The new exponent pairs in Theorem 5.22 may be used to obtain sharper bounds on \(\mu (\sigma )\) in certain ranges. The current sharpest bounds on \(\mu (\sigma )\) are recorded in Table 6.2 and graphed in Figure 6.1.

Table 6.2 Current sharpest known bound on \(\mu (\sigma )\) for \(1/2 \le \sigma \le 1\)

Upper bound on \(\mu (\sigma )\)

Range of \(\sigma \)

Reference

\(\mu (\sigma ) \leq \dfrac {31}{84} - \dfrac {3}{7}\sigma \)

\(\dfrac {1}{2}\leq \sigma \leq \dfrac {88225}{153852} = 0.5734\ldots \)

Theorem 6.13

\(\mu (\sigma ) \leq \dfrac {220633}{620612} - \dfrac {62831}{155153}\sigma \)

\(\dfrac {88225}{153852}\leq \sigma \leq \dfrac {521}{796} = 0.6545\ldots \)

Theorem 6.13

\(\mu (\sigma ) \leq \dfrac {1333}{3825} - \dfrac {1508}{3825}\sigma \)

\(\dfrac {521}{796}\leq \sigma \leq \dfrac {53141}{76066} = 0.6986\ldots \)

Theorem 6.13

\(\mu (\sigma ) \leq \dfrac {405}{1202} - \dfrac {227}{601}\sigma \)

\(\dfrac {53141}{76066}\leq \sigma \leq \dfrac {454}{641} = 0.7082\ldots \)

Theorem 6.13

\(\mu (\sigma ) \leq \dfrac {779}{2590} - \dfrac {423}{1295}\sigma \)

\(\dfrac {454}{641}\leq \sigma \leq \dfrac {1744}{2411} = 0.7234\ldots \)

Theorem 5.22, Corollary 6.8

\(\mu (\sigma ) \leq \dfrac {179}{622} - \dfrac {96}{311}\sigma \)

\(\dfrac {1744}{2411}\leq \sigma \leq \dfrac {951057}{1298878} = 0.7322\ldots \)

Theorem 5.23, Corollary 6.8

\(\mu (\sigma ) \leq \dfrac {157319}{560830} - \dfrac {251324}{841245}\sigma \)

\(\dfrac {951057}{1298878}\leq \sigma \leq \dfrac {1389}{1736} = 0.8001\ldots \)

Theorem 5.22, Corollary 6.8

\(\mu (\sigma ) \leq \dfrac {2841}{10316} - \dfrac {754}{2579}\sigma \)

\(\dfrac {1389}{1736}\leq \sigma \leq \dfrac {587779}{702192} = 0.8370\ldots \)

Theorem 6.13

\(\mu (\sigma ) \leq \dfrac {1691}{6554} - \dfrac {890}{3277}\sigma \)

\(\dfrac {587779}{702192}\leq \sigma \leq \dfrac {7441}{8695} = 0.8557\ldots \)

Theorem 5.22, Corollary 6.8

\(\mu (\sigma ) \leq \dfrac {29}{130} - \dfrac {3}{13}\sigma \)

\(\dfrac {7441}{8695}\leq \sigma \leq \dfrac {277}{300} = 0.9233\ldots \)

Theorem 5.22, Theorem 6.14

\(\mu (\sigma ) \leq \lambda \mu _n + (1 - \lambda )\mu _{n + 1}\)

\(\mu _n = \dfrac {2}{(n - 1)^2(n + 2)}\)

\(\lambda = (\sigma _{n + 1} - \sigma )/(\sigma _{n + 1} - \sigma _n)\)

\(\sigma _n \leq \sigma \leq \sigma _{n + 1}\)

\(\sigma _n = 1 - \dfrac {3 n^2 - 3 n + 2}{n(n - 1)^2(n + 2)},\quad (n \ge 7)\)

Theorem 6.14

Derived in derived.py as:
compute_best_mu_bound()

\includegraphics[width=0.5\linewidth ]{chapter/mu_bound_plot.png} — Figure 6.1 Current sharpest known bound on \(\mu (\sigma )\) for \(1/2 \le \sigma \le 1\).

6.3 Connection to the Riemann hypothesis

It is well known that the Riemann hypothesis implies the Lindelöf hypothesis. Here is a sharper version, essentially due to Backlund [ 2 ] :

Lemma 6.15 Growth exponent and zeroes

Let \(1/2 \leq \sigma _0 {\lt} 1\) be fixed. Then the assertion \(\mu (\sigma _0)=0\) is equivalent to the assertion that for any fixed \(\varepsilon {\gt}0\) and unbounded \(T{\gt}0\), the number of zeroes \(\sigma +it\) of the zeta function with \(\sigma \geq \sigma _0+\varepsilon \) and \(T \leq t \leq T+1\) is \(o(\log T)\).

Proof ▶

This is a routine adaptation of Theorem 2 of https://terrytao.wordpress.com/2015/03/01.