The claims (i), (ii) are obvious. The claim (iii) is clear by setting \(a_n = 1_I\) in Definition 7.2.

Now we turn to (iv). By symmetry it suffices to prove the upper bound. Actually it suffices to just show

as this easily implies the general upper bound.

Let \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) be a zeta large value pattern with \(N\) unbounded, \(T= N^{\frac{\tau }{\tau -1}+o(1)}\), and \(V = N^{\frac{1}{2} + \frac{1}{\tau -1} (\sigma -\frac{1}{2})+o(1)}\). By definition, it suffices to show the bound

for some \(\sigma \leq \sigma ' \leq 1\). By definition, \(a_n = 1_I(n)\). By a Fourier expansion of \((n/N)^{1/2}\) in \(\log n\), we can bound

and hence by the pigeonhole principle, we can find \(t' = t + O(N^{o(1)})\) for each \(t \in W\) such that

for \(t \in W\). By refining \(W\) by \(N^{o(1)}\) if necessary, we may assume that the \(t'\) are \(1\)-separated.

Now we use the approximate functional equation

for \(x \sim N\); see [ 144 , Theorem 4.1 ] . Applying this to the two endpoints of \(I\) and subtracting, we conclude that

where \(J_{t'} := \{ m: t' / 2\pi m \in I \} \). Since \(\chi (1/2+it')\) has magnitude one, we conclude that

Writing \(M := T/N = N^{\frac{1}{\tau -1}+o(1)}\), we see that \(J_r \subset [M/10, 10M]\) and

Performing a Fourier expansion of \((M/m)^{1/2} 1_{J_r}(m)\) (smoothed out at scale \(O(1)\)) in \(\log m\), we can bound

and hence

If we let \(E\) denote the set of \(t_1 \in [T/10, 10T]\) for which \(|\sum _{m \in [M/10,10M]} m^{-it_1}| \geq M^{\sigma -o(1)}\) for a suitably chosen \(o(1)\) error, then we have

Summing in \(t'\), we obtain

and so by dyadic pigeonholing we can find \(M^{\sigma -o(1)} \ll V'' \ll M\) and a \(1\)-separated subset \(W''\) of \(E\) such that

for all \(t'' \in W''\), and

By passing to a subsequence we may assume that \(V'' = M^{\sigma '+o(1)}\) for some \(\sigma \leq \sigma ' \leq 1\). Partitioning \([M/10,10M]\) into a bounded number of intervals each of which lies in a dyadic range \([M',2M']\) for some \(M' \asymp M\), and using Definition 8.1, we have

and 2 follows.

Clearly (i) implies (ii). If (iii) holds, then in any zeta large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N\) unbounded and \(V = N^{\sigma +o(1)}\), \(W\) is necessarily empty, giving (i). Conversely, if (i) fails, then there must be \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N\) unbounded and \(V = N^{\sigma +o(1)}\) with \(W\) non-empty, contradicting (ii).

Suppose one has data \(N, I\) obeying the hypotheses of Lemma 8.4(iii), then by 2 (with \(\alpha = 1/\tau \)) one has

and the claim follows from Lemma 8.4.

From Definition 6.1 one has

for unbounded \(t\). By standard arguments (see [ 144 , (8.13) ] ), this implies that

for unbounded \(N\), if \(I \subset [N,2N]\) and \(|t| = N^{\tau +o(1)}\). By partial summation this gives

The claim now follows from Lemma 8.4.

Immediate from Corollary 8.5 and Lemma 5.3; alternatively, one can use Corollary 8.6 and Corollary 6.8.

Apply Corollary 8.6 with \(\sigma _0=1/2\), so that \(\mu (\sigma _0)\) vanishes from the Lindelof hypothesis.

The upper bound \(\mathrm{LV}_\zeta (1/2,\tau ) \leq \tau \) follows from Lemma 8.3(ii), so it suffices to prove the lower bound. Accordingly, let \(N\) be unbounded, let \(T = CN\) for a large fixed constant \(C\), and set \(I := [N,2N]\). In the case \(\sigma =1\), we see from the \(L^2\) mean value theorem (Lemma 3.1) that the expression \(\sum _{n \in I} n^{-it}\) has an \(L^2\) mean of \(\asymp N^{1/2}\) for \(t \in [T,2T]\); on other hand, from 8 we also have an \(L^\infty \) norm of \(O(N^{1/2+o(1)})\). We conclude that \(|\sum _{n \in I} n^{-it}| \gg N^{1/2+o(1)}\) for \(t\) in a subset of \([T,2T]\) of measure \(T^{1-o(1)}\), and hence on a \(1\)-separated subset of cardinality \(\gg T^{1-o(1)}\). This gives the claim \(\mathrm{LV}(1/2,1) \geq 1\).

Next, we establish the \(\tau \geq 2\) case. Let \(N\) be unbounded, set \(T := N^\tau \), and set \(I := [N,2N]\). From Lemma 3.1 we see that the \(L^2\) mean of \(\sum _{n \in I} n^{-it}\) is \(\asymp N^{1/2}\). Also, by squaring this Dirichlet series and applying Lemma 3.1 again we see that the \(L^4\) mean is \(O(N^{1/2+o(1)})\). We may now argue as before to give the desired claim \(\mathrm{LV}(1/2,\tau ) \geq \tau \).

Finally we need to handle the case \(1 {\lt} \tau {\lt} 2\). By Lemma 8.3(iv) with \(\sigma =1/2\) we have

By the \(\tau \geq 2\) case, the left-hand side is at least \(\tau /(\tau -1)\), thus

On the other hand, from Theorem 7.9 and Lemma 8.3(iii) we have

We conclude that the supremum is in fact attained asymptotically at \(\sigma '=1/2\), in the sense that

By the monotonicity of \(\mathrm{LV}_\zeta \) in \(\sigma \), this implies that \(\mathrm{LV}_\zeta (1/2,\tau ) \geq \tau \), as required.

The first claim follows from Corollary 8.5 and Lemma 4.4. For the second claim, it suffices by Lemma 8.3(ii) to establish the lower bound \(\mathrm{LV}_\zeta (\sigma ,\tau ) \geq \tau \). But this is clear from 5.

It suffices to show that

since the terms with \(\sigma ' {\lt} 2\sigma -1\) are less than the left-hand side and can thus be dropped. We repeat the proof of Lemma 7.10. We can find a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N\) unbounded, \(V = N^{\sigma +o(1)}\), \(T = N^{\tau +o(1)}\), and \(|W| = N^{\mathrm{LV}(\sigma ,\tau )+o(1)}\), and we have

for some \(1\)-bounded \(c_t\), and hence by the triangle inequality

which we rearrange as

As in the proof of Lemma 7.10, the contribution of the case \(|t - t'| \leq N^{1-\varepsilon }\) to the right-hand side is \(N^{2-2\sigma +o(1)}\), so we can restrict attention to the case \(|t - t'| \geq N^{1-o(1)}\). By a dyadic decomposition and the pigeonhole principle, we may then assume that

for some \(N^{1-o(1)} \ll T' \ll T\) and some \(r'\); by passing to a subsequence we may assume that \(T' = N^{\tau '+o(1)}\) for some \(1 \leq \tau ' \leq \tau \). By further dyadic decomposition, we may also assume that \( |\sum _{n \in [N,2N]} n^{i(t-t')}| \asymp N^{\sigma '+o(1)}\) for some \(\sigma ' \leq 1\); the cardinality of the sum is then bounded both by \(|W|\) and by \(N^{\mathrm{LV}_\zeta (\sigma ',\tau ') + o(1)}\), hence

The case \(\sigma ' {\lt} 1/2\) is dominated by that of \(\sigma '=1/2\). The claim now follows.

By Lemma 7.7 it suffices to verify the claim for \(\tau {\lt} \frac{2\sigma -1-\sigma '}{\mu (\sigma ')}\). The claim now follows from Lemma 8.12 and Corollary 8.6.

Immediate from Corollary 8.13, since \(\mu (1/2)=0\) in this case.

We set \(\theta \) to equal

and then from the bounds \(\mu (1/2) \leq 1/6\), \(\mu (5/7) \leq 1/14\), \(\mu (5/6) \leq 1/30\) one can bound \(\mu (\theta )\) by the quantity \(c(\theta )\), defined to equal

By Corollary 8.13, we have \(\mathrm{LV}(\sigma ',\tau ) \leq 2-2\sigma '\) for

The right-hand side can be computed to equal \(f(\sigma )(\sigma '-\sigma ) + 2 - 2\sigma '\), giving the claim.

Write \(\rho := \mathrm{LV}(\sigma ,\tau )\), and let \(\varepsilon {\gt}0\) be arbitrary. By Lemma 8.12, we may assume without loss of generality that

for some \(1/2 \leq \sigma ' \leq 1\) and \(1 \leq \tau ' \leq \tau \). On the other hand, from Lemma 8.11 applied to the exponent pair \((2/7,4/7)\) from Lemma 5.11, and bounding \(\tau '\) by \(\tau \), one has

and thus on taking convex combinations

hence \(\rho \) is bounded by either \(2 - 2 \sigma \), \(1 - 2\sigma + \frac{5}{6}\rho + \frac{1}{6} \tau + \frac{1}{2}\), or \(1-2\sigma + \frac{18}{19} \rho + \frac{3}{19} \tau + \frac{1}{2}\). The claim then follows after solving for \(\rho \).