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Note on the resonance method for the Riemann zeta function
Number Theory · Submitted 18 Jan 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$.
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The abstract, sentence by sentence
We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|\zeta(1/2+it)|$ on long intervals, as well as work of Soundararajan, G\'{a}l, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. | contribution 0.57 | 0.96 | 0.78 | asserted 1.00 |
| s1 | We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. | result 0.77 | 0.95 | 0.14 | asserted 1.00 |
| s2 | We rely on our recent work on lower bounds for maxima of $|\zeta(1/2+it)|$ on long intervals, as well as work of Soundararajan, G\'{a}l, and others. | method 0.62 | 0.28 | 0.03 | asserted 0.99 |
| s3 | The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs. | contribution 0.71 | 0.55 | 0.05 | asserted 0.94 |
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The paper
Original abstract
We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of $|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on intervals of length much larger than $x$. We rely on our recent work on lower bounds for maxima of $|\zeta(1/2+it)|$ on long intervals, as well as work of Soundararajan, G\'{a}l, and others. The paper aims at displaying and clarifying the conceptually different combinatorial arguments that show up in various parts of the proofs.
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References
How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 3 sentences without a confident label
Builds on it 1 paper
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See [ref] .] In the present paper we will adapt the “long resonator” argument to the case $σ=1$, and prove the following theorem.
Extreme values of the Riemann zeta function on the 1-line · Introduction · 0.83
Uses its method or tool 1 paper
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We will use the construction of Bondarenko and Seip in [ref] .
Extreme values of derivatives of the Riemann zeta function · Proof of Proposition [ref] · 1.00
Compares with it 1 paper
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We may in fact obtain a slightly weaker result than [ref] as a direct consequence of our earlier work in [ref] .
Extreme values of the Riemann zeta function and its argument · Introduction · 0.83
Cites it as background 1 paper
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For details see [ref] and the very recent paper [ref] .
Extreme values of the Riemann zeta function on the 1-line · Introduction · 0.89
What it cites · 9 sentences without a confident label
Builds on 1 paper
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This was done for $σ=1/2$ and $M≥N^ε$ for every $ε>0$ in [ref] .
Large GCD sums and extreme values of the Riemann zeta function · The case $1/2+1/log_2 T ≤σ≤3/4$ · 0.87
Uses the method or tool of 1 paper
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Indeed, we may use the construction in [ref] and the estimates established there, corresponding to Lemma [ref] and Lemma [ref].
Large GCD sums and extreme values of the Riemann zeta function · Proof of Theorem [ref] · 0.92
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