Large greatest common divisor sums and extreme values of the Riemann zeta function
By Andriy V. Bondarenko, Kristian Seip. 2015
87 citations recorded by the source corpus.
It is shown that the maximum of $|\zeta(1/2+it)|$ on the interval $T^{1/2}\le t \le T$ is at least $\exp\left((1/\sqrt{2}+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. Our proof uses Soundararajan's resonance method and a certain large GCD sum. The method of proof shows that the absolute constant $A$ in the inequality \[ \sup_{1\le n_1<\cdots < n_N} \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N \exp\left(A\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right), \] established in a recent paper of ours, cannot be taken smaller than $1$.
Subfields: Algebra and Number Theory, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b88ebd55ee2adc34bad0337c36e33bca173b4b84
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In Algebra and Number Theory
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Citations in the corpus
- Sommes de Gál et applications (2018, background): “(i) Dans [7], il est montré que c < √ min{ 1 2 , 1− β} est une condition admissible.”
- Extreme values of the Riemann zeta function and its argument (2017, methodology): “We proceed as in [2] (following an idea from [1]) and let J be the set of integers j such that [ (1 + T−1) j , (1 + T−1) j+1 )⋂ M = ∅,”
- Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, unknown): “)| ≥ exp c s logT loglogT ! in [26] under the Riemann hypothesis, then the constant c was improved in [4] and, unconditionally, in [35]. The best lower bounds on the maximum were recently obtained in [6]: for any c < 1/ √ 2, (2) max s∈[0,T] ζ 1 2 +is ≥ exp c s logT logloglogT loglogT ! . Previo”