Duke Mathematical Journal

Large greatest common divisor sums and extreme values of the Riemann zeta function

Andriy Bondarenko and Kristian Seip

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It is shown that the maximum of |ζ(1/2+it)| on the interval T1/2tT is at least exp((1/2+o(1))logTlogloglogT/loglogT). Our proof uses Soundararajan’s resonance method and a certain large greatest common divisor sum. The method of proof shows that the absolute constant A in the inequality

sup 1n1<<nNk,=1Ngcd(nk,n)nknNexp(AlogNlogloglogNloglogN), established in a recent paper of ours, cannot be taken smaller than 1.

Article information

Duke Math. J., Volume 166, Number 9 (2017), 1685-1701.

Received: 5 August 2015
Revised: 23 July 2016
First available in Project Euclid: 26 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11C20: Matrices, determinants [See also 15B36]

greatest common divisor sums Riemann zeta function extreme values


Bondarenko, Andriy; Seip, Kristian. Large greatest common divisor sums and extreme values of the Riemann zeta function. Duke Math. J. 166 (2017), no. 9, 1685--1701. doi:10.1215/00127094-0000005X. https://projecteuclid.org/euclid.dmj/1485400054

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  • [1] C. Aistleitner, Lower bounds for the maximum of the Riemann zeta function along vertical lines, Math. Ann. 365 (2016), 473–496.
  • [2] C. Aistleitner, I. Berkes, and K. Seip, GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc. 17 (2015), 1517–1546.
  • [3] R. Balasubramanian and K. Ramachandra, On the frequency of Titchmarsh’s phenomenon for $\zeta(s)$, III, Proc. Indian Acad. Sci. Sect. A 86 (1977), 341–351.
  • [4] A. Bondarenko and K. Seip, GCD sums and complete sets of square-free numbers, Bull. Lond. Math. Soc. 47 (2015), 29–41.
  • [5] A. Bondarenko and K. Seip, “Note on the resonance method for the Riemann zeta function” in Operator Theory: Advances and Applications, Birkhäuser, Basel, to appear.
  • [6] J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205–224.
  • [7] T. Dyer and G. Harman, Sums involving common divisors, J. Lond. Math. Soc. 34 (1986), 1–11.
  • [8] I. S. Gál, A theorem concerning Diophantine approximations, Nieuw Arch. Wiskunde 23 (1949), 13–38.
  • [9] A. Granville and K. Soundararajan, “Extreme values of $|\zeta(1+it)|$” in The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra, Ramanujan Math. Soc. Lect. Notes Ser. 2, Ramanujan Math. Soc., Mysore, 2006, 65–80.
  • [10] G. Harman, Metric Number Theory, London Math. Soc. Monogr. N.S. 18, Clarendon Press, Oxford Univ. Press, New York, 1998.
  • [11] T. Hilberdink, An arithmetical mapping and applications to $\Omega$-results for the Riemann zeta function, Acta Arith. 139 (2009), 341–367.
  • [12] J. F. Koksma, On a certain integral in the theory of uniform distribution, Nederl. Akad. Wetensch. Proc. Ser. A. 54/Indag. Math. 13 (1951), 285–287.
  • [13] Y. Lamzouri, On the distribution of extreme values of zeta and L-functions in the strip $1/2<\sigma<1$, Int. Math. Res. Not. IMRN 2011, no. 23, 5449–5503.
  • [14] N. Levinson, $\Omega$-theorems for the Riemann zeta-function, Acta Arith. 20 (1972), 317–330.
  • [15] M. Lewko and M. Radziwiłł, Refinements of Gál’s theorem and applications, Adv. Math. 305 (2017), 280–297.
  • [16] H. L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv. 52 (1977), 511–518.
  • [17] K. Ramachandra and A. Sankaranarayanan, Note on a paper by H. L. Montgomery (Omega theorems for the Riemann zeta-function), Publ. Inst. Math. (Beograd) (N.S.) 50 (64) (1991), 51–59.
  • [18] K. Soundararajan, Extreme values of zeta and $L$-functions, Math. Ann. 342 (2008), 467–486.
  • [19] E. C. Titchmarsh, On an inequality satisfied by the zeta-function of Riemann, Proc. Lond. Math. Soc. 28 (1928), 70–80.
  • [20] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford Univ. Press, New York, 1986.