×

zbMATH — the first resource for mathematics

Study of the multiplicative autocorrelation of the fractional part function. (Étude de l’autocorrélation multiplicative de la fonction ‘partie fractionnaire’.) (French) Zbl 1173.11343
The authors study the function \[ A(\lambda):=\int_0^\infty\{t\}\{\lambda t\}\frac{dt}{t^2},\quad \lambda>0, \] which can be used to reformulate the Riemann hypothesis. They prove that this function has a strict local maximum at every rational point and that its Mellin transform is \[ -\frac{\zeta(-s)\zeta(s+1)}{s(s+1)}. \]

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11L99 Exponential sums and character sums
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alcántara-Bode, J., Some properties of the Beurling function, Pro Mathematica, 14, 5-11, (2000)
[2] Báez-Duarte, L., A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Rend. Mat. Ac. Lincei, S. 9, 14, 1, 5-11, (2003) · Zbl 1097.11041
[3] L. Báez-Duarte, M. Balazard, B. Landreau, et E. Saias, Notes sur la fonction ζ de Riemann, 3, Adv. in Math.149 (2000), 130-144. · Zbl 1008.11032
[4] L. Báez-Duarte, M. Balazard, B. Landreau, et E. Saias, Sur l’autocorrélation multiplicative de la fonction partie fractionnaire, arXiv : math.NT/0306251. · Zbl 1173.11343
[5] A. Beurling, “A closure problem related to the Riemann Zeta-function,” Proc. Nat. Acad. Sci.41 (1955), 312-314. · Zbl 0065.30303
[6] Chowla, S.; Walfisz, A., Über eine Riemannsche Identität, Acta Arith., 1, 87-112, (1936) · Zbl 0010.39201
[7] de la Bretèche, R.; Tenenbaum, G., Séries trigonométriques à coefficients arithmétiques, Journal d’Analyse Mathématique, 92, 1-79, (2004) · Zbl 1171.11319
[8] T. Estermann, “On the representation of a number as the sum of two products,” Proc. London Math. Soc.31(2) (1930), 123-133. · JFM 56.0174.02
[9] Ishibashi, M., The value of the Estermann zeta functions at \(s = 0,\) Acta Arith., 73, 357-361, (1995) · Zbl 0845.11034
[10] M. Jutila, “Lectures on a method in the theory of exponential sums,” TIFR Lectures on Mathematics and Physics, Springer-Verlag, Berlin, 1987. · Zbl 0671.10031
[11] Lehmer, D. H., Euler constants for arithmetical progressions, Acta Arith., 27, 125-142, (1975) · Zbl 0302.12003
[12] B. Nyman, “On some groups and semigroups of translations,” Thèse, Uppsala, 1950. · Zbl 0037.35401
[13] Sylvester, J. J., Sur la fonction \(E(x),\) C. R. A. S., 50, 732-734, (1860)
[14] V. I. Vassiounine, “Sur un système biorthogonal relié à l’hypothèse de Riemann (en russe),” Alg. i An.7 (1995), 118-135; traduction anglaise dans St-Petersburg Math. J.7 (1996), 405-419.
[15] Walfisz, A., Über einige trigonometrische Summen, Math. Z., 33, 564-601, (1931) · Zbl 0001.39001
[16] A. Zygmund, Trigonometric Series, 2nd ed. Vols. I, II., Cambridge University Press, New York, 1959. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.