×

Multiplicative functions in short intervals. (English) Zbl 1339.11084

In the paper under review, the authors relate short averages of the multiplicative function \(f:\mathbb{N}\to[-1,1]\) to its long averages, by proving that there exists absolute constants \(C,C'>1\) such that for any \(2\leq h\leq X\) and \(\delta>0\), one has \[ \left|\frac{1}{h}\sum_{x\leq n\leq x+h}f(n)-\frac{1}{X}\sum_{X\leq n\leq 2X}f(n) \right|\leq\delta+C'\frac{\log\log h}{\log h} \] for all but at most \[ CX\left(\frac{\log^{\frac{1}{3}}h}{\delta^2 h^{\frac{\delta}{25}}}+\frac{1}{\delta^2\log^{\frac{1}{50}}X}\right) \] integers \(x\in[X,2X]\). Moreover, as a bilinear version of this result holding in all intervals of length \(\asymp\sqrt{x}\), they show that for any \(10\leq h\leq x\) one has \[ \begin{aligned} & \frac{1}{h\sqrt{x}\log 2}\sum_{\overset {x\leq n_1n_2\leq x+h\sqrt{x}}{\sqrt{x}\leq n_1\leq 2\sqrt{x}}}f(n_1)f(n_2)\\ &=\left(\frac{1}{\sqrt{x}}\sum_{\sqrt{x}\leq n\leq 2\sqrt{x}}f(n)\right)^2+O\left(\frac{\log\log h}{\log h}+\frac{1}{\log^{\frac{1}{100}}x}\right).\end{aligned} \] The authors use the above results to study \(x^\varepsilon\)-smooth numbers by proving that for given \(\varepsilon>0\) there exists constant \(C(\varepsilon)>0\) such that the number of \(x^\varepsilon\)-smooth numbers in the interval \([x,x+C(\varepsilon)\sqrt{x}]\) is at least \(\sqrt{x}\log^{-4}x\) for all large enough \(x\). As another application, they consider Chowla’s conjecture on the Liouville’s function \(\lambda(n)=(-1)^{\Omega(n)}\) asserting that \(x^{-1}\sum_{n\leq x}\lambda(n)\lambda(n+1)=o(1)\) as \(x\to\infty\), and prove that for every integer \(h\geq 1\) there exists \(\delta(h)>0\) such that \[ \frac{1}{x}\left|\sum_{n\leq x}\lambda(n)\lambda(n+h)\right|\leq 1-\delta(h) \] for all large enough \(x>1\). The authors mention that same result holds for any completely multiplicative function \(f:\mathbb N\to[-1,1]\) with \(f(n)<0\) for some \(n>0\), and they consider the problem of counting sign changes of \(f(n)\).

MSC:

11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions

References:

[1] A. Balog, On the distribution of integers having no large prime factor, Paris: Math. Soc. France, 1987, vol. 147-148. · Zbl 0617.10031
[2] J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, and A. Sárközy, ”On finite pseudorandom binary sequences. III. The Liouville function. I,” Acta Arith., vol. 87, iss. 4, pp. 367-390, 1999. · Zbl 0920.11053
[3] E. S. Croot III, ”On the oscillations of multiplicative functions taking values \(\pm 1\),” J. Number Theory, vol. 98, iss. 1, pp. 184-194, 2003. · Zbl 1090.11062 · doi:10.1016/S0022-314X(02)00024-0
[4] E. Croot, ”Smooth numbers in short intervals,” Int. J. Number Theory, vol. 3, iss. 1, pp. 159-169, 2007. · Zbl 1115.11057 · doi:10.1142/S1793042107000833
[5] P. D. T. A. Elliott, ”On the correlation of multiplicative functions,” Notas Soc. Mat. Chile, vol. 11, iss. 1, pp. 1-11, 1992. · Zbl 0810.11055
[6] P. D. T. A. Elliott, Duality in Analytic Number Theory, Cambridge: Cambridge Univ. Press, 1997, vol. 122. · Zbl 0887.11002 · doi:10.1017/CBO9780511983405
[7] C. Elsholtz and D. S. Gunderson, ”Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations,” Funct. Approx. Comment. Math., vol. 52, iss. 2, pp. 263-281, 2015. · Zbl 1397.11020 · doi:10.7169/facm/2015.52.2.6
[8] J. Friedlander and H. Iwaniec, Opera de Cribro, Providence, RI: Amer. Math. Soc., 2010, vol. 57. · Zbl 1226.11099
[9] J. B. Friedlander and A. Granville, ”Smoothing “smooth” numbers,” Philos. Trans. Roy. Soc. London Ser. A, vol. 345, iss. 1676, pp. 339-347, 1993. · Zbl 0795.11041 · doi:10.1098/rsta.1993.0133
[10] A. Ghosh and P. Sarnak, ”Real zeros of holomorphic Hecke cusp forms,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 14, iss. 2, pp. 465-487, 2012. · Zbl 1287.11054 · doi:10.4171/JEMS/308
[11] A. Granville, ”Smooth numbers: computational number theory and beyond,” in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Cambridge: Cambridge Univ. Press, 2008, vol. 44, pp. 267-323. · Zbl 1230.11157
[12] A. Granville and K. Soundararajan, ”Decay of mean values of multiplicative functions,” Canad. J. Math., vol. 55, iss. 6, pp. 1191-1230, 2003. · Zbl 1047.11093 · doi:10.4153/CJM-2003-047-0
[13] J. Hafner, On smooth numbers in short intervals under the Riemann hypothesis, 1993.
[14] G. Harman, Prime-Detecting Sieves, Princeton, NJ: Princeton Univ. Press, 2007, vol. 33. · Zbl 1220.11118
[15] G. Harman, J. Pintz, and D. Wolke, ”A note on the Möbius and Liouville functions,” Studia Sci. Math. Hungar., vol. 20, iss. 1-4, pp. 295-299, 1985. · Zbl 0544.10041
[16] A. Harper, Sharp conditional bounds for moments of the zeta function, 2013.
[17] A. Hildebrand, Math. Reviews.
[18] A. Hildebrand, ”Multiplicative functions at consecutive integers,” Math. Proc. Cambridge Philos. Soc., vol. 100, iss. 2, pp. 229-236, 1986. · Zbl 0615.10053 · doi:10.1017/S0305004100066056
[19] A. Ivić, The Riemann Zeta-Function. Theory and Applications, Mineola, NY: Dover Publications, 2003. · Zbl 1034.11046
[20] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53. · Zbl 1059.11001
[21] M. Jutila, ”Zero-density estimates for \(L\)-functions,” Acta Arith., vol. 32, iss. 1, pp. 55-62, 1977. · Zbl 0307.10045
[22] E. Kowalski, O. Robert, and J. Wu, ”Small gaps in coefficients of \(L\)-functions and \(\mathfrakB\)-free numbers in short intervals,” Rev. Mat. Iberoam., vol. 23, iss. 1, pp. 281-326, 2007. · Zbl 1246.11099 · doi:10.4171/RMI/496
[23] Y. Lau, J. Y. Liu, and J. Wu, ”Coefficients of symmetric square \(L\)-functions,” Sci. China Math., vol. 53, iss. 9, pp. 2317-2328, 2010. · Zbl 1267.11047 · doi:10.1007/s11425-010-4046-z
[24] H. W. Lenstra Jr., ”Elliptic curves and number-theoretic algorithms,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Providence, RI, 1987, pp. 99-120. · Zbl 0686.14039
[25] S. Lester, K. Matomäki, and M. Radziwill, Small scale distribution of zeros and mass of modular forms, 2015. · Zbl 1404.11041
[26] L. Lucht and F. Tuttas, ”Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen,” Monatsh. Math., vol. 87, iss. 1, pp. 15-19, 1979. · Zbl 0369.10029 · doi:10.1007/BF01470935
[27] K. Matomäki, Another note on smooth numbers in short intervals, 2015. · Zbl 1337.11064 · doi:10.1142/S1793042116500196
[28] K. Matomäki, ”A note on smooth numbers in short intervals,” Int. J. Number Theory, vol. 6, iss. 5, pp. 1113-1116, 2010. · Zbl 1204.11154 · doi:10.1142/S1793042110003381
[29] K. Matomäki and M. Radziwiłl, A note on the Liouville function in short intervals, 2015.
[30] K. Matomäki and M. Radziwiłl, ”Sign changes of Hecke eigenvalues,” Geom. Funct. Anal., vol. 25, iss. 6, pp. 1937-1955, 2015. · Zbl 1359.11040 · doi:10.1007/s00039-015-0350-7
[31] K. Matomäki, M. Radziwiłl, and T. Tao, ”An averaged form of Chowla’s conjecture,” Algebra Number Theory, vol. 9, iss. 9, pp. 2167-2196, 2015. · Zbl 1377.11109 · doi:10.2140/ant.2015.9.2167
[32] H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Providence, RI: Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc., 1994, vol. 84. · Zbl 0814.11001
[33] M. Radziwiłl and K. Soundararajan, ”Moments and distribution of central \(L\)-values of quadratic twists of elliptic curves,” Invent. Math., vol. 202, iss. 3, pp. 1029-1068, 2015. · Zbl 1396.11098 · doi:10.1007/s00222-015-0582-z
[34] K. Ramachandra, ”Some problems of analytic number theory,” Acta Arith., vol. 31, iss. 4, pp. 313-324, 1976. · Zbl 0291.10034
[35] B. Saffari and R. C. Vaughan, ”On the fractional parts of \(x/n\) and related sequences. II,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 27, iss. 2, p. v, 1-30, 1977. · Zbl 0379.10023 · doi:10.5802/aif.649
[36] P. Shiu, ”A Brun-Titchmarsh theorem for multiplicative functions,” J. Reine Angew. Math., vol. 313, pp. 161-170, 1980. · Zbl 0412.10030 · doi:10.1515/crll.1980.313.161
[37] K. Soundararajan, Smooth numbers in short intervals, 2010.
[38] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press, Cambridge, 1995, vol. 46. · Zbl 0831.11001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.