Exponential sums with multiplicative coefficients

Bachman, Gennady¹

¹ Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020

Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 128-135.

Résumé

We provide estimates for the exponential sum \begin{equation*}F(x,\alpha )=\sum _{n\le x} f(n)e^{2\pi i\alpha n}, \end{equation*} where $x$ and $\alpha$ are real numbers and $f$ is a multiplicative function satisfying $|f|\le 1$. Our main focus is the class of functions $f$ which are supported on the positive proportion of primes up to $x$ in the sense that $\sum _{p\le x}|f(p)|/p\gg \log \log x$. For such $f$ we obtain rather sharp estimates for $F(x,\alpha )$ by extending earlier results of H. L. Montgomery and R. C. Vaughan. Our results provide a partial answer to a question posed by G. Tenenbaum concerning such estimates.

Détail

DOI : 10.1090/S1079-6762-99-00071-2

Bibliographie

[1] Bachman, Gennady On the coefficients of cyclotomic polynomials Mem. Amer. Math. Soc. 1993

[2] Bachman, Gennady On exponential sums with multiplicative coefficients 1995 29 38

[3] Bachman, Gennady Some remarks on nonnegative multiplicative functions on arithmetic progressions J. Number Theory 1998 72 91

[4] Daboussi, Hédi Fonctions multiplicatives presque périodiques B 1975 321 324

[5] Daboussi, H. On some exponential sums 1990 111 118

[6] Daboussi, Hédi, Delange, Hubert Quelques propriétés des fonctions multiplicatives de module au plus égal à 1 C. R. Acad. Sci. Paris Sér. A 1974 657 660

[7] Daboussi, Hédi, Delange, Hubert On multiplicative arithmetical functions whose modulus does not exceed one J. London Math. Soc. (2) 1982 245 264

[8] Dupain, Y., Hall, R. R., Tenenbaum, G. Sur l’équirépartition modulo 1 de certaines fonctions de diviseurs J. London Math. Soc. (2) 1982 397 411

[9] Elliott, P. D. T. A. Multiplicative functions on arithmetic progressions. VI. More middle moduli J. Number Theory 1993 178 208

[10] Fouvry, É., Tenenbaum, G. Entiers sans grand facteur premier en progressions arithmetiques Proc. London Math. Soc. (3) 1991 449 494

[11] Halberstam, H., Richert, H.-E. On a result of R. R. Hall J. Number Theory 1979 76 89

[12] Hall, R. R. Halving an estimate obtained from Selberg’s upper bound method Acta Arith. 1973/74 347 351

[13] Hildebrand, Adolf Multiplicative functions on arithmetic progressions Proc. Amer. Math. Soc. 1990 307 318

[14] Hildebrand, Adolf, Tenenbaum, Gérald Integers without large prime factors J. Théor. Nombres Bordeaux 1993 411 484

[15] Goubin, Louis Sommes d’exponentielles et principe de l’hyperbole Acta Arith. 1995 303 324

[16] Montgomery, H. L., Vaughan, R. C. The order of magnitude of the 𝑚th coefficients of cyclotomic polynomials Glasgow Math. J. 1985 143 159

[17] Montgomery, H. L., Vaughan, R. C. Exponential sums with multiplicative coefficients Invent. Math. 1977 69 82

[18] Shiu, P. A Brun-Titchmarsh theorem for multiplicative functions J. Reine Angew. Math. 1980 161 170

[19] Tenenbaum, Gérald Facteurs premiers de sommes d’entiers Proc. Amer. Math. Soc. 1989 287 296

[20] Vaughan, R. C. A new iterative method in Waring’s problem Acta Math. 1989 1 71