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Positive radial solutions involving nonlinearities with zeros
A generalization of Kátai's orthogonality criterion with applications
1. | The Ohio State University, 231 W 18th Ave, MA 410, Columbus, OH 43210, USA |
2. | Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland |
3. | Northwestern University, 2033 Sheridan Rd, B4, Evanston, IL 60208, USA |
$ a\colon \mathbb{N} \to \mathbb{C} $ |
$ \begin{equation*} \sum\limits_{n\leq x} a(pn)\overline{a(qn)} = {\rm{o}} (x),\;\; {\mathit{for\;all\;distinct\;primes}\;\; p \;\;\mathit{and}\;\; q}. \end{equation*} $ |
$ f $ |
$ z\in \mathbb{C} $ |
$ E = \{n\in \mathbb{N} :f(n) = z\} $ |
$ \begin{equation*} \sum\limits_{n\leq x} {1_{E}(n)a(n)} = {\rm{o}} (x). \end{equation*} $ |
$ E = \{n_1<n_2<\ldots\} $ |
References:
[1] |
V. Bergelson and I. J. Håland Knutson,
Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416.
doi: 10.1017/S0143385708000862. |
[2] |
V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, and F. K. Richter, A structure theorem for level sets of multiplicative functions and applications, International Mathematics Research Notices, (2018).
doi: 10.1093/imrn/rny040. |
[3] |
M. Boshernitzan,
An extension of Hardy's class L of "orders of infinity", J. Anal. Math., 39 (1981), 235-255.
doi: 10.1007/BF02803337. |
[4] |
M. Boshernitzan,
New "orders of infinity", J. Anal. Math., 41 (1982), 130-167.
doi: 10.1007/BF02803397. |
[5] |
M. Boshernitzan,
Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240.
doi: 10.1007/BF02835955. |
[6] |
J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier Analysis and Number Theory to Radon Transforms and Geometry, vol. 28 of Dev. Math., Springer, New York (2013), 67–83.
doi: 10.1007/978-1-4614-4075-8_5. |
[7] |
H. Daboussi,
Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324.
|
[8] |
H. Daboussi and H. Delange,
On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2), 26 (1982), 245-264.
doi: 10.1112/jlms/s2-26.2.245. |
[9] |
H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837. |
[10] |
H. Delange,
Un théorème sur les fonctions arithmétiques multiplicatives et ses applications, Ann. Sci. École Norm. Sup. (3), 78 (1961), 1-29.
doi: 10.24033/asens.1097. |
[11] |
P. D. T. A. Elliott, Probabilistic Number Theory. I, vol. 239 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin 1979, Mean-value theorems. |
[12] |
N. Frantzikinakis and B. Host,
Multiple ergodic theorems for arithmetic sets, Trans. Amer. Math. Soc., 369 (2017), 7085-7105.
doi: 10.1090/tran/6870. |
[13] |
A. Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach. In preparation – Available from: http://www.dms.umontreal.ca/ andrew/PDF/BookChaps1n2.pdf. |
[14] |
G. Halász,
Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403.
doi: 10.1007/BF01894515. |
[15] |
G. H. Hardy,
Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10 (1912), 54-90.
doi: 10.1112/plms/s2-10.1.54. |
[16] |
G. H. Hardy, Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge tracts in mathematics and mathematical physics, 12, Cambridge University Press, London, 1954. |
[17] |
I. Kátai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[18] |
I. Z. Ruzsa,
General multiplicative functions, Acta Arith., 32 (1977), 313-347.
doi: 10.4064/aa-32-4-313-347. |
[19] |
I. Schoenberg,
Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199.
doi: 10.1007/BF01181156. |
[20] |
E. Wirsing,
Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102.
doi: 10.1007/BF01351892. |
show all references
References:
[1] |
V. Bergelson and I. J. Håland Knutson,
Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416.
doi: 10.1017/S0143385708000862. |
[2] |
V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, and F. K. Richter, A structure theorem for level sets of multiplicative functions and applications, International Mathematics Research Notices, (2018).
doi: 10.1093/imrn/rny040. |
[3] |
M. Boshernitzan,
An extension of Hardy's class L of "orders of infinity", J. Anal. Math., 39 (1981), 235-255.
doi: 10.1007/BF02803337. |
[4] |
M. Boshernitzan,
New "orders of infinity", J. Anal. Math., 41 (1982), 130-167.
doi: 10.1007/BF02803397. |
[5] |
M. Boshernitzan,
Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240.
doi: 10.1007/BF02835955. |
[6] |
J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier Analysis and Number Theory to Radon Transforms and Geometry, vol. 28 of Dev. Math., Springer, New York (2013), 67–83.
doi: 10.1007/978-1-4614-4075-8_5. |
[7] |
H. Daboussi,
Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324.
|
[8] |
H. Daboussi and H. Delange,
On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2), 26 (1982), 245-264.
doi: 10.1112/jlms/s2-26.2.245. |
[9] |
H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837. |
[10] |
H. Delange,
Un théorème sur les fonctions arithmétiques multiplicatives et ses applications, Ann. Sci. École Norm. Sup. (3), 78 (1961), 1-29.
doi: 10.24033/asens.1097. |
[11] |
P. D. T. A. Elliott, Probabilistic Number Theory. I, vol. 239 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin 1979, Mean-value theorems. |
[12] |
N. Frantzikinakis and B. Host,
Multiple ergodic theorems for arithmetic sets, Trans. Amer. Math. Soc., 369 (2017), 7085-7105.
doi: 10.1090/tran/6870. |
[13] |
A. Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach. In preparation – Available from: http://www.dms.umontreal.ca/ andrew/PDF/BookChaps1n2.pdf. |
[14] |
G. Halász,
Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403.
doi: 10.1007/BF01894515. |
[15] |
G. H. Hardy,
Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10 (1912), 54-90.
doi: 10.1112/plms/s2-10.1.54. |
[16] |
G. H. Hardy, Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge tracts in mathematics and mathematical physics, 12, Cambridge University Press, London, 1954. |
[17] |
I. Kátai,
A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.
doi: 10.1007/BF01949145. |
[18] |
I. Z. Ruzsa,
General multiplicative functions, Acta Arith., 32 (1977), 313-347.
doi: 10.4064/aa-32-4-313-347. |
[19] |
I. Schoenberg,
Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199.
doi: 10.1007/BF01181156. |
[20] |
E. Wirsing,
Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102.
doi: 10.1007/BF01351892. |
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