Geodesic
The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a~Problem of Chowla
Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 204-222.

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The double trigonometric series $U(x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{e^{2\pi imnx}}{\pi mn}$ and $U(\chi,x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\chi_{m,n}\frac{e^{2\pi imnx}}{\pi mn}$ with the hyperbolic phase and coordinate-wise slow multipliers $\chi_{m,n}$ are studied. Complete descriptions of the $\mathcal K$-convergence (summability) sets of the sine series $\Im U(x)$ and the cosine series $\Re U(x)$ are given. The $\mathcal K$-sum of a double series is defined as the common value of the limits of partial sums over expanding families of kites in $\mathbb N^2$. The latter include convex domains in the usual sense, such as rectangles, as well as nonconvex domains, for example, hyperbolic crosses $\{(m,n):1\le mn\le N\}$. 
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K. I. Oskolkov. The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a~Problem of Chowla. Informatics and Automation, Studies on function theory and differential equations, Tome 248 (2005), pp. 204-222. http://geodesic.mathdoc.fr/item/TRSPY_2005_248_a19/

[1] Arkhipov G.I., Oskolkov K.I., “Ob odnom spetsialnom trigonometricheskom ryade i ego primeneniyakh”, Mat. sb., 134:2 (1987), 147–157 | MR

[2] Berry M.V., “Quantum fractals in boxes”, J. Phys. A: Math. and Gen., 29 (1996), 6617–6626 | DOI | MR

[3] Berry M., Marzoli I., Schleich W., “Quantum carpets, carpets of light”, Phys. World., 2001, June, 39–44

[4] The collected papers of Sarvadaman Chowla, V. 1 (1925–1935), eds. J. G. Huard, K. S. Williams, Univ. Montreal, Publ. CRM, Montreal, 1999 ; V. 2 (1936–1961) ; V. 3 (1962–1986) | MR | MR | MR

[5] Chowla S.D., “Some problems of diophantine approximation. (I)”, Math. Ztschr., 33 (1931), 544–563 | DOI | MR | Zbl

[6] Friesch O., Marzoli I., Schleich W.P., “Quantum carpets woven by Wiegner functions”, New J. Phys., 2 (2000), 4.1–4.11 | DOI | Zbl

[7] Garaev M.Z., “On a multiple trigonometric series”, Acta arith., 102:2 (2002), 183–187 | DOI | MR | Zbl

[8] Collected papers of G. H. Hardy, including joint papers with J. E. Littlewood and others, v. 1, Clarendon Press, Oxford, 1966 | MR

[9] Hecke E., “Über analytische Funktionen und die Verteilung von Zahlen mod. eins”, Abhandl. Math. Sem. Hamburg. Univ., 1 (1921), 54–76 | DOI

[10] Hua L.K., Introduction to number theory, Springer, Berlin; Heidelberg; New York, 1982 | MR

[11] Oskolkov K.I., “Ob odnom rezultate Telyakovskogo i kratnykh preobrazovaniyakh Gilberta s polinomialnoi fazoi”, Mat. zametki, 74:2 (2003), 242–256 | MR | Zbl

[12] Oskolkov K.I., “Ryady Vinogradova v zadache Koshi dlya uravnenii tipa Shrëdingera”, Tr. MIAN, 200 (1991), 265–288 | Zbl

[13] Oskolkov K.I., “A class of I. M. Vinogradov's series and its applications in harmonic analysis”, Progress in approximation theory. An international prospective, Proc. Intern. Conf. on Approximation Theory (Tampa, Univ. South Florida, March 19–22, 1990), Springer Ser. Comput. Math., 19, eds. A. A. Gonchar, E. B. Saff, Springer, New York, 1992, 353–402 | MR | Zbl

[14] Oskolkov K.I., “Schrödinger equation and oscillatory Hilbert transforms of second degree”, J. Fourier Anal. and Appl., 4:3 (1998), 341–356 | DOI | MR | Zbl

[15] Oskolkov K.I., “On functional properties of incomplete Gaussian sums”, Canad. J. Math., 43:1 (1991), 182–212 | MR | Zbl

[16] Oskolkov K.I., “Continued fractions and the convergence of a double trigonometric series”, East J. Approx., 9:3 (2003), 375–383 | MR | Zbl

[17] Oskolkov K.I., The valleys of shadow in Schrödinger landscape, Res. Rept. Columbia, Univ. South Carolina, Industr. Math. Inst., no. 11, 2003 | Zbl

[18] Stein E.M., Wainger S., “Discrete analogues in harmonic analysis. I: $l^2$ estimates for singular Radon transforms”, Amer. J. Math., 121 (1999), 1291–1336 | DOI | MR | Zbl

[19] Stein E.M., Wainger S., “Discrete analogues in harmonic analysis. II: Fractional integration”, J. anal. math., 80 (2000), 335–354 | DOI | MR

[20] Stein E.M., Wainger S., “Two discrete fractional integral operators revisited”, J. anal. math., 87 (2002), 451–479 | DOI | MR | Zbl

[21] Stein E.M., Wainger S., “Discrete analogs of singular Radon transforms”, Bull. Amer. Math. Soc., 23 (1990), 537–544 | DOI | MR | Zbl

[22] Stein E.M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, With the assistance of T. S. Murphy, Princeton Univ. Press, Princeton (NJ), 1993 | MR | Zbl

[23] Talbot H.F., “Facts relating to optical science. IV”, London and Edinburgh Philos. Mag. and J. Sci. Ser. 3, 9:56 (1836), 401–407

[24] Telyakovskii S.A., “Ob otsenkakh proizvodnykh ot trigonometricheskikh polinomov mnogikh peremennykh”, Sib. mat. zhurn., 4 (1963), 1404–1411

[25] Telyakovskii S.A., “Ravnomernaya ogranichennost nekotorykh kratnykh trigonometricheskikh polinomov”, Mat. zametki, 42:1 (1987), 33–39 | MR | Zbl

[26] Telyakovskii S.A., Temlyakov V.N., “O skhodimosti kratnykh ryadov Fure funktsii ogranichennoi variatsii”, Mat. zametki, 61:4 (1997), 583–595 | MR | Zbl

[27] Vinogradov I.M., Osnovy teorii chisel, Nauka, M., 1981 | MR

[28] Zigmund A., Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR