Abstract
In number theory, one often encounters sums of the form
Where D is a bounded domain in R k and e(w) =e2πiw. We shall Refer to the case k = 1 as the one-dimensional case, k = 2 as the two- dimensional case, etc. Our objective here is to give an exposition of van der Corput’s method for estimating the sums in (1). The one- dimensional case is well understood. Our knowledge of the two-dimensional case is fragmentary, and dimensions higher than two are terra incognita We shall review the one-dimensional case in Section 2. In Section 3 we will give an outline of what is known and what is conjectured about the two-dimensional case.
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References
F. V. Atkinson, The mean value of the Riemann zeta-function, Acta. Math. 81 (1949), 353–376.
Chen Jing-Run, The lattice points in a circle, Sci. Sinica 12 (1963),633–649.
S. W. Graham, The distribution of squarefree numbers, J. London Math. Soc. (2) 24 (1981), 54–64.
W. Haneke, Verscharfung der Abschätzung von ζ(1/2 + it), Acta Arith. 8 (1963), 357–430.
D. R. Heath-Brown, The Pjateckii-Sapiro prime number theorem, J. No. Theory 16 (1963), 242–266.
F. Herzog and G. Piranian, Sets of convergence of Taylor Series I, Duke Math. Jnl. 16 (1949) 529–534.
L. K. Hua, The lattice points in a circle, Quke. J. Math. (Oxford) 12 (1941), 193–200.
G. Kolesnik, Improvement of remainder term for the divisors problem,Math. Zametki 6 (1969), 545–554.
———, On the estimation of multiple exponential sums, Recent Progress in Analytic Number Theory, Vol. 1 (eds. H. Halberstam and C. Hooley, Academic Press, New York, 1981) 247–256.
———, On the number of abelian groups of a given order J. Reine Angew. Math. 329 (1981), 164–175.
———, On the order of ζ(1/2 + it) and Δ(R), Pac. Jnl. of Math., 98 (1982) 107–122.
S. H. Min, On the order of ζ(1/2 + it), Trans. Amer. Math. Soc. 65 (1949) 448–472.
E. Phillips, The zeta-function of Riemann; further developments of van der Corput’s method, Quart. J. Math. (Oxford) 4 (1933) 209–225.
R. A. Rankin, Van der Corput’s method and the theory of exponent pairs, Quart. J. Math. Oxford (2), 6 (1955) 147–153.
H. E. Richert, Verschrfüng der Abschatzüng beim Dirichletsehen Teilerproblem, Math. Z. 58 (1953) 204–218.
P. G. Schmidt, Zur Anzahl Abelscher Gruppen gegebner Ordnung I, Acta Anith. 13 (1968) 405–417.
B. R. Srinivasan, The lattice point problem in many dimensional hyperboloids, III, Math. Ann. 160 (1965) 280–311.
E. C. Titchmarsh, The lattice points in a circle, Proc. London Math. Soc. (2) 38 (1934) 96–155; see also “Corrigendum”, op. cit. 55 (1935).
———, On the order of ζ(l/2 + it), Quart. J. Math. (Oxford) 13 (1942) 11–17.
———, The theory of the Riemann-zeta function, Clarendon Press, Oxford 1951.
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© 1987 Birkhäuser Boston
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Graham, S.W., Kolesnik, G. (1987). One and Two Dimensional Exponential Sums. In: Adolphson, A.C., Conrey, J.B., Ghosh, A., Yager, R.I. (eds) Analytic Number Theory and Diophantine Problems. Progress in Mathematics, vol 70. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4816-3_11
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DOI: https://doi.org/10.1007/978-1-4612-4816-3_11
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