Abstract
Let f be any arithmetic function and define \({S_{f}}(x):=\sum\nolimits_{{n \le x}}f([x/n])\). If the function f is small, namely, f(n) ≪ nε, then the error term Ef(x) in the asymptotic formula of Sf(x) has the form O(x1/2+ε). In this paper, we shall study the mean square of Ef(x) and establish some new results of Ef(x) for some special functions.
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References
Bordellés, O.: On certain sums of number theory, Int. J. Number Theory, 18(9), 2053–2074 (2022)
Bordellés, O., Dai, L., Heyman, R., et al.: On a sum involving the Euler function, J. Number Theory, 202, 278–297 (2019)
Fouvry, E., Iwaniec, H.: Exponential sums with monomials, J. Number Theory, 33, 311–333 (1989)
Heath-Brown, D. R.: The Pjateckiĭ–Šapiro prime number theorem, J. Number Theory, 16, 242–266 (1983)
Heyman, R.: Primes in floor function sets, Integers, 22, Paper No. A59, 10 pp (2022)
Graham, S. W., Kolesnik, G.: Van der Corput’s method of Exponential Sums, London Mathematical Society Lecture Note Series, Vol. 126, Cambridge University Press, Cambridge, 1991
Liu, K., Wu, J., Yang, Z.: A variant of the Prime Number Theorem, Indag. Math. (N.S.), 33(2), 388–396 (2022)
Liu, K., Wu, J., Yang, Z.: On some sums involving the integral part function, Int. J. Number Theory, 20(3), 831–847 (2024)
Ma, J., Sun, H.: On a sum involving certain arithmetic functions and the integral part function, Ramanujan J., 60(4), 1025–1032 (2023)
Ma, J., Sun, H.: On a sum involving the divisor function, Period. Math. Hungar., 83, 185–191 (2021)
Ma, J., Wu, J.: On a sum involving the Mangoldt function, Period. Math. Hungar., 83, 39–48 (2021)
Ma, R., Wu, J.: On the primes in floor functions sets, Bull. Aust. Math. Soc., 2, 236–243 (2023)
Robert, O., Sargos, P.: Three-dimensional exponential sums with monomials, J. Reine Angew. Math., 591, 1–20 (2006)
Stucky, J.: The fractional sum of small arithmetic functions, J. Number Theory, 238, 731–739 (2022)
Vaaler, J. D.: Some extremal problems in Fourier analysis, Bull. Amer. Math. Soc., 12(2), 183–216 (1985)
Wu, J.: On a sum involving the Euler totient function, Indagationes Mathematicae, 30, 536–541 (2019)
Wu, J.: Note on a paper by Bordellés, Dai, Heyman, Pan, and Shparlinski, Period. Math. Hungar., 80(1), 95–102 (2020)
Wu, J.: On the average number of direct factors of a finite abelian group, Monatsh. Math., 131(1), 79–89 (2000)
Zhai, W.: On a sum involving the Euler function, J. Number Theory, 211, 199–219 (2020)
Zhai, W.: On a sum involving small arithmetic functions, Int. J. Number Theory, 18(9), 2029–2052 (2022)
Zhai, W.: On the average number of unitary factors of finite abelian groups. II. Acta Math. Sin. (Engl. Ser.), 16(4), 549–554 (2000)
Zhai, W., Cao, X.: On the average number of direct factors of finite abelian groups, Acta Arith., 82(1), 45–55 (1997)
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The author deeply thanks the referees for careful reading of the manuscript and for many valuable suggestions.
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Supported by the National Natural Science Foundation of China (Grant No. 11971476)
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Zhai, W.G. On Some Sums Involving Small Arithmetic Functions. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2129-y
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DOI: https://doi.org/10.1007/s10114-024-2129-y