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ON THE VOLTERRA INTEGRAL EQUATION FOR THE REMAINDER TERM IN THE ASYMPTOTIC FORMULA ON THE ASSOCIATED EULER TOTIENT FUNCTION
Keywords:
Volterra integral equation of second type, the remainder term in the asymptotic formula, twisted Euler $\varphi$-function, associated Euler totient functionDOI:
https://doi.org/10.17654/0972555522017Abstract
Kaczorowski and Wiertelak considered the integral equation for remainder terms in the asymptotic formula for the Euler totient function and for the twisted Euler $\varphi$-function. In [4], Kaczorowski defined the associated Euler totient function which extends the above two functions and proved an asymptotic formula for it. In the present paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the asymptotic formula for the associated Euler totient function into two summands called arithmetic and analytic part, respectively.
Received: April 2, 2022
Accepted: May 9, 2022
References
H. Iwata, On the solution of the Volterra integral equation of second type for the error team in an asymptotic formula for arithmetic functions, Advanced Studies: Euro-Tbilisi Mathematical Journal (to appear).
J. Kaczorowski and K. Wiertelak, Oscillations of the remainder term related to the Euler totient function, J. Number Theory 130 (2010), 2683-2700.
J. Kaczorowski and K. Wiertelak, On the sum of the twisted Euler function, Int. J. Number Theory 8(7) (2012), 1741-1761.
J. Kaczorowski, On a generalization on the Euler totient function, Monatsh Math. 170 (2013), 27-48.
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