A family of multiple integrals connected with relatives of the Dickman function

Abstract

In this paper we present an asymptotic expansion for a family of multiple integrals connected with relatives of the Dickman function. The coefficients of this expansion have a similar arithmetic structure as those appearing in Soundararajan's work on an analogous expansion for the Dickman

A note on an asymptotic expansion related to the Dickman function

“…The convolution identity in (6) was proved in [2, see (17)], while the contour integral representation in (7) was shown in [2, see (23)]. Using this last representation, the author established in [2] an asymptotic expansion for K ℓ (u, κ), given below. The proof follows that of (1), but requires some additional calculations.…”

“…See [2,Lemma 4] to verify that the constants Cr,κ defined in (16) are generated by the function given above in (8). This completes the sketch of the proof of Theorem 2.…”

“…Proofs of these observations can be found in [2]. For our purpose, we provide a different recursive formula for Cr,κ, whose proof uses only the integral definition of Cr,κ given above in (16).…”

“…We will deduce Theorem 1 from an asymptotic expansion for a related family of integrals, K ℓ (u, κ), previously investigated by the author in [2]. To elaborate on this family, for integers ℓ ≥ 1, we define…”

“…These integrals arise when one investigates the mean value of the generalized divisor function, dκ(n), over the smooth numbers, S(x, y). For instance, letting u = log x/ log y, it was shown in [2] that as x, y → ∞ with u bounded, n∈S(x,y)…”