Hecke's Functional Equation and Arithmetical Identities

Abstract

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Weighted Divisor Sums and Bessel Function Series

“…If a(n) is generated by a Dirichlet series satisfying a functional equation involving Γ(s), is there an analogue of (1.8) for n≤x a(n); if so, can it be extended to provide a representation for n≤x a(n)(x − n) q , for complex q, as a double series of Bessel functions? Bessel function identities for n≤x a(n)(x − n) q are, in fact, equivalent to the corresponding Dirichlet series satisfying a functional equation involving Γ(s) [6]. We derive Theorem 1.2 from Theorem 1.1.…”

“…x n 1/2 6) where the prime on the summation sign on the left side indicates that if x is an integer, only 1 2 r 2 (x) is counted. Observe that the series on the right side of (1.6) is similar to the inner series on the right side of (1.3).…”

“…However, in many cases n≤x a(n) may not be representable in terms of an infinite series of Bessel functions, but, for sufficiently large positive numbers q, n≤x a(n)(x − n) q can be so represented. See, for example, [6], [1], and [2]. Is there an analogue of (1.8) for 1≤n≤x d χ (n)(x − n) q for complex q?…”

Questionable claims found in Ramanujan’s lost notebook

“…We only briefly sketch the background and hypotheses needed for the statement of the Voronoï summation formula. For complete details, see the papers [4], [5], [6], and [15]. Table 3 x Let s = σ + it, with σ and t real, and let…”

“…For such arithmetical functions, K. Chandrasekharan and R. Narasimhan [15] and Berndt [4], [5] have established theorems providing representations for the Riesz sum…”