Functional equations with multiple gamma factors and the average order of arithmetical functions.

*(English)*Zbl 0211.37901This is a further development of earlier work by the same authors [Ann. Math. (2) 74, 1–23 (1961; Zbl 0107.03702); Acta Arith. 6, 487–503 (1961; Zbl 0101.03703); C. R. Acad. Sci., Paris 251, 1333–1335 (1960; Zbl 0093.05203); erratum, p. 2547].

It is assumed that a functional equation \(\Delta(s)\varphi(s)= \Delta(\delta - s)\psi( \delta - s)\) is satisfied, where

\[ \Delta(s) = \prod_{\nu=1}^N \Gamma(\alpha_\nu s + \beta_\nu) \qquad (\alpha_\nu> 0,\ A\equiv \sum_{\nu=1}^N \alpha_\nu \ge 1), \]

\(\delta\) is real, \(\varphi(s)\) and \(\psi(s)\) have absolutely convergent Dirichlet expansions \(\sum_{n=1}^N a_n\lambda_n^{-s}\) and \(\sum_{n=1}^N b_n \mu_n^{-s}\) in some half-plane, and the equation is to be understood in the sense that the two sides have a common analytic continuation \(\chi(s)\) in a domain \(\mathcal D\) which is the exterior of a bounded closed set \(S\), such that \(\chi(s)\to 0\) uniformly in every fixed finite interval \(\sigma_1\le\sigma\le\sigma_2\) as \(t\to\pm\infty\) \((s = \sigma+ it)\).

The main object is to obtain \(\Omega_\pm\) and \(O\)-theorems relating to

\[ R_\lambda^\rho(x) = A_\lambda^\rho(x) - Q_\rho(x),\]

where \(\Gamma(\rho+1) A_\lambda^\rho(x)\) is the Riesz sum of order \(\rho\ge 0\) and type \(\{\lambda_n\}\) of the coefficients \(a_n\) and \(Q_\rho(x)\) is a certain function arising from the singularities of \(\varphi(s)\) and \(\Gamma(s)\).

It is proved (Theorem 3.2) that the real (or imaginary) par of \(R_\lambda^\rho(x)\) is \(\Omega_\pm(x^\theta)\), where \(2A\theta = A\delta + \rho(2A -1) - \frac12\), if \(b'_n\ne 0\) for some \(n\), where \(b'_n\) is the real (or imaginary) part of \(b_n\); and (Theorem 3.1) that the conclusion may be strengthened to \(\Omega_\pm(x^\theta \omega(x))\), where \(\omega(x)\to\infty\) with \(x\), if the set \(\{n\}\) of positive integers has a subset \(M= \{n_k\}\) such that (i) no number \(\mu_n^{1/(2A)}\) is representable as a linear combination of numbers \(\mu_n^{1/(2A)}\) \((m\in M)\) with coefficients \(\pm 1\), apart from a trivial representation \(\mu_n^{1/(2A)} = \mu_n^{1/(2A)}\) 4/(2”) if \(n\in M\), and (ii) \(\sum_{m\in M} \vert b'_n\vert n_m^{-\tau} = \infty\), where \(2A\tau = A\delta + \rho + \frac12\)i.

It is also proved that Theorem 3.2 is best possible for large p\(\rho\), in that \(R_\lambda^\rho(x) = O(x^\theta)\) if \(\sum_{n=1} ^\infty \vert b_n\vert \mu_n^{-\tau} < \infty\).

The proofs make use of explicit formulae involving functions \(I(x)\) that have convenient approximations by Bessel and trigonometric functions, combined with a known adaptation to \(\Omega\)-problems of a device used in an analytical proof of Kronecker’s theorem on Diophantine approximation. For smaller \(\rho\) the \(O\)-results are less precise. The authors concentrate on \(R_\lambda^0(x)\), for which they use Landau’s method of taking a \(\rho\)-th difference of \(A_\lambda^\rho(x)\) with a sufficiently large integer \(\rho\). The full result (Theorem 4.1) is rather elaborate, but it includes classical estimates in special cases, such as the errors \(O(x^{1/3}\log x)\) and \(O(x^{1/3})\) in Dirichlet’s divisor problem and the problem of the lattice points of a circle, the corresponding \(\Omega\)-results of Theorem 3.1 being \(\Omega_\pm(x^{1/4}\omega(x))\).

(Mention maybe made of the work of K. S. Gangadharan [Proc. Camb. Philos. Soc. 57, 699–721 (1961; Zbl 0100.03901)] which, in the parts of these two problems not already covered by stronger results of G. M. Hardy, replaces the unspecified \(\omega(x)\) by explicit functions tending to infinity with \(x\).)

The authors note that, although the condition \(A\ge 1\) is not essential for results of the above general character, one cannot have a functional equation with \(A<0\) and the same main features as before, since the explicit formula would imply unlimited differentiability of \(A_\lambda^\rho(x)\) for suitable \(\rho\). As an incidental consequence, the existence of an infinity of complex zeros of \(\zeta(s)\) (for example) is inferred from a consideration of \(1/\zeta(s)\).

It is assumed that a functional equation \(\Delta(s)\varphi(s)= \Delta(\delta - s)\psi( \delta - s)\) is satisfied, where

\[ \Delta(s) = \prod_{\nu=1}^N \Gamma(\alpha_\nu s + \beta_\nu) \qquad (\alpha_\nu> 0,\ A\equiv \sum_{\nu=1}^N \alpha_\nu \ge 1), \]

\(\delta\) is real, \(\varphi(s)\) and \(\psi(s)\) have absolutely convergent Dirichlet expansions \(\sum_{n=1}^N a_n\lambda_n^{-s}\) and \(\sum_{n=1}^N b_n \mu_n^{-s}\) in some half-plane, and the equation is to be understood in the sense that the two sides have a common analytic continuation \(\chi(s)\) in a domain \(\mathcal D\) which is the exterior of a bounded closed set \(S\), such that \(\chi(s)\to 0\) uniformly in every fixed finite interval \(\sigma_1\le\sigma\le\sigma_2\) as \(t\to\pm\infty\) \((s = \sigma+ it)\).

The main object is to obtain \(\Omega_\pm\) and \(O\)-theorems relating to

\[ R_\lambda^\rho(x) = A_\lambda^\rho(x) - Q_\rho(x),\]

where \(\Gamma(\rho+1) A_\lambda^\rho(x)\) is the Riesz sum of order \(\rho\ge 0\) and type \(\{\lambda_n\}\) of the coefficients \(a_n\) and \(Q_\rho(x)\) is a certain function arising from the singularities of \(\varphi(s)\) and \(\Gamma(s)\).

It is proved (Theorem 3.2) that the real (or imaginary) par of \(R_\lambda^\rho(x)\) is \(\Omega_\pm(x^\theta)\), where \(2A\theta = A\delta + \rho(2A -1) - \frac12\), if \(b'_n\ne 0\) for some \(n\), where \(b'_n\) is the real (or imaginary) part of \(b_n\); and (Theorem 3.1) that the conclusion may be strengthened to \(\Omega_\pm(x^\theta \omega(x))\), where \(\omega(x)\to\infty\) with \(x\), if the set \(\{n\}\) of positive integers has a subset \(M= \{n_k\}\) such that (i) no number \(\mu_n^{1/(2A)}\) is representable as a linear combination of numbers \(\mu_n^{1/(2A)}\) \((m\in M)\) with coefficients \(\pm 1\), apart from a trivial representation \(\mu_n^{1/(2A)} = \mu_n^{1/(2A)}\) 4/(2”) if \(n\in M\), and (ii) \(\sum_{m\in M} \vert b'_n\vert n_m^{-\tau} = \infty\), where \(2A\tau = A\delta + \rho + \frac12\)i.

It is also proved that Theorem 3.2 is best possible for large p\(\rho\), in that \(R_\lambda^\rho(x) = O(x^\theta)\) if \(\sum_{n=1} ^\infty \vert b_n\vert \mu_n^{-\tau} < \infty\).

The proofs make use of explicit formulae involving functions \(I(x)\) that have convenient approximations by Bessel and trigonometric functions, combined with a known adaptation to \(\Omega\)-problems of a device used in an analytical proof of Kronecker’s theorem on Diophantine approximation. For smaller \(\rho\) the \(O\)-results are less precise. The authors concentrate on \(R_\lambda^0(x)\), for which they use Landau’s method of taking a \(\rho\)-th difference of \(A_\lambda^\rho(x)\) with a sufficiently large integer \(\rho\). The full result (Theorem 4.1) is rather elaborate, but it includes classical estimates in special cases, such as the errors \(O(x^{1/3}\log x)\) and \(O(x^{1/3})\) in Dirichlet’s divisor problem and the problem of the lattice points of a circle, the corresponding \(\Omega\)-results of Theorem 3.1 being \(\Omega_\pm(x^{1/4}\omega(x))\).

(Mention maybe made of the work of K. S. Gangadharan [Proc. Camb. Philos. Soc. 57, 699–721 (1961; Zbl 0100.03901)] which, in the parts of these two problems not already covered by stronger results of G. M. Hardy, replaces the unspecified \(\omega(x)\) by explicit functions tending to infinity with \(x\).)

The authors note that, although the condition \(A\ge 1\) is not essential for results of the above general character, one cannot have a functional equation with \(A<0\) and the same main features as before, since the explicit formula would imply unlimited differentiability of \(A_\lambda^\rho(x)\) for suitable \(\rho\). As an incidental consequence, the existence of an infinity of complex zeros of \(\zeta(s)\) (for example) is inferred from a consideration of \(1/\zeta(s)\).

Reviewer: A. E. Ingham (M.R. 25 # 3911)