## Functional equations with multiple gamma factors and the average order of arithmetical functions.(English)Zbl 0211.37901

This 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 $$1/\zeta(s)$$.

### MSC:

 11N37 Asymptotic results on arithmetic functions 11M41 Other Dirichlet series and zeta functions

### Citations:

Zbl 0107.03702; Zbl 0101.03703; Zbl 0093.05203; Zbl 0100.03901
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