Inverse problems for partition functions (Q2762715)

Given some nonnegative weight-function \(w(n)\), the weighted partition function \(p_w(n)\) is defined by the identity NEWLINE\[NEWLINE \sum_0^\infty p_w(n) \cdot x^n = \prod_{m=1}^\infty (1-x^m)^{-w(m)}. NEWLINE\]NEWLINE Starting from papers of \textit{P. Erdős} [Ann. Math. (2) 43, 437-450 (1942; Zbl 0061.07905)], \textit{E. E. Kohlbecker} [Trans. Am. Math. Soc. 88, 346-365 (1958; Zbl 0173.04203)], \textit{G. A. Freiman} [Izv. Akad. SSSR, Ser. Mat. 19, 275-284 (1955; Zbl 0065.02906)], Meinardus, Geluk, and others, the author is interested in the ``inverse'' problem of deducing asymptotic results for NEWLINE\[NEWLINE N_w(u) = \sum_{n\leq u} w(n) NEWLINE\]NEWLINE [with good remainder terms] from knowledge about the asymptotic behaviour of NEWLINE\[NEWLINE P_w(u) = \sum_{n\leq u} p_w(n),NEWLINE\]NEWLINE with remainder terms. Generalizing a result of E. E. Kohlbecker, the author proves: NEWLINENEWLINENEWLINELet \(\lambda(u)\) be positive, differentiable, \(\lambda(u) \nearrow \infty\), \(u^{-a} \lambda(u)\log u \searrow 0\) for some \(a\), \(0<a<1\), as \(u\to\infty\). Assume that NEWLINE\[NEWLINE \log P_w(u) = A u^\alpha \left\{ 1 + {\mathcal O}\left({1\over {\lambda(u)}} \right)\right\},NEWLINE\]NEWLINE then NEWLINE\[NEWLINE N_w(u) = B u^b \left\{ 1 + {\mathcal O}\left({1\over {\log \lambda(u)}} \right)\right\},NEWLINE\]NEWLINE where \(b={a\over{1-a}}\), and \(B\) is explicitly given in terms of \(A, a\). NEWLINENEWLINENEWLINEThis result is a special case of a more general result, formulated in terms of an ``admissible'' pair \((\Phi(u), \Phi^\ast(u))\) of functions [we omit the complicated definition of this notion]. The asymptotic formula NEWLINE\[NEWLINE \log P_w(u) = \Phi(u) \left\{ 1 + {\mathcal O}\left({1\over {\lambda(u)}} \right)\right\}NEWLINE\]NEWLINE implies NEWLINE\[NEWLINE N_w(u) = \Phi^\ast(u) \left\{ 1 + {\mathcal O}\left({1\over {\log \lambda(u)}} \right)\right\}.NEWLINE\]NEWLINE Next, the author is interested in better remainder terms in the formula for \(N_w(u)\), under stronger assumptions. NEWLINENEWLINENEWLINEAssume that \(\log P_w(u) = A u^\alpha + {\mathcal O}(u^{a_1})\), where \(A>0, 0<a_1<a<1\), and that the Dirichlet series NEWLINE\[NEWLINE f_w(s) := \sum_1^\infty {{w(n)}\over {n^s}} \quad\text{equals}\quad {1\over{\Gamma(s)\zeta(1+s)}}\cdot \left\{ {D\over{s-b}} + h_w(s)\right\}, NEWLINE\]NEWLINE where \( b = {a\over{1-a}}\) and \( D= A^{1/(1-a)} a^{a/(1-a)} (1-a)\), and where \(h_w(s)\) is analytic in \(\displaystyle \Re(s)> {{a_1}\over{(1-a)}}\). Suppose, furthermore, that \(w(n) = {\mathcal O}_\varepsilon( n^{a_1/(1-a)+\varepsilon})\), and \(|f_w(\sigma + it)|= {\mathcal O}_{\delta,\varepsilon} (t^\varepsilon)\) uniformly in \(\displaystyle\sigma \geq {{a_1}\over{(1-a)}}+\delta\). Then NEWLINE\[NEWLINE N_w(u) = B u^b + {\mathcal O}(u^{b_1 + \varepsilon}), NEWLINE\]NEWLINE where \(b_1 = a_1/(1-a)\), \(b = a/(1-a)\). NEWLINENEWLINENEWLINEAs a corollary, the author obtains a characterization of Riemann's hypothesis by asymptotic properties of partitions, using the weights NEWLINE\[NEWLINEw_{k,\ell}(n) = \Lambda(n), \quad\text{if } n\equiv \ell \bmod k, \quad\text{otherwise } = 0.NEWLINE\]NEWLINE Fix some \(\theta\), \({1\over 2} \leq \theta < 1\). Then all the Dirichlet \(L\)-functions \(L(s,\chi)\) [for any character \(\chi\) modulo \(k\)] are nonzero in the half-plane \(\{s\); \(\Re(s) > \theta\}\) if and only if for any \(\ell\), \((k,\ell) = 1\) and any \(\varepsilon > 0\) NEWLINE\[NEWLINE \log P_{w_{k,\ell}} = 2 \sqrt{ {\zeta(2)\over{\varphi(k)}} } \cdot u^{{1\over 2}} + {\mathcal O}_{\varepsilon, k, \ell} \left( u^{{\theta\over 2}+ \varepsilon}\right).NEWLINE\]