Class numbers and short sums of Kronecker symbols (Q1306697)

Let \(F= \{-3,-4,5,-7, \pm 8,-11, 12, 13,\dots\}\) denote the set of fundamental discriminants \(\neq 1\). For \(D\in F\) we set \(\chi= \chi_D= (\frac{D}{\cdot})\) (the Kronecker symbol). Let \(q_1, q_2\) be rational numbers such that \(0\leq q_1< q_2< 1\) and let \(r\) denote the least common denominator of \(q_1\) and \(q_2\). The authors consider short sums of Kronecker symbols of the form \[ S(D,q_1,q_2)= \sum_{q_1|D|< n< q_2|D|} (\tfrac{D}{n}). \] By a result of \textit{J. Szmidt, J. Urbanowicz} and \textit{B. D. Zagier} [Acta Arith. 71, 273-278 (1995; Zbl 0829.11011)] it is known that if \(D\in F\) and \((D,r)=1\) then \[ S(D,q_1,q_2)= \sum_{\psi\in A} c_\psi B_{1,\chi\psi}, \] where \(A=A(D,q_1,q_2)=\{\)primitive characters space \(\psi\): conductor \((\psi)|r\), \(\psi(-1)= \chi(-1)\}\), \(B_{1,\chi,\psi}\) denotes the generalized first Bernoulli number attached to \(\chi\psi\), and \(c_\psi= c_\psi(D, q_1,q_2)\) can be given explicitly. Set \(C(D,q_1,q_2)= \{\psi\in A(D,q_1,q_2)\mid c_\psi\neq 0\}\). The authors consider the pairs \((q_1,q_2)\in \mathbb{Q}^2\) satisfying \[ 0\leq q_1< q_2< 1, \qquad q_1+ q+2\leq 1,\tag{1} \] for which \(\text{card } C(D_0,q_1,q_2)=0\) for at least one \(D_0\in F\) coprime with \(r\). they show that there are 55 such pairs \((q_1,q_2)\) satisfying the further condition \[ q_2\leq \tfrac 12 \quad\text{if }q_1=0\text{ or }D_0< 0.\tag{2} \] This extra condition is needed only to retain essentially different cases. For each of these 55 pairs it is the case that \(S(d,q_1,q_2)=0\) for all \(D\in F\) coprime with \(r\) in certain arithmetic progressions. For example \((\frac 25, \frac{17}{30})\) is such a pair and \(S(\frac 25, \frac{17}{30})= 0\) for all fundamental discriminants \(D\) such that \(D<0\), \(D\equiv 23,47\pmod {120}\). Similarly the authors consider the pairs \((q_1,q_2)\in \mathbb{Q}\) satisfying (1) for which \(\text{card } C(D_0,q_1,q_2)= 1\) for at least one \(D_0\in F\) coprime with \(r\). They show that there are 116 such pairs \((q_1,q_2)\) satisfying (2). For these pairs it is shown that \[ S(D,q_1,q_2)= \text{constant}\times h(DE), \] where \(E\) is a fundamental discriminant satisfying \(E|r,\chi_E(-1)= -\chi_D(-1)\), so that \((D,E)= 1\), \(DE< 0\), and \(h(DE)\) is the class number of the field \(\mathbb{Q}(\sqrt{DE})\), for all \(D\in F\) coprime with \(r\) in certain arithmetic progressions. For example \((\frac{1}{12}, \frac{1}{10})\) is such a pair and \[ \sum_{\frac{1}{12}|D|< n< \frac{1}{10}|D|} (\tfrac{D}{n})= \tfrac{1}{4} h(12D) \] for all \(D\in F\), \(|D|\equiv 11\) or \(59\pmod {60}\). The evaluation of short sums of Kronecker symbols is a difficult problem and this paper makes an important contribution to the subject.