On the fourth power mean of a sum analogues to character sums over short intervals (Q2430154)

The paper under review deals with the sum \(\sum_{1\leq n\leq N}(-1)^n\chi(n)\), where \(\chi\) denotes a Dirichlet character modulo \(q\geq 2\). The first theorem gives the following asymptotic formula \[ \sum_{\chi\bmod q}\Bigl|\sum_{n\leq N}(-1)^nn^k\chi(n)\Bigr|^4=\frac{80\ln N-50\ln 2}{(2k+1)^3n^2}\frac{\varphi^4(q)}{q^3}N^{4k+2}\prod_{p|q}\frac{p}{p+1}+O\left(\varphi(q)N^{4k+2}2^{\omega(q)}\right) \] for \(k>0\), an odd \(q\geq 3\) and \(1<N<\sqrt{q}\); here \(\omega(q)\) denotes the number of prime divisors of \(q\). As the second main result the author proves an asymptotic formula for the sum \[ \sum_{\chi\bmod q}|K(r,s,\chi;q)|^2\,\Bigl|\sum_{n\leq N}(-1)^nn^k\chi(n)\Bigr|^4, \] where \(q\geq 3\) is odd, \(N\leq q^{1/6}\), \((rs,q)=1\), and \(K(r,s,\chi;q)\) denotes the general Kloosterman sum \(\sum_{a\leq q}\chi(a)\exp\left(2\pi i\frac{ra+s\overline{a}}{q}\right)\); here \(\overline{a}\) is such that \(a\overline{a}\equiv 1\pmod q\).