Exceptional characters and prime numbers in short intervals
In the second of their series of papers concerning the consequences for the distribution of primes which follow from the assumption of the existence of exceptional Dirichlet characters \(\chi\pmod D\), the authors deal with the problem of the distribution of primes in short intervals \(x-y<p\leq x\). The Riemann Hypothesis implies that \(\psi(x)-\psi(x-y)=y+O(x^{{1\over2}+\varepsilon})\), which has content for \(x^{{1\over2}+\varepsilon}<y<x\), and their goal is to reach a fixed power below \(\sqrt x\). The following is their main theorem: Let \(\chi\) be the real primitive character of conductor \(D\). Let \(x\geq D^r\), with \(r=18290\), and \(x^{39\over79}\leq y\leq x\). Then \[ \psi(x)-\psi(x-y)=y\{1+O(L(1,\chi)(\log x)^{r^r})\}, \] where \(L(s,\chi)\) is the Dirichlet \(L\)-function, and the implied constant is absolute and computable. Although the result here is unconditional, the asymptotic formula has meaning only when \(L(1,\chi)\) is exceptionally small, so that \(\chi\) is exceptional; thus, for an error term \(O(1/\log x)\) the requirement is \(L(1,\chi)\ll(\log x)^{-{r^r}-1}\). Moreover, because of the restriction on~\(D\) relative to~\(x\), the result does not hold for every interval, although there will be such intervals if \(L(1,\chi)\) is really that small. The proof follows the lines introduced in their first paper [Int. Math. Res. Not. 2003, No. 37, 2033--2050 (2003; Zbl 1038.11060)]. The method involves the arithmetic function \(\rho=\chi\star1\star1\); in particular the proof requires an asymptotic formula for the average value for \(\rho(n)\). The authors show that, in the presence of exceptional characters, it suffices to deal with some twist of the divisor function \(\tau_2(n)\) to obtain the main term, and a similar twist of \(\tau_3(n)\) to estimate the error term. Thus one may say that the possibility for an argument which leads to a result with a fixed power below \(\sqrt x\) follows from the fact that the error term associated with the asymptotic formula for the average value for \(\tau_3(n)\) has an exponent less than \({1\over2}\).
scientific article; zbMATH DE number 2103785