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I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for all $k\ge 0, x\ge1 $ $$ \sum_{n0$ $$ L(\sigma,\chi)>0. $$ I expect one is meant to use a Mellin transform with Cesàro weighting, S 5.1 in M&V. The difficulty is that $\chi(n)/n$ are the Dirichlet series coefficients of $L(s+1,\chi)$, not $L(s,\chi)$. Thus (5.18) gives $$ L(\sigma+1,\chi)>0 $$ for all $\sigma>0$.

Am I missing something obvious? Alternately, the method of part a will show that $$ \sum_{n0$ for all quadratic $\chi$.

EDIT: The reason one cares which version of part a is used, is that the numerics for small $k$ and moderate $x$ indicate that positivity is at least plausible for the original part a. It is not for the revised version.

Of course, one expects that no such $k$ exists in either case. For $k=0$, the positivity of $\sum_{nnegativity of $\sum_{n

  • 0
    Maybe you should write to the authors.2011-09-10
  • 2
    I did, no reply (yet).2011-09-10

0 Answers 0