6

Gram points

**6.1 Definition and order of Gram points**

As already mentioned in Chapter 1, the function *θ*(*t*) (see(1.19)) is monotonic increasing for *t* ≥ 7. For *n* ≥ –1, we define the *n*th *Gram point g _{n}* to be the unique solution

or equivalently sin *θ*(*g _{n}*) = 0. Numerical calculations (see, for example, C. B. Haselgrove [Has]) reveal that

etc. The significance of Gram points is that (1.20) shows that

hence *ζ*( + *i**g _{n}*) is real. Therefore ...

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