mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Turing's method is used to verify that for any given Gram point there lie ''m'' + 1 zeros of , in the region , where is the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

. It was discovered by

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical com ...

and published in 1953, although that proof contained errors and a correction was published in 1970 by R. Sherman Lehman. For every integer ''i'' with we find a list of Gram points

\

and a complementary list

\

, where is the smallest number such that :

(-1)^i Z(g_i + h_i) > 0,

where ''Z''(''t'') is the Hardy Z function. Note that may be negative or zero. Assuming that

h_m = 0

and there exists some integer ''k'' such that

h_k = 0

, then if :

1 + \frac < 2,

and :

-1 - \frac > -2,

Then the bound is achieved and we have that there are exactly ''m'' + 1 zeros of , in the region .

References

{{reflist 1953 introductions 1953 in science Alan Turing Zeta and L-functions