mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 and its analytic c ...

. 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. He was highly influential in the development of theoretical computer ...

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