فرضية ريمان
الجذور على المستقيم الحرج > حسابات عددية

| السنة | عدد الأصفار | عالم الرياضيات |
| --- | --- | --- |
| 1859? | 3 | استعمل برنارد ريمان صيغة ريمان-سيغل  (دالة لم تنشر ولكنها ذُكرت في سيغل 1932). |
| 1903 | 15 | J. P. غرام (1903) استعمل صيغة أويلر-ماكلورين فاكتشف قانون غرام. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he found. |
| 1914 | 79 (γn ≤ 200) | R. J. Backlund (1914) introduced a better method of checking all the zeros up to that point are on the line, by studying the argument S(T) of the zeta function. |
| 1925 | 138 (γn ≤ 300) | J. I. Hutchinson (1925) found the first failure of Gram's law, at the Gram point g126. |
| 1935 | 195 | E. C. Titchmarsh (1935) used the recently rediscovered صيغة ريمان-سيغل , which is much faster than Euler–Maclaurin summation.It takes about O(T3/2+ε) steps to check zeros with imaginary part less than T, while the Euler–Maclaurin method takes about O(T2+ε) steps. |
| 1936 | 1041 | E. C. Titchmarsh (1936) and L. J. Comrie were the last to find zeros by hand. |
| 1953 | 1104 | A. M. Turing (1953) found a more efficient way to check that all zeros up to some point are accounted for by the zeros on the line, by checking that Z has the correct sign at several consecutive Gram points and using the fact that S(T) has average value 0. This requires almost no extra work because the sign of Z at Gram points is already known from finding the zeros, and is still the usual method used. This was the first use of a digital computer to calculate the zeros. |
| 1956 | 15000 | D. H. Lehmer (1956) discovered a few cases where the zeta function has zeros that are "only just" on the line: two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between them. This is called "Lehmer's phenomenon", and first occurs at the zeros with imaginary parts 7005.063 and 7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1. |
| 1956 | 25000 | D. H. Lehmer |
| 1958 | 35337 | N. A. Meller |
| 1966 | 250000 | R. S. Lehman |
| 1968 | 3500000 | Rosser, Yohe & Schoenfeld (1969) stated Rosser's rule (described below). |
| 1977 | 40000000 | ريتشارد بي. برنت |
| 1979 | 81000001 | R. P. Brent |
| 1982 | 200000001 | R. P. Brent, J. van de Lune, H. J. J. te Riele  ‏, D. T. Winter |
| 1983 | 300000001 | J. van de Lune, H. J. J. te Riele |
| 1986 | 1500000001 | van de Lune, te Riele & Winter (1986) gave some statistical data about the zeros and give several graphs of Z at places where it has unusual behavior. |
| 1987 | A few of large (~1012) height | A. M. Odlyzko (1987) computed smaller numbers of zeros of much larger height, around 1012, to high precision to check Montgomery's pair correlation conjecture. |
| 1992 | A few of large (~1020) height | A. M. Odlyzko (1992) computed a 175 million zeroes of heights around 1020 and a few more of heights around 2×1020, and gave an extensive discussion of the results. |
| 1998 | 10000 of large (~1021) height | A. M. Odlyzko (1998) computed some zeros of height about 1021 |
| 2001 | 10000000000 | J. van de Lune (unpublished) |
| 2004 | 900000000000 | S. Wedeniwski (ZetaGrid distributed computing) |
| 2004 | 10000000000000 and a few of large (up to ~1024) heights | X. Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm. They also checked two billion zeros around heights 1013, 1014, ... , 1024. |