The de Bruijn–Newman constant, denoted by and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function , where is a real parameter and is a complex variable. More precisely,
where is the super-exponentially decaying function
and is the unique real number with the property that has only real zeros if and only if .
The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of are real, the Riemann hypothesis is equivalent to the conjecture that . [1] Brad Rodgers and Terence Tao proved that , so Riemann's hypothesis is equivalent to . [2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner. [3]
De Bruijn showed in 1950 that has only real zeros if , and moreover, that if has only real zeros for some , also has only real zeros if is replaced by any larger value. [4] Newman proved in 1976 the existence of a constant for which the "if and only if" claim holds; and this then implies that is unique. Newman also conjectured that , [5] which was then proven by Brad Rodgers and Terence Tao in 2018.
De Bruijn's upper bound of was not improved until 2008, when Ki, Kim and Lee proved , making the inequality strict. [6]
In December 2018, the 15th Polymath project improved the bound to . [7] [8] [9] A manuscript of the Polymath work was submitted to arXiv in late April 2019, [10] and was published in the journal Research In the Mathematical Sciences in August 2019. [11]
This bound was further slightly improved in April 2020 by Platt and Trudgian to . [12]
Year | Lower bound on Λ | Authors |
---|---|---|
1987 | −50 [13] | Csordas, G.; Norfolk, T. S.; Varga, R. S. |
1990 | −5 [14] | te Riele, H. J. J. |
1991 | −0.0991 [15] | Csordas, G.; Ruttan, A.; Varga, R. S. |
1993 | −5.895×10−9 [16] | Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. |
2000 | −2.7×10−9 [17] | Odlyzko, A.M. |
2011 | −1.1×10−11 [18] | Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick |
2018 | ≥0 [2] | Rodgers, Brad; Tao, Terence |
Year | Upper bound on Λ | Authors |
---|---|---|
1950 | ≤ 1/2 [4] | de Bruijn, N.G. |
2008 | < 1/2 [6] | Ki, H.; Kim, Y-O.; Lee, J. |
2019 | ≤ 0.22 [7] | Polymath, D.H.J. |
2020 | ≤ 0.2 [12] | Platt, D.; Trudgian, T. |
The de Bruijn–Newman constant, denoted by and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function , where is a real parameter and is a complex variable. More precisely,
where is the super-exponentially decaying function
and is the unique real number with the property that has only real zeros if and only if .
The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of are real, the Riemann hypothesis is equivalent to the conjecture that . [1] Brad Rodgers and Terence Tao proved that , so Riemann's hypothesis is equivalent to . [2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner. [3]
De Bruijn showed in 1950 that has only real zeros if , and moreover, that if has only real zeros for some , also has only real zeros if is replaced by any larger value. [4] Newman proved in 1976 the existence of a constant for which the "if and only if" claim holds; and this then implies that is unique. Newman also conjectured that , [5] which was then proven by Brad Rodgers and Terence Tao in 2018.
De Bruijn's upper bound of was not improved until 2008, when Ki, Kim and Lee proved , making the inequality strict. [6]
In December 2018, the 15th Polymath project improved the bound to . [7] [8] [9] A manuscript of the Polymath work was submitted to arXiv in late April 2019, [10] and was published in the journal Research In the Mathematical Sciences in August 2019. [11]
This bound was further slightly improved in April 2020 by Platt and Trudgian to . [12]
Year | Lower bound on Λ | Authors |
---|---|---|
1987 | −50 [13] | Csordas, G.; Norfolk, T. S.; Varga, R. S. |
1990 | −5 [14] | te Riele, H. J. J. |
1991 | −0.0991 [15] | Csordas, G.; Ruttan, A.; Varga, R. S. |
1993 | −5.895×10−9 [16] | Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. |
2000 | −2.7×10−9 [17] | Odlyzko, A.M. |
2011 | −1.1×10−11 [18] | Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick |
2018 | ≥0 [2] | Rodgers, Brad; Tao, Terence |
Year | Upper bound on Λ | Authors |
---|---|---|
1950 | ≤ 1/2 [4] | de Bruijn, N.G. |
2008 | < 1/2 [6] | Ki, H.; Kim, Y-O.; Lee, J. |
2019 | ≤ 0.22 [7] | Polymath, D.H.J. |
2020 | ≤ 0.2 [12] | Platt, D.; Trudgian, T. |