Riemann Zeta Function Zeros
Zeros of the Riemann zeta function 
come in two different types. So-called "trivial zeros" occur at all negative even integers 
, 
, 
, ..., and "nontrivial zeros" occur at certain values of 
satisfying
come in two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" occur at certain values of satisfying |
(1)
|
for 
in the "critical strip" 
. In general, a nontrivial zero of 
is denoted 
, and the 
th nontrivial zero with 
is commonly denoted 
(Brent 1979; Edwards 2001, p. 43), with the corresponding value of 
being called 
.
in the "critical strip" . In general, a nontrivial zero of is denoted , and the th nontrivial zero with is commonly denoted (Brent 1979; Edwards 2001, p. 43), with the corresponding value of being called . |
The plots above show the real and imaginary parts of 
plotted in the complex plane together with the complex modulus of 
. As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of 
lie.
plotted in the complex plane together with the complex modulus of . As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of lie.
The position of the complex zeros can be seen slightly more easily by plotting the contours of zero real (red) and imaginary (blue) parts, as illustrated above. The zeros (indicated as black dots) occur where the curves intersect.
The figures above highlight the zeros in the complex plane by plotting 
(where the zeros are dips) and 
(where the zeros are peaks).
(where the zeros are dips) and (where the zeros are peaks).
The above plot shows 
for 
between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros less than 10, 
, 
, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).
for between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros less than 10, , , ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko). | Sloane |  |
| 1 | A058303 | 14.134725 |
| 2 | 21.022040 | |
| 3 | 25.010858 | |
| 4 | 30.424876 | |
| 5 | 32.935062 | |
| 6 | 37.586178 |
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