RIEMANN ZETA FUNCTION across the FULL COMPLEX PLANE.
Try it out immediately on: WWW.ZETA-CALCULATOR.COM
How to use the vanilla_zeta() function - Guide:
Note: I take no responsibility for the correctness of the results! As one should understand, calculating zeta values and verifying them is a challenging task. Therefore, any output from vanilla_zeta() should be considered as a value recommendation, and it may differ significantly from the actual results.
Execute the vanilla_zeta function / Step 1: Declare Inputs.
The complex part of the Riemann Zeta function is defined by its real and imaginary part.
Real part: -Number.MAX_VALUE <= input_Real <= Number.MAX_VALUE
var input_Real = 0.5;
Imaginary part: -Number.MAX_VALUE <= input_Imag <= Number.MAX_VALUE
var input_Imag = 14.134725141734693;
You must choose the amount of computational effort you want to invest, where 0 represents minimal effort and 1 represents maximum effort.
Naturally, this will affect the calculation time.
Effort level: 0 <= input_Effort <= 1
var input_Effort = 0.67;
The input for a special feature called 'Trust'.
If the feature is enabled, vanilla_zeta() performs a test that verifies whether the calculation has converged, which is essential for ensuring the correctness of the result.
'Trust' (verify result) enabled: true, false
var input_TrustEnabled = true;
If 'Trust' is disabled (false), the output listed in [4] will always be '-1', indicating that it is not enabled.
If 'Trust' is enabled, you will get a value from '0' up to '0.999999999999999'. The meaning of this value will be explained shortly below.
Be aware that the result verification 'Trust' requires a lot of computational power! If the output[4] is not relevant, set the feature to 'false' to save time.
Execute the vanilla_zeta function / Step 2: Execute.
var output_array = vanilla_zeta(input_Real, input_Imag, input_Effort, input_TrustEnabled);
Interpreting the vanilla_zeta Output 'output_array' / Step 3: Read Output.
Output Descriptions:
output_array[0] - Real result of Zeta
output_array[1] - Imaginary result of Zeta
output_array[2] - Polar radius result of Zeta
output_array[3] - Polar angle result of Zeta in radians
output_array[4] - Trust value of the calculation
output_array[5] - Boolean indicating if the calculation was successful
output_array[6] - Process time of 'vanilla_zeta()' in milliseconds
Outputs [0] to [3]:
The interpretation is self-explanatory.
With the given input, the first nontrivial zero of the zeta function should approximately be obtained, i.e., = 0 + 0i.
Output[5]:
If nothing went wrong during the calculation, output[5] is 'true'.
If there were issues during the calculation, output[5] is 'false', and output[0] will be marked with '#err' or '#ins'. '#err' typically means there were overflows or underflows. '#ins' means the inputs were not suitable. Check if the types and tolerances conform to the requirements!
For automated processing, it makes sense to check this indicator first after the calculation to avoid passing inaccurate results further along. This primarily serves to prevent disruptions.
Output[6]:
This is self-explanatory. Be aware that only a few calculations may not reflect real-time performance. If only a few calculations are performed, your computer may remain in standby mode, causing the time to be much higher. Perform many calculations to get the time of a fully active CPU!
Additionally, the times can vary significantly depending on the region (real-imaginary position)!
Output[4] - Result Verification / Trust:
Possible results are '-1' if disabled, '0' (zero), or a value greater than '0' up to the maximum of '0.999999999999999'.
The meaning of '=0' (exact zero) is that no convergence has been registered. Assume that the result is incorrect in this case.
Now if the value is greater than '0' and at most '0.999999999999999', convergence has happened and the result should be correct. To obtain better values, always choose a high 'input_Effort'.
The exact interpretation of the value is simple - the closer it approaches '1', the better the convergence, which often indicates more accurate results.
ATTENTION to Output[4] - Result Verification / Trust:
This feature provides good assistance in getting a feel for the quality of the results – however, the statement about the correctness of the results may be incorrect! To gain additional confidence, it makes sense to use other algorithms for deriving the zeta values and make comparisons.