The authors have declared that no competing interests exist.
This paper performs a forensic study of the Peru’s presidential election on June 6^{th}, 2021 between Pedro Castillo and Keiko Fujimori, where excandidate Keiko Fujimori claimed there had been irregularities. We calculate three pvalues that help us determine if there was fraud. The consensus of the results indicates that there was no manipulation of the results.
On April 11^{th}, 2021 the first round presidential election was held with 24 political parties, where it was projected as potential winners the Peru Libre candidate Pedro Castillo and the Fuerza Popular candidate Keiko Fujimori with 2,724,752 and 1,930,762 votes, respectively. The second round was held on June 6^{th}, 2021, where Pedro Castillo was chosen as winner with 8,835,579 votes against the 8,791,521 obtained by Fujimori. As a matter of fact, the National Office of Electoral Processes (known as ONPE, by Spanish initials) declared that there was 17,620,000 valid votes, 121,477 blank votes and 25.43% of abstention (details in ONPE, https://www.resultadossep.eleccionesgenerales2021.pe/SEP2021/).
However, on June 8^{th} the rightwing conservative candidate Keiko Fujimori denounced there had been irregularities in favor of Pedro Castillo
For this reason, we performed a study based on statistical techniques according to the Benford’s Law
The forensic analysis was performed with the same computational methodology employed both in the Covid19 registered cases study
The pvalue(χ^{2}), pvalue(Man) and pvalue(FW), refer collectively to as pvalues, were calculated in the following way
Where i goes from 1 to 9 (excluding zero). With these probabilities, the Pearson value (χ
where P(k) and b(k) are the distributions obtained from the votes and the expected from Benford’s Law, respectively. Thanks to this value, it was possible to determine the pvalue(χ^{2}) which indicates us whenever data is correct, as long as it is greater than or equal to 0.05
The next value, pvalue(Man), employs the Mantissa Arc test, and to do so we must find the mass center of data according to the following mathematical relation
where the x_{i} are the votes to validate, and N is the total number of them. We then calculated the L^{2} term given by:
where L^{2} should be almost zero, it means, while greater than zero, it is possible to suspect a manipulation of the elections.
So the pvalue(Man) equals to:
Finally, the pvalue (FW) is known as the FreedmanWatson test (FW), designed to compare discrete distributions based on the following mathematical relation
However, it is recommended to check Freedman’s original paper
So, there would be no suspicion of manipulation of the election, if any of the pvalues is greater than or equal to 0.05. Nevertheless, if all three values are less than 0.05, it is a sign of inconsistency or fraud
Finally, we want to validate this methodology according to the total number of voters inscribed in twenty five Peru states according the ONPE data,
In
We show the pvalues for the total of all registered votes in the electoral roll of Peru in the




Amazonas  161,890  98,716  52,913 
Ancash  597,055  314,394  233,325 
Apurimac  209,560  160,943  36,737 
Arequipa  900,759  549,681  299,759 
Ayacucho  287,140  223,383  49,130 
Cajamarca  690,285  456,128  190,041 
Callao  642,766  195,098  403,813 
Cusco  718,117  561,406  116,299 
Huancavelica  174,567  139,498  26,243 
Huanuco  367,857  229,059  114,648 
Ica  515,652  231,546  225,920 
Junin  693,301  377,083  271,117 
La Libertad  1,022,886  376,424  570,558 
Lambayeque  711,954  274,662  387,053 
Lima  6,418,172  2,127,809  3,903,451 
Loreto  366,268  176,864  171,514 
Madre de Dios  76,770  50,244  20,533 
Moquegua  114,448  78,009  28,926 
Pasco  130,700  80,358  42,140 
Piura  996,743  363,786  560,618 
Puno  733,093  624,592  76,280 
San Martin  430,319  222,029  177,108 
Tacna  219,577  150,672  57,187 
Tumbes  131,348  41,464  80,064 
Ucayali  258,435  115,356  126,116 




pValor(χ2)  0,19  0,97  0,53 
pValor(Man)  0,21  0,44  0,22 
L 
0,06  0,03  0,06 
pValor(FW)  0,19  0,88  0,44 
This paper determined a forensic analysis of the Peru’s presidential elections on June 6, 2021. We determined three pvalues that can help us determine if there have been manipulations of the results. The results indicated that there is no fraud. Moreover, we show how valid is the methodology when we analyzed the electoral register in twenty five states, and therefore, we concluded that there was no fraud in Peru election.