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 right-wing conservative candidate Keiko Fujimori denounced there had been irregularities in favor of Pedro Castillo1, and even claimed that the elections had to be held again2 Nevertheless, Keiko Fujimori recognized the victory of her opponent on July 26^th , and finally the ONPE declared as winner to Pedro Castillo.

For this reason, we performed a study based on statistical techniques according to the Benford’s Law3 to determine if there actually was any manipulation in results. This methodology has been employed in some many studies34567. Moreover, this law has even been used in presidential elections as explained by Mebane8 and Deckert et al9 as explained in the next section.

The forensic analysis was performed with the same computational methodology employed both in the Covid-19 registered cases study6, and in the recent US elections held in 20207. This methodology is based in the Benford’s Law, and we determine three critical values.

The p-value(χ²), p-value(Man) and p-value(FW), refer collectively to as p-values, were calculated in the following way67. The first value [ p-value(χ²) ] comes from the occurrence frequency of first digit of the data (left to right), which is based on the Benford’s Law of the first digit, and the probability should follow the following mathematical relation10:

Where i goes from 1 to 9 (excluding zero). With these probabilities, the Pearson value (χ2) is calculated to detect discrepancies in data, which is given by:

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 p-value(χ²) which indicates us whenever data is correct, as long as it is greater than or equal to 0.0567.

The next value, p-value(Man), employs the Mantissa Arc test, and to do so we must find the mass center of data according to the following mathematical relation10:

where the x_i are the votes to validate, and N is the total number of them. We then calculated the L² term given by:

where L² should be almost zero, it means, while greater than zero, it is possible to suspect a manipulation of the elections.

So the p-value(Man) equals to:

Finally, the p-value (FW) is known as the Freedman-Watson test (FW), designed to compare discrete distributions based on the following mathematical relation11:

However, it is recommended to check Freedman’s original paper11to know every mathematical detail of this last equation.

So, there would be no suspicion of manipulation of the election, if any of the p-values 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 fraud67.

Finally, we want to validate this methodology according to the total number of voters inscribed in twenty five Peru states according the ONPE data, i.e. in the states of Amazonas, Ancash, Apurimac, Arequipa, Ayacucho, Cajamarca, Callao, Cusco, Huancavelica, Huanuco, Ica, Junin, La Libertad, Lambayeque, Lima, Loreto, Madre de Dios, Moquegua, Pasco, Piura, Puno, San Martin, Tacna, Tumbes, and Ucayal. Finally we analyzed the second round election results grouped by political party.

In Table 1 we present all the data collected for the study according to the ONPE results. In this table we can observe that Pedro Castillo loss in six of twenty five states (ie., Callao, La Libertad, Lambayeque, Lima, Piura, and Tumbes). In the Loreto state, for instance, Castillo won only for 5,350 votes (which represent about 1.46% of the electoral roll), and in the Ica state while the votes difference was a little higher (5626 votes ahead for Castillo), this difference is only 1.09% of the electoral register in this last state. In contrast, there were some states where the difference was way higher, such as Puno: where Castillo won over with more than 580 thousand votes (a difference of almost 75% of the electoral roll).The most favorable state for Fujimori was Lima, where she outnumbered Castillo by almost two million votes, which only represent 27,67% of difference in between them.

We show the p-values for the total of all registered votes in the electoral roll of Peru in the Table 2, and also the analysis of the votes obtained by Pedro Castillo and Keiko Fujimori, respectively. As we can see in this table, there’s no evidence of data manipulation as the all surpass the threshold value of 0.05. Furthermore, the L²value must be close to zero and there are also similarities between all of them (see Table 2)

This paper determined a forensic analysis of the Peru’s presidential elections on June 6, 2021. We determined three p-values 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.