**August 4 ^{th}, 2017**

**Despacito will reach 4 billion views this year and… **

In the first part of this article, published on July 11^{th}, 2017, using basic mathematical notions we predicted that Despacito would reach 3 billion views after August 2^{nd}, with a strong probability on the threshold of the second week. Officially it happened on August 4^{th} in America and on August 5^{th} in Europe and Asia. The route taken was as follow (UTC +2):

- Surpassed Uptown funk – Bruno Mars, on July 14
^{th}, 2017 at 7:30 PM. - Surpassed Sorry – Justin Bieber, on July 19
^{th}, 2017 at 11:30 AM. - Surpassed Gangnam style – PSY, on July 31
^{st}at 5:00 PM. - Surpassed See you again – Wiz Khalifa on August 4
^{th}at 7:30 PM. - Reached 3 billion views on August 5
^{th}at 00:30 AM.

In this second part, we are going to answer, when will it reach 4 billion views and be considered the only song achieving this number? The answer, once again, lies in mathematics…

Despacito started growing at 6 million views per day in January, increased its rate to 16 million views per day in May, and reached 21 million views per day in July. Will it reach higher values? No, it will not, Despacito will decay reaching a stable level in January 2018 (6 months after reaching its highest peak, which was in July), this is likely to happen because other songs have shown this outcome. What would this stable value be? Probably around 3 million views per day. Why? Because it has been the stable value of many iconic songs. Here are some examples:

Song / Artist | Published in Youtube (year/month) | Number of Youtube views per day in 2017/08 (millions) |

Bailando (Enrique Iglesias) |
2014/04 | 2 |

Uptown funk (Bruno Mars) | 2014 /11 | 2 |

Sorry (Justin Bieber) |
2015 /10 | 2 |

See you again (Wiz Khalifa) |
2015/04 | 3.5 |

Chantaje (Shakira) |
2016/11 | 4.5 |

This data wil be enough to make a rough estimation of the future of the song using Integral Calculus (developed by Isaac Newton and Gottfried Leibniz in the XVII century). Probably in that time, these geniuses had to face a similar problem and that’s how they developed Calculus, so let’s go back in time and see what they would have done…

First, Newton would have locked himself in his room, he was a shy person and preferred solitude. Then, he would have drawn a graph similar to the following:

Showing that in 6 months (from August 2017 to January 2018) Despacito would fall linearly from 21 million to 3 million views per day (see blue line in Figure 1). With this, can we derive when Despacito will reach 4 billion views? It looks a bit cumbersome, because of this, Newton and Leibniz developed a trick… Instead of studying the line, they traced rectangles for each month as shown in Figure 1. Why? Because by doing so, things become much simpler, we now can assume that in August Despacito would have a constant value of 18 million views per day, in September of 15 million views per day, in October 12, in November 9, in December 6, and in January 3; hence, considering that each month has 30 days we can generate the following table:

Month | Total number of Youtube views each month (millions) |

August | 18 x 30 = 540 |

September | 15 x 30 = 450 |

October | 12 x 30 = 360 |

November | 9 x 30 = 270 |

December | 6 x 30 = 180 |

January | 3 x 30 = 90 |

Total | 1890 |

From the table we can see that:

- Despacito reaches 4 billion views on October 5
^{th}, 2017. - Despacito reaches 5 billion views in March 2018 (if it keeps a stable value after January).

The curious thing about the trick proposed by Newton and Leibniz is that the area of each rectangle gives the total number of views in the time interval of the rectangle (30 days for the example given in the table). “Amazing!” -would have said Newton while pulling his hair in his little dorm in Woolsthorpe. Leibniz would have done the same in Paris.

Both were fascinated with the concept of the infinitesimal, to the point that they were immersed in philosophical debates regarding the so called monads. Because of this, both reached a point where they questioned themselves: “what if we draw more rectangles and make them thinner, infinitesimally thinner so that they would exactly match the area under the curve (the blue line in our case).” This simple thought gave a marvelous conclusion: “The area under the curve, no matter how complicated is this curve, gives us the total number views.”

To visualize this thought, let’s consider Figure 2, where we show the actual evolution of the number of views of Despacito from January to August, 2017. As you can see, the behavior is not linear, but Newton and Leibniz showed us that in order to get the total number of views, all we need is to find the area under the curve (the area under the green line in this case). By drawing thin rectangles in the month of March (as shown in Fig. 2) and summing up their areas we can obtain approximately the total number of views in that month. The value will be exact if these rectangles are infinitesimally thin, so thin that when they are drawn they describe exactly the area under the curve. This conclusion allowed them to develop Integral Calculus, which in simple words means: “To find the total area under any curve we just need to sum up the area of infinitesimally thin rectangles.” For this it is necessary to know the equation of our curve. In Fig. 1 is a line, its equation was mentioned in the first part of this article. In Fig. 2 it is more complex but not impossible to obtain. In this case we consider Fig. 1, its mathematical equation takes the following form:

On the left hand side we have the equation of our line (the blue line), where t represents the time expressed in days. The symbol in the form of an elongated *S* tells us that we have to add infinitesimally thin rectangles starting on day *0* and culminating on day *d*, date for which we will accumulate 1 billion views. Solving the equation we have:

- Despacito reaches 4 billion views on September 30
^{th}, 2017. - Despacito reaches 5 billion views on January 1
^{st}, 2018.

How accurate are these results?

With time we will know, Despacito may decay faster (or slower) than a linear slope, in fact, it will be a chaotic decay, as seen from Fig. 2, we are just making a rough approximation, but everything seems to indicate that in October Despacito will reach a new record. What about 5 billion views? Most likely, Despacito will grasp this number between January and April, 2018. The imminent question is: Is any song capable of beating this record? It is very difficult at the moment. The great advantage of new songs coming by is that the global connection to the internet is increasing; however, if the world population is meant to decrease for any reason in the coming years, most likely Despacito will remain unbeaten for a long long time.

Christian Ortiz Ph.D.

SPINTEC laboratory, CEA Grenoble