#### IMSc Webinar

#### Local distribution of values of Euler's totient function

#### J.-M. Deshouillers

##### University of Bordeaux, France

*In this joint work with Pramod Eyyunni and Sanoli Gun, we pursue the study initiated by P. E., M. K. Das and B. R. Patil on the study of N(x, H) which is the number of values of Euler’s totient function in the interval (x, x+H].*

It is a standard result that the set of values of Euler’s function has a zero asymptotic density, which implies that N(x, H) is usually o(H). However, K. Ford, S. Konyagin and C. Pomerance have shown that N(x, H) is at most H(1/4 + o(1)) as H tends to infinity, uniformly in x. The question has been raised to know whether the constant can be replaced by 0. We give some support to the conjecture that the constant 1/4 in the above mentioned result is best possible by showing that this is the case if one takes for granted a standard conjecture of L. Dickson that suitable linear functions may simultaneously take prime values.

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