## On the average value of divisor sums in arithmetic progressions

##### Abstract

We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both
additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in particular, we obtain
upper bounds for such twisted sums.