##
**Early study in which Leonhard Euler made use of the mathematical analysis to the research of number theory.**
*(Chinese.
English summary)*
Zbl 1174.01309

Summary: This paper presents a brief introduction to Euler’s study. There are many exciting formulas and theorems such as Euler’s infinite summation formula about the reciprocal sum with powers of natural numbers and Euler’s infinite product representation. Another example is Euler’s thinking and proving about the proof of Fermat’s four-square theorem, yielding the arithmetical function, the partition function, and prime ideal. These conceptions, theorems, and formulas are all first discovered accurately by Euler’s demonstrations. Euler is extraordinary at converting a problem of number theory into mathematical analysis. In fact, Euler’s ideas have become more generalized. These facts are enough to prove that he has extensive and deep knowledge of his subject. Finally, we quote a few famous examples of power series, and the law of quadratic reciprocity which Euler found. They are all part of our precious legacy in the library of number theory from Euler.