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The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact Abelian groups, all the results of commutative harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.

The main object studied in (abstract) harmonic analysis are locally compact groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact Abelian groups, all the results of commutative harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.

The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact groups, all the results on locally compact groups known from harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.

The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact Abelian groups, all the results of commutative harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.

The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact groups, all the results on locally compact groups known from harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that discusses similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question wasn't, but I think that looking into the aforementioned books is the best way to give you more intuition regarding the applications of harmonic analysis in number theory.

The main object studied in harmonic analysis are locally compact (Abelian) groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact groups, all the results on locally compact groups known from harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.