Analytic Number Theory/Useful summation formulas

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Analytic number theory is so abysmally complex that we need a basic toolkit of summation formulas first in order to prove some of the most basic theorems of the theory.

Abel's summation formula[edit | edit source]

Theorem 1.1 (Abel's summation formula, also called Abel's identity):

Let be a sequence and let be a differentiable function such that is Riemann integrable. If we define


then we have


Note: We need the Riemann integrability to be able to apply the fundamental theorem of calculus.

Proof 1:

We prove the theorem by induction on .

1. :

First, we have in this case


Then, we have

by the fundamental theorem of calculus.

2. Induction step:

Define . We have

by the induction hypothesis. Further,


Putting things together, we obtain

and thus the desired formula.

The method of proof we applied here was using induction and then trying to express the terms from the induction hypothesis in terms of the terms from the desired formula.

Proof 2:

We prove the theorem by direct manipulation of the term on the left.

Define .

Proof 3:

We prove the formula by the means of the Riemann-Stieltjes integral. Indeed, by integration by parts, we have


Corollary 1.2:


Proof 1:

We deduce the formula from integration by parts for the Riemann-Stieltjes integral.

Proof 2:

We directly manipulate the LHS (left hand side).

Define and .

Two further proofs are given in exercises 1.1.1 and 1.1.5.

We note that induction and direct manipulation are quicker proofs for theorem 1.1, while corollary 1.2 is quicker proven from theorem 1.1 or Riemann-Stieltjes integration.

Exercises[edit | edit source]

  • Exercise 1.1.1: Prove corollary 1.2 from theorem 1.1. Hint: .
  • Exercise 1.1.2: Compute . Hint: Use , , apply Abelian summation and split the resulting integral into pieces where is constant. Then apply a similar process.
  • Exercise 1.1.3: Prove that the limit exists. This limit is called the Euler–Mascheroni constant. Hint: Use and .
  • Exercise 1.1.4: Prove theorem 1.1 from corollary 1.2.
  • Exercise 1.1.5: Prove corollary 1.2 using induction on .

Euler's summation formula[edit | edit source]

Definition 1.3:

For , we define


Theorem 1.4 (Euler's summation formula):

Let be a differentiable function, such that is Riemann integrable. Then



We prove the theorem from Corollary 1.2, setting and using integration by parts (integration by parts is proven using the fundamental theorem of calculus).



where in the last line we used integration by parts on the integral .

Corollary 1.5:

Exercises[edit | edit source]

  1. Prove corollary 1.5.

Euler–Maclaurin formula[edit | edit source]

Theorem 1.6 (Euler–Maclaurin formula):

Define the functions and . Then for any twice continuously differentiable function such that is Riemann integrable, we have


Proof 1:

We prove the theorem by direct computation.

Proof 2:

We prove the theorem from Euler's summation formula.