The image contains a mathematical explanation about summation formulas involving exponential functions and modular arithmetic. Let us analyze the content and explain step by step:

Key Points from the Image

1. The Formula for $S_{a,p}$: $S_{a,p} = 1 + 2 \sum_r e^{2\pi i \frac{ar}{p}}$ This is a summation involving exponential terms that depend on the variable $r$, a parameter $a$, and a modulus $p$.

2. Manipulation of the Formula: Subtracting $S_{a,p}$ from another form gives: $0 = 1 + \sum_r e^{2\pi i \frac{ar}{p}} + \sum_n e^{2\pi i \frac{an}{p}},$ which is used to prove certain properties of the summation.

3. Squared Magnitude of $S_{a,m}$: The squared magnitude $|S_{a,m}|^2$ is calculated as: $|S_{a,m}|^2 = \sum_{t=0}^{m-1} \sum_{x=0}^{m-1} e^{2\pi i \frac{a(t^2 + 2tx)}{m}}.$ This formula examines the properties of summation by separating $t$ and $x$.

4. Case Analysis:

   • When $m$ is even ($m = 2m_1$): $|S_{a,m}|^2 = m \left( e^{2\pi i \frac{a \cdot 0^2}{m}} + e^{2\pi i \frac{a \cdot m_1^2}{m}} \right).$
     • The second exponential term determines whether the sum is zero or $2m$.
   • When $m$ is odd, the sum simplifies differently.

5. Generalization for Any Integer $b$: Another expression for $|S_{a,m}|$ is given: $|S_{a,m}| = \left| \sum_{x=0}^{m-1} e^{2\pi i \frac{Ax^2 + 2Abx}{m}} \right|,$ where $A$ and $b$ are chosen such that $2Ab \equiv a \mod m$.

Explanation of Summation Cases

Case 1: When $m$ is Even

• The exponential terms simplify based on whether $m_1$ is odd or even.
• If $m_1$ is odd, certain symmetry properties of the exponential function cause the sum to vanish.

Case 2: When $m$ is Odd

• The properties of $t^2$ modulo $m$ and how $x$ contributes to the sum are critical. The simplifications are different due to lack of the symmetry present in even $m$.

Case 3: General Expression

• The expression involving $b$ is highly flexible and allows control over how $a$ contributes to the result by solving $2Ab \equiv a \pmod{m}$.

Application

The results are often used in number theory and modular arithmetic, particularly in sums over roots of unity and Gaussian sums. These sums have connections to quadratic residues and character sums.

Would you like me to expand further on:

• Derivations of specific cases,
• Properties of the summation formula, or
• Applications in number theory?

Here are 5 related questions to explore:

1. How do we prove that $S_{a,m}$ vanishes when $m$ is odd?
2. What is the significance of the parameter $A$ in the generalized summation formula?
3. How does the symmetry of exponential functions simplify these sums for even $m$?
4. What is the connection between these sums and quadratic residues?
5. Can this method be used to compute other sums involving modular exponentials?

Tip: Modular arithmetic often simplifies complex sums by exploiting periodicity and symmetry properties of exponential functions.