# Initial Sums of the Legendre Symbol: Is $$\min + \max \ge 0$$ ?

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

## Abstract

Dirichlet famously proved that for primes p of the form $$4n+3$$, the half-interval $$(0,\frac{1}{2}p)$$ contains more quadratic residues modulo p than nonresidues. An elementary argument then uses this to prove an inequality for an initial sum of the Legendre symbol $$\bigl (\frac{a}{p}\bigr )$$ for any odd prime p, namely $$\sum _{0<a<\frac{1}{2}p}\bigl (\frac{a}{p}\bigr ) \ge 0$$, with strict inequality if and only if $$p\equiv 3\!\pmod 4$$. From computations with the first 25000 primes, Sondow conjectured that

\begin{aligned} \min _{0<k<p} \sum _{a=1}^k \left( \frac{a}{p}\right) + \max _{0<k<p}\sum _{a=1}^k \left( \frac{a}{p}\right) \ge 0, \end{aligned}

also with strict inequality if and only if $$p\equiv 3\!\pmod 4$$. In this note, we prove that equality holds when $$p\equiv 1\!\pmod 4$$, and that if $$3\ne p\equiv 3\!\pmod 4$$ then $$\displaystyle \max _{0<k<p}\!\!$$ $$\sum _{a=1}^k \bigl (\frac{a}{p}\bigr )$$ exceeds the class number $$h(-p)$$. We also give extensions to the Jacobi and Kronecker symbols $$\left( \frac{a}{n}\right)$$.

### Keywords

• Legendre symbol
• Class number
• Jacobi symbol
• Kronecker symbol

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## Acknowledgements

We are grateful to Patrick Gallagher for stimulating discussions on quadratic residues and to Tauno Metsänkylä for information on class numbers.

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Correspondence to Kieren MacMillan .

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