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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

  • Quadratic residue
  • Legendre symbol
  • Class number
  • Jacobi symbol
  • Kronecker symbol

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References

  1. B.C. Berndt, Classical theorems on quadratic residues. Enseign. Math. 2(22), 261–304 (1976)

    MathSciNet  MATH  Google Scholar 

  2. R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, 2nd edn. (Springer, New York, 2005)

    MATH  Google Scholar 

  3. Davenport, H.: Multiplicative Number Theory, vol. 74, 2nd edn., ed. by H.L. Montgomery. Graduate Texts in Mathematics (Springer, New York, 1980)

    Google Scholar 

  4. P.G.L. Dirichlet, Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres. J. Reine Angew. Math. 19, 324–369 (1839), https://doi.org/10.1515/crll.1840.21.1

  5. K. Girstmair, A “popular” class number formula. Amer. Math. Monthly 101, 997–1001 (1994)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory (Springer, New York, 2000)

    MATH  Google Scholar 

  7. N.J.A. Sloane, The on-line encyclopedia of integer sequences (2015), http://oeis.org/

  8. I.M. Vinogradov, Elements of Number Theory (trans. S. Kravetz), 5th revised ed. (Dover, New York, 1954)

    Google Scholar 

  9. A.L. Whiteman, Theorems on quadratic residues. Math. Mag. 23, 71–74 (1949)

    MathSciNet  CrossRef  MATH  Google Scholar 

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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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MacMillan, K., Sondow, J. (2017). Initial Sums of the Legendre Symbol: Is \(\min + \max \ge 0\) ?. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_14

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