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<p>We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with the help of the Hermite polynomials. Then we apply this result to deduce, under the Riemann hypothesis, a formula describing the local distribution of zeros of the Riemann zeta function lying along the critical line. For this application we make use of the Prime Number Theorem in conjunction with Kac’s formula for the distribution of values of trigonometric polynomials.</p>

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An explicit formula for the zeros of the Riemann zeta function and the statistics of their local distribution

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月22日

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  • Eugenio P. Balanzario,
  • Daniel Eduardo Cárdenas Romero

摘要

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with the help of the Hermite polynomials. Then we apply this result to deduce, under the Riemann hypothesis, a formula describing the local distribution of zeros of the Riemann zeta function lying along the critical line. For this application we make use of the Prime Number Theorem in conjunction with Kac’s formula for the distribution of values of trigonometric polynomials.