Geodesic
On Popov's explicit formula and the Davenport expansion
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 869-883
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We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function $a_n$ with the periodic Bernoulli polynomial weight $\bar {B}_\varkappa (nx)$ and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order $0$ or $1$ gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order $1$ of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order $1$ subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer's Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function.
We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function $a_n$ with the periodic Bernoulli polynomial weight $\bar {B}_\varkappa (nx)$ and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order $0$ or $1$ gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order $1$ of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order $1$ subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer's Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function.
DOI : 10.21136/CMJ.2023.0322-22
Classification : 11J54, 11M41, 11N05
Keywords: explicit formula; Davenport expansion; Kummer's Fourier series; Riemann zeta-function; functional equation 
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Yang, Quan; Mehta, Jay; Kanemitsu, Shigeru. On Popov's explicit formula and the Davenport expansion. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 869-883. doi: 10.21136/CMJ.2023.0322-22

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