<abstract xmlns="http://www.w3.org/1999/xhtml">This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>𝒨</mi><msub><mrow><mi>i</mi></mrow><mi>s</mi></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mrow><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mi>s</mi></msup></mrow></mfrac><msup><mrow><mi>z</mi></mrow><mi>n</mi></msup></mrow></mrow></math><tex-math>$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $</tex-math></alternatives></inline-formula>. A significant result is that the real part <italic>f</italic> of<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>∑</mo><mrow><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mi>n</mi></mfrac><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>i</mi><mi>n</mi><mi>π</mi><mi>θ</mi></mrow></msup></mrow></mrow></math><tex-math>$$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$</tex-math></alternatives></disp-formula>is an example of a non-trivial real-valued continuous function <italic>f</italic> on the real line which is 1-periodic, is not odd and has the property <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mi>f</mi><mo>(</mo><mi>h</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mrow></math><tex-math>$\sum\nolimits_{h = 1}^n {f(h/k) = 0}$</tex-math></alternatives></inline-formula> for every positive integer <italic>k</italic>.</abstract>

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series 𝒨is(z)=∑n=1∞μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $. A significant result is that the real part f of∑μ(n)ne2inπθ$$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$is an example of a non-trivial real-valued continuous function f on the real line which is 1-periodic, is not odd and has the property ∑h=1nf(h/k)=0$\sum\nolimits_{h = 1}^n {f(h/k) = 0}$ for every positive integer k.

Harmonic numbers, harmonic series and zeta function

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Harmonic numbers, harmonic series and zeta function"}

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the...

Harmonic numbers, harmonic series and zeta function

Publié en ligne: 16 mai 2019

Pages: 122 - 157

Reçu: 20 févr. 2019

Accepté: 07 avr. 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0012

Mots clés
Harmonic numbers, Möbius function, Zeta functions, Explicit Formulas, Sums of squares

© 2018 Ahmed Sebbar, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Harmonic numbers, harmonic series and zeta function

Publié en ligne: 16 mai 2019

Pages: 122 - 157

Reçu: 20 févr. 2019

Accepté: 07 avr. 2019

DOI: https://doi.org/10.1515/mjpaa-2018-0012

Mots clésHarmonic numbers, Möbius function, Zeta functions, Explicit Formulas, Sums of squares

© 2018 Ahmed Sebbar, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
Harmonic numbers, Möbius function, Zeta functions, Explicit Formulas, Sums of squares