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The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≡</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><msup><mi>n</mi><mrow><mo>−</mo><mi>s</mi></mrow></msup><mo>=</mo><msub><mo>∏</mo><mrow><mi>p</mi><mspace width="0.166667em"></mspace><mi>p</mi><mi>r</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></msub><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msup><mi>p</mi><mrow><mo>−</mo><mi>s</mi></mrow></msup></mrow></mfrac></mrow></semantics></math></inline-formula>, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> to the complex plane <i>z</i> and conjectured that all nontrivial zeros are in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula> axis. The nonadditive entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>q</mi></msub><mo>=</mo><mi>k</mi><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><msub><mo form="prefix">ln</mo><mi>q</mi></msub><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msub><mi>p</mi><mi>i</mi></msub><mo>)</mo></mrow><mrow><mspace width="0.277778em"></mspace><mo>(</mo><mi>q</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>;</mo><mspace width="0.166667em"></mspace></mrow><msub><mi>S</mi><mn>1</mn></msub><mo>=</mo><msub><mi>S</mi><mrow><mi>B</mi><mi>G</mi></mrow></msub><mo>≡</mo><mo>−</mo><mi>k</mi><msub><mo>∑</mo><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub><mo form="prefix">ln</mo><msub><mi>p</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo form="prefix">ln</mo><mi>q</mi></msub><mi>z</mi><mo>≡</mo><mfrac><mrow><msup><mi>z</mi><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac><mspace width="0.277778em"></mspace><mrow><mo>(</mo><msub><mo form="prefix">ln</mo><mn>1</mn></msub><mi>z</mi><mo>=</mo><mo form="prefix">ln</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is already known that this function paves the way for the emergence of a <i>q</i>-generalized algebra, using <i>q</i>-numbers defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>⟨</mo><mi>x</mi><mo>⟩</mo></mrow><mi>q</mi></msub><mo>≡</mo><msup><mi>e</mi><mrow><msub><mo form="prefix">ln</mo><mi>q</mi></msub><mi>x</mi></mrow></msup></mrow></semantics></math></inline-formula>, which recover the number <i>x</i> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The <i>q</i>-prime numbers are then defined as the <i>q</i>-natural numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo>⟨</mo><mi>n</mi><mo>⟩</mo></mrow><mi>q</mi></msub><mo>≡</mo><msup><mi>e</mi><mrow><msub><mo form="prefix">ln</mo><mi>q</mi></msub><mi>n</mi></mrow></msup><mspace width="0.277778em"></mspace><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>⋯</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>n</i> is a prime number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mo>⋯</mo></mrow></semantics></math></inline-formula> We show that, for any value of <i>q</i>, infinitely many <i>q</i>-prime numbers exist; for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula> they diverge for increasing prime number, whereas they converge for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>; the standard prime numbers are recovered for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we generalize the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> function as follows: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ζ</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≡</mo><msub><mrow><mo>⟨</mo><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>⟩</mo></mrow><mi>q</mi></msub></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>). We show that this function appears to diverge at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>q</mi></mrow></semantics></math></inline-formula>. Also, we alternatively define, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>ζ</mi><mi>q</mi><mo>∑</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≡</mo><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mfrac><mn>1</mn><msubsup><mrow><mo>⟨</mo><mi>n</mi><mo>⟩</mo></mrow><mi>q</mi><mi>s</mi></msubsup></mfrac><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msubsup><mrow><mo>⟨</mo><mn>2</mn><mo>⟩</mo></mrow><mi>q</mi><mi>s</mi></msubsup></mfrac><mo>+</mo><mo>⋯</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>ζ</mi><mi>q</mi><mo>∏</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>≡</mo><msub><mo>∏</mo><mrow><mi>p</mi><mspace width="0.166667em"></mspace><mi>p</mi><mi>r</mi><mi>i</mi><mi>m</mi><mi>e</mi></mrow></msub><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mo>⟨</mo><mi>p</mi><mo>⟩</mo></mrow><mi>q</mi><mrow><mo>−</mo><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mo>⟨</mo><mn>2</mn><mo>⟩</mo></mrow><mi>q</mi><mrow><mo>−</mo><mi>s</mi></mrow></msubsup></mrow></mfrac><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mo>⟨</mo><mn>3</mn><mo>⟩</mo></mrow><mi>q</mi><mrow><mo>−</mo><mi>s</mi></mrow></msubsup></mrow></mfrac><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msubsup><mrow><mo>⟨</mo><mn>5</mn><mo>⟩</mo></mrow><mi>q</mi><mrow><mo>−</mo><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>⋯</mo></mrow></semantics></math></inline-formula>, which, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, generically satisfy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>ζ</mi><mi>q</mi><mo>∑</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo><</mo><msubsup><mi>ζ</mi><mi>q</mi><mo>∏</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, in variance with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> case, where of course <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>ζ</mi><mn>1</mn><mo>∑</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mi>ζ</mi><mn>1</mn><mo>∏</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.