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Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of <inline-formula> <math display="inline"> <semantics> <msup> <mi>e</mi> <mi>α</mi> </msup> </semantics> </math> </inline-formula>, for algebraic values of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as <inline-formula> <math display="inline"> <semantics> <msup> <mi>e</mi> <mrow> <mi>γ</mi> <mo form="prefix">log</mo> <mo> </mo> <mi>n</mi> </mrow> </msup> </semantics> </math> </inline-formula>. This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called <i>γ-th arithmetic zeta function</i> <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ζ</mi> <mi>γ</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msup> <mi>n</mi> <mi>γ</mi> </msup> </mrow> </semantics> </math> </inline-formula> (<inline-formula> <math display="inline"> <semantics> <mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>γ</mi> <mo form="prefix">log</mo> <mo> </mo> <mi>n</mi> </mrow> </msup> </mrow> </semantics> </math> </inline-formula>), for a positive integer <i>n</i> and a complex number <inline-formula> <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math> </inline-formula>. Moreover, we raise a conjecture about the exceptional set of <inline-formula> <math display="inline"> <semantics> <msub> <mi>ζ</mi> <mi>γ</mi> </msub> </semantics> </math> </inline-formula>, in the case in which <inline-formula> <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math> </inline-formula> is transcendental, and we connect it to the famous Schanuel’s conjecture.