<p>For a rational number <i>a</i>/<i>N</i> with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1 \le a &lt; N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>&lt;</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, we call the value of the digamma function, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\psi (a/N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">/</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, an <i>N</i>-division value. An <i>N</i>-division value is said to be primitive if in addition, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((a,N)=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we derive an explicit expression for an <i>N</i>-division value as a rational linear combination of primitive <i>N</i>-division values and logarithms of primes that divide <i>N</i>. For a periodic arithmetic function <i>f</i>, let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L(s,f) = \sum _{n\ge 1} f(n)\, n^{-s}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. Due to the connection between division values of the digamma function and <i>L</i>(1,&#xa0;<i>f</i>), our approach leads to new observations regarding the latter.</p>