1 Introduction

Let p be a prime, let \(q=p\) if p is odd and \(q=4\) if \(p=2\), and let \(\chi \) be a Dirichlet character of conductor f. A p-adic L-function \(L_p(s,\chi )\) for a Dirichlet character \(\chi \) is a p-adic meromorphic function and an analogue of the complex L-function. For powers of the Teichmüller character \(\omega \) of conductor q, one obtains the p-adic zeta functions \(\zeta _{p,i}=L_p(s,\omega ^{1-i})\), where \(i=0,\, 1,\, \dots ,\, p-2\) (\(i=0, 1\) if \(p=2\)). It is well known that \(L_p(s,\chi )\) is identically zero for odd \(\chi \). p-adic L-functions have a long history and the primary constructions going back to Kubota-Leopoldt [6] and Iwasawa [3] are via the interpolation of special values of complex L-functions.

It can also be shown that p-adic L-functions are in fact Iwasawa functions.

It is well known that for \(Re(s) > 0\),

$$\begin{aligned} (1-2^{1-s}) \zeta (s) = \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n^s} \end{aligned}$$

and, more generally, if \(c\ge 2\) is an integer,

$$\begin{aligned} (1-\chi (c)c^{1-s})L(s, \chi ) = \sum _{n=1}^{\infty } \chi (n) \frac{a_{c,n}}{n^s}, \end{aligned}$$

where \(a_{c,n} = 1-c\) if \(n\equiv 0 \text { mod}\ c\) and \(a_{c,n} = 1\) if \(n\not \equiv 0 \text { mod}\ c\). In the following, we derive similar, but slightly different, expansions for p-adic L-functions.

An explicit formula for \(L_p(s,\chi )\) is given in [9] (Theorem 5.11): let F be any multiple of q and f. Then \(L_p(s,\chi )\) is a meromorphic function (analytic if \(\chi \ne 1)\) on \(\{ s \in {\mathbb {C}}_p\ |\ |s| < qp^{-1/(p-1)} \}\) such that

$$L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} \sum\limits_{{j = 0}}^{\infty } {\left( {\begin{array}{*{20}c} {1 - s} \\ j \\ \end{array} } \right)} \left( {\frac{F}{a}} \right)^{j} B_{j} .{\text{ }}$$

In Sect. 2, we will use formula (1) to derive a Dirichlet series expansion of \(L_p(s,\chi )\).

p-adic L-functions can be also be defined using distributions and measures. Let \(\chi \) have conductor \(f=dp^m\) with \((d,p)=1\). Choose an integer \(c \ge 2\), where \((c,dp)=1\). Then there is a measure \(E_{1,c}\) on \(({\mathbb {Z}}/d {\mathbb {Z}})^{\times } \times \, {\mathbb {Z}}_p^{\times }\) (the regularized Bernoulli distribution) such that

$$\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \int _{({\mathbb {Z}}/d {\mathbb {Z}})^{\times } \times \, {\mathbb {Z}}_p^{\times }} \chi \omega ^{-1}(a)\langle a\rangle ^{-s} \ dE_{1,c} \end{aligned}$$

(see [9] Theorem 12.2). In Sect. 3, we give an explicit formula for the values of \(E_{1,c}\) and derive the Dirichlet series expansion from (2).

The expansion is particularly simple for \(c=2\), and this parameter can be used for \(p \ne 2\) and Dirichlet characters with odd conductor. For this case we obtain similar results as in [1, 2], and [4]. In Sect. 4, we provide examples for different parameters c.

2 Expansions of p-adic L-functions

First, we derive an approximation of \(L_p(s,\chi )\) that is close to the original definition of Kubota-Leopoldt (see [6]).

For \(r \in {\mathbb {C}}_p^{\times }\) we write \(\delta (r)\) for a term with p-adic absolute value \(\le |r|\).

Proposition 2.1

Let p be a prime number, \(\chi \) an even Dirichlet character of conductor f, and F a multiple of q and f. For \(s \in {\mathbb {C}}_p\) with \(|s| < qp^{-1/(p-1)}\), we have

$$L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} + \delta (F/qp).$$


We use formula (1) above and look at the series \(\sum _{j=0}^{\infty } \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \left( \frac{F}{a} \right) ^j B_j\). The first two terms are \(1+(1-s) \frac{-F}{2a}\). We claim that the p-adic absolute value of the other terms (\(j \ge 2\)) is less than or equal to \( | (s-1)F^2/qp |\). To this end, we note that \(|1/j!|\le p^{(j-1)/(p-1)}\) and

$$\begin{aligned} \left| \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \right| \le |1-s|\, p^{(j-1)/(p-1)}(qp^{-1/(p-1)})^{j-1}= |1-s|\, q^{j-1} \end{aligned}$$

since we assumed that \(|s| < qp^{-1/(p-1)}\). Since \(|F|\le \frac{1}{q}\), \(|a|=1\), and \(|B_j| \le p\), we obtain

$$\begin{aligned} \left| \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \left( \frac{F}{a} \right) ^j B_j \right| \le |1-s| \, q^{j-1} q^{2-j} |F|^2\, p = |1-s| \, |F|^2\, q p. \end{aligned}$$

Then (1) implies

$$ L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} + \frac{1}{2}\sum\limits_{\substack{\mathop{a = 1} \\ {p\not\mid a}}}^{F} \chi \omega ^{{ - 1}} (a)\langle a\rangle ^{{ - s}} + \delta \left( {F/qp} \right). $$

It remains to show that the second sum can be absorbed into \(\delta (F/qp)\). We have

$$\begin{aligned} \sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi \omega ^{{ - 1}} (a)\left\langle a \right\rangle ^{{ - s}} & = \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (F - b)\left\langle {F - b} \right\rangle ^{{ - s}} \\ & = - \sum\limits_{\substack{\mathop {b=1}\\{p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\left\langle {b - F} \right\rangle ^{{ - s}} \\ & = - \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\left\langle b \right\rangle ^{{ - s}} + \delta (F/qp^{{ - 1/(p - 1)}} ). \end{aligned}$$

The last step can be justified by noting that

$$\begin{aligned} \frac{\langle b-F\rangle ^{-s}}{\langle b\rangle ^{-s}} = \left( 1-\frac{F}{b}\right) ^{-s} = 1 +\sum _{j=1}^{\infty } \left( {\begin{array}{c}-s\\ j\end{array}}\right) \left( \frac{-F}{b}\right) ^j = 1 + \delta (F/qp^{-1/(p-1)}), \end{aligned}$$

since \(|s|< qp^{-1/(p-1)}\) (this is the same estimate as earlier, without the presence of the Bernoulli number). This proves the proposition. \(\square \)

Remark 2.2

For \(F=fp^n\) and \(n \rightarrow \infty \), formula (3) gives the original definition of \(L_p(s,\chi )\) by Kubota and Leopoldt (see [6]).

Remark 2.3

Suppose that \(p\ne 2\). Then the error term in the above Proposition (as well as in the following Theorem 2.4) can be improved to \(\delta (F/p^{2-(p-2)/(p-1)})\). First we note that \(B_j=0\) for odd \(j \ge 3\). By the von Staudt–Clausen Theorem (see [9] 5.10), we have for even \(j \ge 2\): \(|B_j| =p\) iff \((p-1) \mid j\), and otherwise \(|B_j| \le 1\). Furthermore, \(|1/j!| = p^{(j-S_j)/(p-1)}\), where \(S_j\) is the sum of the digits of j, written to the base p (see [5]). Since \(j \equiv S_j \text { mod}\ (p-1)\), \(j \equiv 0 \text { mod}\ (p-1)\) is equivalent to \(S_j \equiv 0 \text { mod}\ (p-1)\). We conclude that \(|B_j |=p\) yields \(S_j \ge p-1\) and \(|1/j!| \le p^{(j-1)/(p-1) } p^{-(p-2)/(p-1)}\). This implies the above error term. We also see that this error term cannot be further improved.\(\lozenge \)

Now we give the Dirichlet expansion of \(L_p(s,\chi )\). For \(m \in {\mathbb {N}}\), we denote by \(\{x\}_{m}\) the unique representative of \(x \text { mod}\ m{\mathbb {Z}}\) between 0 and \(m-1\).

Theorem 2.4

Let p be a prime number, \(\chi \) be an even Dirichlet character of conductor f, and F a multiple of q and f. Let \(c>1\) be an integer satisfying \((c,F)=1\). For \(a \in {\mathbb {Z}}\), define

$$\begin{aligned} \epsilon _{a,c,F} = \frac{c-1}{2} - \{ -a F^{-1} \}_{c} \, \in \left\{ - \frac{c-1}{2} ,\, - \frac{c-1}{2} + 1,\, \dots ,\ \frac{c-1}{2}\right\} . \end{aligned}$$

Then we have for \(s \in {\mathbb {C}}_p\) with \(|s| < qp^{-1/(p-1)}\) the formula

$$\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^F \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,F} + \delta (F/qp) . \end{aligned}$$


Use (3) with cF in place of F, and subtract \(\chi (c) \langle c \rangle ^{1-s}\) times (3) with F, to obtain

$$\begin{aligned} (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) & = \frac{1}{{cF}}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^{cF} \chi (a)\langle a\rangle ^{{1 - s}} \\ & - \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (ac)\langle ac\rangle ^{{1 - s}} + \delta (F/qp). \end{aligned}$$

Let \(0<a_0 < F\) with \((a_0,\, p)=1\). Since we assumed \((c,\, F)=1\) and \(p \mid F\), there is a unique number of the form \(a_0 c\) with \(0<a_0c < cF\) and \((a_0 c,\, p)=1\) in each congruence class modulo F relatively prime to p. The first sum in (4) can be written as

$$\begin{aligned} \frac{1}{cF} \frac{1}{s-1} \sum\limits_{\substack{\mathop {a_0=1}\\ {p\not\mid a_0}}}^F \chi (a_0 c) \langle a_0 c \rangle ^{1-s} \left( \sum\limits_{\substack{\mathop {a=1} \\ \mathop{a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \left\langle 1+ \frac{a-a_0 c}{a_0 c} \right\rangle ^{1-s} \right) \\ \quad = \frac{1}{cF} \frac{1}{s-1} \sum\limits_{\substack{\mathop {a_0=1} \\ {p\not\mid a_0}}}^F \chi (a_0 c) \langle a_0 c \rangle ^{1-s} \left( \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \left( 1+ (1-s) \frac{a-a_0 c}{a_0 c} \right) \right) + \delta (F/q) . \end{aligned}$$

Note that \(| \frac{a-a_0 c}{a_0 c} | \le |F|\), so this is the same type of estimate used in the proof of Proposition 2.1. Subtracting the second sum in (4) yields

$$\begin{gathered} (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) \hfill \\ = \frac{{ - 1}}{{cF}}\sum\limits_{\substack{\mathop {a_0=1}\\ {p\not\mid a_0}}}^F \chi (a_{0} c)\langle a_{0} c\rangle ^{{1 - s}} \left( \sum\limits_{\substack{\mathop{a=1} \\ {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{a_0 c} \right) + \delta (F/qp) \hfill \\ = \frac{{ - 1}}{c} \sum\limits_{\substack{\mathop {a_0=1}\\{p\not\mid a_0}}}^F \chi \omega ^{{ - 1}} (a_{0} c)\langle a_{0} c\rangle ^{{ - s}} \left( \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{F} \right) + \delta (F/qp) . \hfill \\ \end{gathered}$$

We compute the inner sum. Let \(b=\{a_0 c\}_F\). Then \(a_0 c = b + \{ -F^{-1} b \}_c\, F\), since the latter sum is congruent to b modulo F and congruent to 0 modulo c. If a satisfies \(a \equiv a_0 c \text { mod}\ F\) and \(0<a<cF\), then \(a=b+jF\) with \(0\le j <c\). Hence

$$\begin{aligned} \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{F} = \sum _{j=0}^{c-1} (j - \{ - F^{-1} b \}_c) = c\ \epsilon _{b,c,F}\ . \end{aligned}$$

Since \(b \equiv a_0 c \text { mod}\ F\), we have \(\chi \omega ^{-1}(b) \langle b \rangle ^{-s} = \chi \omega ^{-1}(a_0 c) \langle a_0 c \rangle ^{-s} + \delta (F/q)\) by the same estimate as earlier, so

$$- (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) = \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\langle b\rangle ^{{ - s}} \epsilon _{{b,c,F}} + \delta (F/qp).{\text{ }}$$

This completes the proof. \(\square \)

We can take the limit of \(F=fp^n\) as \(n \rightarrow \infty \) and obtain:

Corollary 2.5

Let p be a prime number, \(\chi \) an even Dirichlet character of conductor f, and \(c>1\) an integer satisfying \((c,pf)=1\). Then we have for \(s \in {\mathbb {C}}_p\) with \(|s| < qp^{-1/(p-1)}\),

$$\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{fp^n} \chi \omega ^{-1}(a) \frac{ \epsilon _{a,c,fp^n} }{ \langle a \rangle ^{s}}. \end{aligned}$$

The next Theorem shows that a finite number of Euler factors can be factored off in a similar way as in [8], where a weak Euler product was obtained. The main statement is that the remaining Dirichlet series has the expected form, similar to the complex case.

Theorem 2.6

Let p be a prime number and let \(\chi \) be an even Dirichlet character of conductor f. Let S be any finite (or empty) set of primes not containing p and set \(S^+ = S \cup \{p\}\). Let F be a multiple of q, f and all primes in S. Let \(c>1\) be an integer satisfying \((c,F)=1\). Then we have for \(s \in {\mathbb {C}}_p\) with \(|s| < qp^{-1/(p-1)}\) the formula

$$- (1 - \chi (c)\langle c\rangle ^{{1 - s}} ) \cdot \prod\limits_{{l \in S}} {(1 - \chi \omega ^{{ - 1}} (l)\langle l\rangle ^{{ - s}} )} \; \cdot \;L_{p} (s,\chi ) = \sum\limits_{\substack{\mathop {a=1}\\{ \left( {a,S^{ + } } \right) = 1 }}}^F\chi \omega ^{{ - 1}} (a)\frac{{ \epsilon_{{a,c,F}} }}{{\langle a\rangle ^{s} }}\; + \;\delta (F/qp).$$


We prove the statement by induction on |S|. By Theorem 2.4, the formula is true for \(S=\varnothing \). Now assume the formula is true for S, and \(l \ne p\) is a prime with \(l \notin S\) and \((c,l)=1\). It suffices to prove the following formula:

$$\begin{aligned} \begin{aligned} (1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} ) \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^F \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,F}\ = \\ \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } \cup \left\{ l \right\}} \right) = 1 }}^{lF} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF}\ + \ \delta (F/qp) . \end{aligned} \end{aligned}$$

Note that \(|1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} | \le 1\) and \(|lF|=|F|\), so we can keep the error term. We can use lF in place of F and write the left side of (5) as

$$\begin{aligned} \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^{lF} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF} \ - \sum\limits_{\substack{ \mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^{F} \chi \omega ^{-1}(la) \langle la \rangle ^{-s} \epsilon _{a,c,F} + \delta (F/qp) . \end{aligned}$$

Now we have

$$\begin{aligned} \epsilon _{la,c,lF} = \frac{c-1}{2} - \{ -la (lF)^{-1} \}_{c} = \frac{c-1}{2} - \{ -a F^{-1} \}_{c} = \epsilon _{a,c,F}. \end{aligned}$$

Thus (6) is equal to

$$\begin{aligned} \sum\limits_{\substack{\mathop {a=1}\\ \left( {a,S^{ + } } \right) = 1 }}^{lF} \chi \omega ^{- 1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF} \ - \sum\limits_{\substack{\mathop {a=1}\\ \left( {a,S^{ + } } \right) = 1 }}^F \chi \omega ^{-1}(la) \langle la \rangle ^{-s} \epsilon _{la,c,lF} + \ \delta (F/qp) \\ & = \sum\limits_{\substack{\mathop{a = 1} \\ \left( {a,S^{ + } } \right) = 1 \\ l\not\mid a} }^{{lF}} {} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF}\ + \ \delta (F/qp), \end{aligned}$$

which shows equation (5).

\(\square \)

Remark 2.7

What happens if S contains more and more primes? It is well known that the Euler product does not converge p-adically (see [2]), since the factors \((1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} )\) have absolute value \(\le 1\) and do not converge to 1 as \(l \rightarrow \infty \). Furthermore, there are infinitely many primes l with \(\chi \omega ^{-1} (l) = 1\) and \( (1 - \langle l \rangle ^{-s})^{-1}\) has a pole at \(s=0\). We have for \(l \ne p\) and \(|s| < qp^{-1/(p-1)}\),

$$\begin{aligned} 1 - \langle l \rangle ^{-s} = - \sum _{j=1}^{\infty } \left( {\begin{array}{c}-s\\ j\end{array}}\right) ( \langle l \rangle -1 )^j . \end{aligned}$$

The p-adic absolute value of each term of the above series is less than

$$\begin{aligned} (qp^{-1/(p-1)})^j p^{(j-1)/(p-1)} q^{-j} = p^{-1/(p-1)} < 1 . \end{aligned}$$

Hence the product \(\prod _{l \in S} (1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} )\) approaches 0 as S expands to include all primes.

3 Regularized Bernoulli distributions

Let p be a prime number and let d be a positive integer with \((d,p)=1\). Define \(X_n = ({\mathbb {Z}}/dp^n {\mathbb {Z}})\) and \(X = \varprojlim X_n \cong {\mathbb {Z}}/d {\mathbb {Z}}\times {\mathbb {Z}}_p\). Let \(k \ge 1\) be an integer. Then the Bernoulli distribution \(E_k\) on X is defined by

$$\begin{aligned} E_k( a + dp^ n X) = (dp^n)^{k-1} \frac{1}{k} B_k \left( \frac{ \{ a \}_{d p^n} }{d p^n} \right) , \end{aligned}$$

where \(B_k(x)\) is the k-th Bernoulli polynomial and \(B_k=B_k(0)\) are the Bernoulli numbers (see [5, 7]). For \(k=1\), one has \(B_1(x)= x - \frac{1}{2}\). Choose \(c \in {\mathbb {Z}}\) with \(c \ne 1\) and \((c,dp)=1\). Then the regularization \(E_{k,c}\) of \(E_k\) is defined by

$$\begin{aligned} E_{k,c} ( a + dp^n X ) = E_k ( a + dp^n X) - c^k E_k \left( \left\{ \frac{a}{c} \right\} _{d p^n} + d p^n X \right) . \end{aligned}$$

One shows that the regularized Bernoulli distributions \(E_{k,c}\) are measures (see [7]). In the following, we consider only \(k=1\); the cases \(k \ge 2\) are similar.

Theorem 3.1

Let p be a prime, \(c,\, d \in {\mathbb {N}}\), and \(c \ge 2\) such that \((c,dp)=1\). Let X be as above, and let \(E_{1,c}\) be the regularized Bernoulli distribution on X. For \(a \in \{0,1,\dots , dp^n-1\}\), we have

$$\begin{aligned} E_{1,c} ( a + dp^n X) = \frac{c-1}{2} - \{ -a (dp^n)^{-1} \}_{c} = \epsilon _{a,c,dp^n}. \end{aligned}$$


By definition,

$$\begin{aligned} E_{1,c}(a+dp^n X) = E_1(a + dp^n X) - c E_1(c^{-1}a + dp^n X) = \frac{a}{dp^n} - \frac{1}{2} - c \left( \frac{ \{c^{-1} a \}_{dp^n}}{dp^n} \right) + \frac{c}{2}. \end{aligned}$$

We give the standard representative of \(c^{-1} a \text { mod}\ dp^n\):

$$\begin{aligned} \{c^{-1} a \}_{dp^n} = \frac{ \{ -a (dp^n)^{-1} \}_{c}\ dp^n + a}{c} \end{aligned}$$

Note that the numerator is divisible by c, since \( \{ -a (dp^n)^{-1} \}_{c}\ dp^n \equiv -a \text { mod}\ c\). Hence the quotient is an integer between 0 and \(dp^n-1\). Furthermore, the numerator is congruent to a modulo \(dp^n\), and so the quotient has the desired property. We obtain

$$\begin{aligned} E_{1,c}(a+dp^n X) = \frac{a}{dp^n} + \frac{c-1}{2} - \frac{ \{ -a (dp^n)^{-1} \}_{c}\ dp^n + a}{dp^n} = \frac{c-1}{2} - \{ -a (dp^n)^{-1} \}_{c} \end{aligned}$$

which is the assertion. \(\square \)

Now the Dirichlet series expansion in Corollary 2.5 follows from Theorem 3.1 and the integral formula (2).

4 Expansions for different regularization parameters

We look at the coefficients \(\epsilon _{a,c,dp^n}\) for different parameters c and the resulting Dirichlet series expansions. The following observation follows directly from the definition.

Remark 4.1

The sequence of values \(E_{1,c} ( a + dp^n X) = \epsilon _{a,c,dp^n}\) for \(a=\) 0,1, 2, \(\dots \), \(dp^n-1\) is periodic with period c. The sequence begins with \(\frac{c-1}{2}\) and continues with a permutation of \(\frac{c-3}{2}, \dots , - \frac{c-1}{2}\). If we restrict to values of n such that \(dp^n\) lies in a fixed congruence class modulo c, then the values do not change as \(n \rightarrow \infty \). \(\lozenge \)

The measure \(E_{1,c}\) and the Dirichlet series expansion are particularly simple for \(c=2\). Note that we assumed that d and p are odd in this case. If a is even, then \( \{ -a (dp^n)^{-1} \}_{2} = 0\) and

$$\begin{aligned} E_{1,2}(a+dp^n X) = \epsilon _{a,2,dp^n} = \frac{1}{2}. \end{aligned}$$

If a is odd, then \(-a(dp^n)^{-1}\) is odd, \(\{ -a (dp^n)^{-1} \}_{2} = 1\) and

$$\begin{aligned} E_{1,2}(a+dp^n X) = \epsilon _{a,2,dp^n} = - \frac{1}{2}. \end{aligned}$$

Hence \(E_{1,2}\) is up to the factor \(\frac{1}{2}\) equal to the following simple measure:

Definition 4.2

Let \( p \ne 2\) be a prime, and let \(X \cong {\mathbb {Z}}/d{\mathbb {Z}}\times {\mathbb {Z}}_p\) be as above. Then

$$\begin{aligned} \mu (a + dp^n X) = (-1)^{\{a\}_{dp^n}} \end{aligned}$$

defines a measure on X. We call \(\mu \) the alternating measure, since the measure of all clopen balls is \(\pm 1\).\(\lozenge \)

The corresponding integral is also called the fermionic p-adic integral (see [4]).

Now we obtain the following Dirichlet series expansion from Corollary 2.5.

Corollary 4.3

Let \(p \ne 2\) be a prime number, and let \(\chi \) be an even Dirichlet character of odd conductor f. Then we have for \(s \in {\mathbb {C}}_p\) with \(|s| < p^{(p-2)/(p-1)}\),

$$\begin{aligned} (1- \chi (2) \langle 2 \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \frac{1}{2} \sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^{fp^{n}} (-1)^{a+1} \chi \omega ^{-1}(a) \frac{1}{\langle a \rangle ^{s}} . \end{aligned}$$

For \(\chi =\omega ^{1-i}\) and odd \(i=1,\, \dots ,\, p-2\), we obtain the branches of the p-adic zeta function:

$$\begin{aligned} \zeta _{p,i}(s) = L_p(s,\, \omega ^{1-i}) = \frac{1}{1-\omega (2)^{1-i} \langle 2 \rangle ^{1- s}} \cdot \lim _{n \rightarrow \infty } \frac{1}{2}\sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{p^{n}} (-1)^{a+1}\omega (a)^{-i}\frac{1}{ \langle a \rangle ^{s}} \end{aligned}$$

Remark 4.4

Dirichlet series expansions of p-adic L-functions were studied by D. Delbourgo in [1] and [2]. He considers Dirichlet characters \(\chi \) satisfying \((p,\, 2 f \phi (f))=1\) and their Teichmüller twists. We obtain the same expansion for \(c=2\) and \(\chi = \omega ^{1-i}\). However, we require \((c,\, fp)=1\) and use other methods for the proof.

Similar expansions for a slightly different p-adic L-function using a fermionic p-adic integral (i.e., \(c=2\)) were also obtained by M.-S. Kim and S. Hu (see [4]).

Example 4.5

We look at the case \(c=3\). The sequence of values \(\epsilon _{a,3,dp^n}\) is periodic with period 3. If \(dp^n \equiv 1 \text { mod}\ 3\), then the sequence is \(1,\ -1,\ 0,\, \dots \) . If \(dp^n \equiv 2 \text { mod}\ 3\), then we obtain the sequence \(1,\ 0,\ -1,\, \dots \) .

Corollary 4.6

Let p be a prime number, and let \(\chi \) be an even Dirichlet character of conductor \(f=dp^m\) such that \((3,\, dp)=1\). If \(d \equiv 1 \text { mod}\ 3\), then define a sequence \(\epsilon _0=1\), \(\epsilon _1=-1\), \(\epsilon _2=0\), \(\dots \) with period 3. Otherwise, set \(\epsilon _0=1\), \(\epsilon _1=0\), \(\epsilon _2=-1\) and extend it with period 3. Then we have for \(s \in {\mathbb {C}}_p\) with \(|s| < qp^{-1/(p-1)}\),

$$\begin{aligned} -(1- \chi (3) \langle 3 \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{ dp^{{2n}} } \chi \omega ^{-1}(a)\frac{\epsilon _a}{ \langle a \rangle ^{s} }. \end{aligned}$$

Example 4.7

For \(c=5\), we get a periodic sequence with period 5 and we have \(\epsilon _{a,5,dp^n}=2\) for \(a \equiv 0 \text { mod}\ 5\). The next four coefficients are a permutation of the values \(-2\), \(-1\), 0 and 1, depending on the class of \(dp^n \text { mod}\ 5\).

Example 4.8

Let \(c=7\). Then \(\epsilon _{0,7,dp^n} = 3\). Now suppose, for example, that \(dp^n \equiv 3 \text { mod}\ 7\). Then \((dp^n)^{-1} \equiv 5 \text { mod}\ 7\). This yields the values

$$\begin{aligned} \epsilon _{1,7,dp^n} = 1,\ \epsilon _{2,7,dp^n} = -1,\ \epsilon _{3,7,dp^n} = -3,\ \epsilon _{4,7,dp^n} = 2,\ \epsilon _{5,7,dp^n} = 0,\ \epsilon _{6,7,dp^n} = -2, \end{aligned}$$

and these are extended with period 7.